1.
Sphere
–
A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
Sphere
–
Circumscribed cylinder to a sphere
Sphere
–
A two-dimensional
perspective projection of a sphere
Sphere
Sphere
–
Deck of playing cards illustrating engineering instruments, England, 1702.
King of spades: Spheres
2.
Plane (geometry)
–
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
Plane (geometry)
–
Vector description of a plane
Plane (geometry)
–
Two intersecting planes in three-dimensional space
3.
Spherical geometry
–
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean, two practical applications of the principles of spherical geometry are navigation and astronomy. In plane geometry, the concepts are points and lines. On a sphere, points are defined in the usual sense, the equivalents of lines are not defined in the usual sense of straight line in Euclidean geometry, but in the sense of the shortest paths between points, which are called geodesics. On a sphere, the geodesics are the circles, other geometric concepts are defined as in plane geometry. Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry, for example, it shares with that geometry the property that a line has no parallels through a given point. An important geometry related to that of the sphere is that of the projective plane. Locally, the plane has all the properties of spherical geometry. In particular, it is non-orientable, or one-sided, Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher-dimensional spherical geometries exist, see elliptic geometry, the earliest mathematical work of antiquity to come down to our time is On the rotating sphere by Autolycus of Pitane, who lived at the end of the fourth century BC. The book of unknown arcs of a written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the law of sines. The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe, however, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15,1771, pp. 195–216, Opera Omnia, Series 1, Volume 28, pp. 142–160. L. Euler, De mensura angulorum solidorum, Acta academiae scientarum imperialis Petropolitinae 2,1781, p. 31–54, Opera Omnia, Series 1, vol. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientarum imperialis Petropolitinae 4,1783, p. 91–96, Opera Omnia, Series 1, vol. L. Euler, Geometrica et sphaerica quaedam, Mémoires de lAcademie des Sciences de Saint-Petersbourg 5,1815, p. 96–114, Opera Omnia, Series 1, vol. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientarum imperialis Petropolitinae 3,1782, p. 72–86, Opera Omnia, Series 1, vol
Spherical geometry
–
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore, it is a two dimensional
manifold.
4.
Non-Euclidean geometry
–
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
Non-Euclidean geometry
–
On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Non-Euclidean geometry
–
Projecting a
sphere to a
plane.
5.
Analytic geometry
–
In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete, usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane and Euclidean space, the numerical output, however, might also be a vector or a shape. That the algebra of the numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is thought to have anticipated the work of Descartes by some 1800 years. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves and that is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation, analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the term used for analytic geometry, is named after Descartes. This work, written in his native French tongue, and its philosophical principles, initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartess masterpiece receive due recognition, Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a form of Ad locos planos et solidos isagoge was circulating in Paris in 1637. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry, as a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonard Euler who first applied the method in a systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the point of origin. These are typically written as an ordered pair and this system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates. In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ from the polar axis
Analytic geometry
–
Cartesian coordinates
6.
Riemannian geometry
–
This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture Ueber die Hypothesen and it is a very broad and abstract generalization of the differential geometry of surfaces in R3. Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century and it deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of Non-Euclidean geometry. Any smooth manifold admits a Riemannian metric, which helps to solve problems of differential topology. It also serves as a level for the more complicated structure of pseudo-Riemannian manifolds. Other generalizations of Riemannian geometry include Finsler geometry, there exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and Disclinations produce torsions and curvature, the choice is made depending on its importance and elegance of formulation. Most of the results can be found in the monograph by Jeff Cheeger. The formulations given are far from being very exact or the most general and this list is oriented to those who already know the basic definitions and want to know what these definitions are about. Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ where χ denotes the Euler characteristic of M and this theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem. Nash embedding theorems also called fundamental theorems of Riemannian geometry and they state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn. If M is a connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is diffeomorphic to a sphere. Given constants C, D and V, there are finitely many compact n-dimensional Riemannian manifolds with sectional curvature |K| ≤ C, diameter ≤ D. There is an εn >0 such that if an n-dimensional Riemannian manifold has a metric with sectional curvature |K| ≤ εn, G. Perelman in 1994 gave an astonishingly elegant/short proof of the Soul Conjecture, M is diffeomorphic to Rn if it has positive curvature at only one point. There is a constant C = C such that if M is a compact connected n-dimensional Riemannian manifold with sectional curvature then the sum of its Betti numbers is at most C. Given constants C, D and V, there are finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D. It implies that any two points of a connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic
Riemannian geometry
–
Bernhard Riemann
Riemannian geometry
–
Projecting a
sphere to a
plane.
7.
Differential geometry
–
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century, since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas, Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. These unanswered questions indicated greater, hidden relationships, initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric and this is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Various concepts based on length, such as the arc length of curves, area of plane regions, the notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds, a distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, i. e. for small neighborhoods of points, any two regular curves are locally isometric. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat, an important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant. These are the closest analogues to the plane and space considered in Euclidean and non-Euclidean geometry. Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite, a special case of this is a Lorentzian manifold, which is the mathematical basis of Einsteins general relativity theory of gravity. Finsler geometry has the Finsler manifold as the object of study. This is a manifold with a Finsler metric, i. e. a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M is a function F, TM → [0, ∞) such that, F = |m|F for all x, y in TM, F is infinitely differentiable in TM −, symplectic geometry is the study of symplectic manifolds. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed, a diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, in dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism
Differential geometry
–
A triangle immersed in a saddle-shape plane (a
hyperbolic paraboloid), as well as two diverging
ultraparallel lines.
8.
Finite geometry
–
A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points, a geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are systems that could be called finite geometries, attention is mostly paid to the finite projective. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field, Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of three or greater is isomorphic to a projective space over a finite field. However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes, similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes, There are two main kinds of finite plane geometry, affine and projective. In an affine plane, the sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, For every two distinct points, there is exactly one line that contains both points. Playfairs axiom, Given a line ℓ and a point p not on ℓ, There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry, the simplest affine plane contains only four points, it is called the affine plane of order 2. Since no three are collinear, any pair of points determines a line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered parallel, or a square where not only opposite sides, but also diagonals are considered parallel. More generally, an affine plane of order n has n2 points and n2 + n lines, each line contains n points. The affine plane of order 3 is known as the Hesse configuration. A projective plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, the intersection of any two distinct lines contains exactly one point
Finite geometry
–
Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are "parallel".
9.
Angle
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
Angle
–
An angle enclosed by rays emanating from a vertex.
10.
Diagonal
–
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal, in matrix algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also other, non-mathematical uses, diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or on a diagonal, hence the name. A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the cross over the poles at an angle. In association football, the system of control is the method referees. As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices, therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, in a convex polygon, if no three diagonals are concurrent at a single point, the number of regions that the diagonals divide the interior into is given by + =24. The number of regions is 1,4,11,25,50,91,154,246, in a polygon with n angles the number of diagonals is given by n ∗2. The number of intersections between the diagonals is given by, in the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix A with row index specified by i and column index specified by j, the off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero, a superdiagonal entry is one that is directly above and to the right of the main diagonal. Just as diagonal entries are those A i j with j = i and this plays an important part in geometry, for example, the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. In geometric studies, the idea of intersecting the diagonal with itself is common, not directly and this is related at a deep level with the Euler characteristic and the zeros of vector fields. For example, the circle S1 has Betti numbers 1,1,0,0,0, a geometric way of expressing this is to look at the diagonal on the two-torus S1xS1 and observe that it can move off itself by the small motion to. Topics In Algebra, Waltham, Blaisdell Publishing Company, ISBN 978-1114541016 Nering, linear Algebra and Matrix Theory, New York, Wiley, LCCN76091646 Diagonals of a polygon with interactive animation Polygon diagonal from MathWorld. Diagonal of a matrix from MathWorld
Diagonal
–
A stand of basic scaffolding on a house construction site, with diagonal braces to maintain its structure
Diagonal
–
The diagonals of a
cube with side length 1. AC' (shown in blue) is a
space diagonal with length, while AC (shown in red) is a
face diagonal and has length.
11.
Orthogonal
–
The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, and engineering. The word comes from the Greek ὀρθός, meaning upright, and γωνία, the ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle, in the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i. e. they form a right angle, two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y, two vector subspaces, A and B, of an inner product space, V, are called orthogonal subspaces if each vector in A is orthogonal to each vector in B. The largest subspace of V that is orthogonal to a subspace is its orthogonal complement. Given a module M and its dual M∗, an element m′ of M∗, two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent, a set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set, nonzero pairwise orthogonal vectors are always linearly independent. In certain cases, the normal is used to mean orthogonal. For example, the y-axis is normal to the curve y = x2 at the origin, however, normal may also refer to the magnitude of a vector. In particular, a set is called if it is an orthogonal set of unit vectors. As a result, use of the normal to mean orthogonal is often avoided. The word normal also has a different meaning in probability and statistics, a vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality, in the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ. In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i. e. they make an angle of 90°, hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces
Orthogonal
–
The line segments AB and CD are orthogonal to each other.
12.
Perpendicular
–
In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects, a line is said to be perpendicular to another line if the two lines intersect at a right angle. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, for this reason, we may speak of two lines as being perpendicular without specifying an order. Perpendicularity easily extends to segments and rays, in symbols, A B ¯ ⊥ C D ¯ means line segment AB is perpendicular to line segment CD. A line is said to be perpendicular to an if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines, two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. Perpendicularity is one instance of the more general mathematical concept of orthogonality, perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the perpendicular is sometimes used to describe much more complicated geometric orthogonality conditions. The word foot is used in connection with perpendiculars. This usage is exemplified in the top diagram, above, the diagram can be in any orientation. The foot is not necessarily at the bottom, step 2, construct circles centered at A and B having equal radius. Let Q and R be the points of intersection of two circles. Step 3, connect Q and R to construct the desired perpendicular PQ, to prove that the PQ is perpendicular to AB, use the SSS congruence theorem for and QPB to conclude that angles OPA and OPB are equal. Then use the SAS congruence theorem for triangles OPA and OPB to conclude that angles POA, to make the perpendicular to the line g at or through the point P using Thales theorem, see the animation at right. The Pythagorean Theorem can be used as the basis of methods of constructing right angles, for example, by counting links, three pieces of chain can be made with lengths in the ratio 3,4,5. These can be out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, the chains can be used repeatedly whenever required. If two lines are perpendicular to a third line, all of the angles formed along the third line are right angles
Perpendicular
–
The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.
13.
Congruence (geometry)
–
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
Congruence (geometry)
–
The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same
perimeter and
area. (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)
14.
Similarity (geometry)
–
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling and this means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other zoomed in or out at some level. For example, all circles are similar to other, all squares are similar to each other. On the other hand, ellipses are not all similar to other, rectangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure and it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem, due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of which is necessary and sufficient for two triangles to be similar,1, the triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is, If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′, all the corresponding sides have lengths in the same ratio, AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other, two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance, AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′ and this is known as the SAS Similarity Criterion. When two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′, there are several elementary results concerning similar triangles in Euclidean geometry, Any two equilateral triangles are similar. Two triangles, both similar to a triangle, are similar to each other. Corresponding altitudes of similar triangles have the ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, the statement that the point F satisfying this condition exists is Walliss Postulate and is logically equivalent to Euclids Parallel Postulate
Similarity (geometry)
–
Sierpinski triangle. A space having self-similarity dimension ln 3 / ln 2 = log 2 3, which is approximately 1.58. (from
Hausdorff dimension.)
Similarity (geometry)
–
Figures shown in the same color are similar
15.
Symmetry
–
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
Symmetry
–
Symmetric arcades of a portico in the
Great Mosque of Kairouan also called the Mosque of Uqba, in
Tunisia.
Symmetry
Symmetry
–
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
Symmetry
–
The ceiling of
Lotfollah mosque,
Isfahan,
Iran has 8-fold symmetries.
16.
One-dimensional space
–
In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n =1, the set of all locations is called a one-dimensional space. An example of a space is the number line, where the position of each point on it can be described by a single number. In algebraic geometry there are structures which are technically one-dimensional spaces. For a field k, it is a vector space over itself. Similarly, the line over k is a one-dimensional space. In particular, if k = ℂ, the complex plane, then the complex projective line P1 is one-dimensional with respect to ℂ. More generally, a ring is a module over itself. Similarly, the line over a ring is a one-dimensional space over the ring. In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, the only regular polytope in one dimension is the line segment, with the Schläfli symbol. The hypersphere in 1 dimension is a pair of points, sometimes called a 0-sphere as its surface is zero-dimensional and its length is L =2 r where r is the radius. The most popular systems are the number line and the angle
One-dimensional space
–
Number line
17.
Point (geometry)
–
In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
Point (geometry)
–
Projecting a
sphere to a
plane.
18.
Line (geometry)
–
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
Line (geometry)
–
The red and blue lines on this graph have the same
slope (gradient); the red and green lines have the same
y-intercept (cross the
y-axis at the same place).
19.
Ray (geometry)
–
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
Ray (geometry)
–
The red and blue lines on this graph have the same
slope (gradient); the red and green lines have the same
y-intercept (cross the
y-axis at the same place).
20.
Two-dimensional space
–
In physics and mathematics, two-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called length and width, both directions lie in the same plane. A sequence of n numbers can be understood as a location in n-dimensional space. When n =2, the set of all locations is called two-dimensional space or bi-dimensional space. Each reference line is called an axis or just axis of the system. The coordinates can also be defined as the positions of the projections of the point onto the two axes, expressed as signed distances from the origin. The idea of system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided and this was known as the complex plane. The complex plane is called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin and they are usually labeled x and y. Another widely used system is the polar coordinate system, which specifies a point in terms of its distance from the origin. In two dimensions, there are infinitely many polytopes, the polygons, the first few regular ones are shown below, The Schläfli symbol represents a regular p-gon. The regular henagon and regular digon can be considered degenerate regular polygons and they can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus. There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers and they are called star polygons and share the same vertex arrangements of the convex regular polygons
Two-dimensional space
–
Bi-dimensional
Cartesian coordinate system
21.
Polygon
–
In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
Polygon
–
Historical image of polygons (1699)
Polygon
–
Some different types of polygon
Polygon
–
The
Giant's Causeway, in
Northern Ireland
22.
Triangle
–
A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
Triangle
–
The
Flatiron Building in New York is shaped like a
triangular prism
Triangle
–
A triangle
23.
Altitude (triangle)
–
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the side is called the extended base of the altitude. The intersection between the base and the altitude is called the foot of the altitude. The length of the altitude, often called the altitude, is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude of that vertex and it is a special case of orthogonal projection. Altitudes can be used to compute the area of a triangle, one half of the product of an altitudes length, thus the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometric functions, in an isosceles triangle, the altitude having the incongruent side as its base will have the midpoint of that side as its foot. Also the altitude having the incongruent side as its base will form the angle bisector of the vertex and it is common to mark the altitude with the letter h, often subscripted with the name of the side the altitude comes from. In a right triangle, the altitude with the hypotenuse c as base divides the hypotenuse into two lengths p and q. If we denote the length of the altitude by hc, we then have the relation h c = p q For acute, the three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute, if one angle is a right angle, the orthocenter coincides with the vertex of the right angle. The product of the distances from the orthocenter to a vertex and this product is the squared radius of the triangles polar circle. The orthocenter H, the centroid G, the circumcenter O, and the center N of the nine-point circle all lie on a single line, known as the Euler line. The orthocenter is closer to the incenter I than it is to the centroid, the isogonal conjugate and also the complement of the orthocenter is the circumcenter. Four points in the plane such that one of them is the orthocenter of the triangle formed by the three are called an orthocentric system or orthocentric quadrangle. Let A, B, C denote the angles of the reference triangle, and let a = |BC|, b = |CA|, c = |AB| be the sidelengths. In the complex plane, let the points A, B and C represent the numbers z A, z B and respectively z C and assume that the circumcenter of triangle A B C is located at the origin of the plane. Then, the number z H = z A + z B + z C is represented by the point H
Altitude (triangle)
–
Three altitudes intersecting at the orthocenter
24.
Hypotenuse
–
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. For example, if one of the sides has a length of 3. The length of the hypotenuse is the root of 25. The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timaeus 54d, a folk etymology says that tenuse means side, so hypotenuse means a support like a prop or buttress, but this is inaccurate. The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle are a and b and that of the hypotenuse is c, many computer languages support the ISO C standard function hypot, which returns the value above. The function is designed not to fail where the straightforward calculation might overflow or underflow, some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. This gives both the length of the hypotenuse and the angle the hypotenuse makes with the line at the same time when given x and y. The angle returned will normally be given by atan2. Orthographic projections, The length of the hypotenuse equals the sum of the lengths of the projections of both catheti. And The square of the length of a cathetus equals the product of the lengths of its projection on the hypotenuse times the length of this. Given the length of the c and of a cathetus b. The adjacent angle of the b, will be α = 90° – β One may also obtain the value of the angle β by the equation. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. Hypotenuse
Hypotenuse
–
A right-angled triangle and its hypotenuse.
25.
Parallelogram
–
In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length, by comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped, rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles Rectangle – A parallelogram with four angles of equal size. Rhombus – A parallelogram with four sides of equal length, square – A parallelogram with four sides of equal length and angles of equal size. A simple quadrilateral is a if and only if any one of the following statements is true. Two pairs of opposite angles are equal in measure, one pair of opposite sides are parallel and equal in length. Each diagonal divides the quadrilateral into two congruent triangles, the sum of the squares of the sides equals the sum of the squares of the diagonals. It has rotational symmetry of order 2, the sum of the distances from any interior point to the sides is independent of the location of the point. Thus all parallelograms have all the properties listed above, and conversely, if just one of statements is true in a simple quadrilateral. Opposite sides of a parallelogram are parallel and so will never intersect, the area of a parallelogram is twice the area of a triangle created by one of its diagonals. The area of a parallelogram is also equal to the magnitude of the cross product of two adjacent sides. Any line through the midpoint of a parallelogram bisects the area, any non-degenerate affine transformation takes a parallelogram to another parallelogram. A parallelogram has rotational symmetry of order 2, if it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong. If it has four lines of symmetry, it is a square. The perimeter of a parallelogram is 2 where a and b are the lengths of adjacent sides, unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area, the diagonals of a parallelogram divide it into four triangles of equal area. All of the formulas for general convex quadrilaterals apply to parallelograms
Parallelogram
–
This parallelogram is a
rhomboid as it has no right angles and unequal sides.
26.
Rectangle
–
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa
Rectangle
–
Running bond
Rectangle
–
Rectangle
Rectangle
–
Basket weave
27.
Quadrilateral
–
In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices or corners. Sometimes, the quadrangle is used, by analogy with triangle. The origin of the quadrilateral is the two Latin words quadri, a variant of four, and latus, meaning side. Quadrilaterals are simple or complex, also called crossed, simple quadrilaterals are either convex or concave. The interior angles of a simple quadrilateral ABCD add up to 360 degrees of arc and this is a special case of the n-gon interior angle sum formula × 180°. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges, any quadrilateral that is not self-intersecting is a simple quadrilateral. In a convex quadrilateral, all angles are less than 180°. Irregular quadrilateral or trapezium, no sides are parallel, trapezium or trapezoid, at least one pair of opposite sides are parallel. Isosceles trapezium or isosceles trapezoid, one pair of sides are parallel. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, parallelogram, a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of length, that opposite angles are equal. In other words, parallelograms include all rhombi and all rhomboids, rhombus or rhomb, all four sides are of equal length. An equivalent condition is that the diagonals bisect each other. Rhomboid, a parallelogram in which adjacent sides are of unequal lengths, not all references agree, some define a rhomboid as a parallelogram which is not a rhombus. Rectangle, all four angles are right angles, an equivalent condition is that the diagonals bisect each other and are equal in length. Square, all four sides are of length, and all four angles are right angles. An equivalent condition is that opposite sides are parallel, that the diagonals bisect each other. A quadrilateral is a if and only if it is both a rhombus and a rectangle
Quadrilateral
–
Six different types of quadrilaterals
28.
Kite (geometry)
–
In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of sides, but they are opposite to each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape, kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be convex or concave. A concave kite is called a dart or arrowhead, and is a type of pseudotriangle. If all four sides of a kite have the same length, if a kite is equiangular, meaning that all four of its angles are equal, then it must also be equilateral and thus a square. A kite with three equal 108° angles and one 36° angle forms the hull of the lute of Pythagoras. The kites that are cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two angles on opposite sides of the symmetry axis are each 90 degrees. These shapes are called right kites and they are in fact bicentric quadrilaterals, among all the bicentric quadrilaterals with a given two circle radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the deltoidal trihexagonal tiling, among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, 5π/12. Its four vertices lie at the three corners and one of the midpoints of the Reuleaux triangle. In non-Euclidean geometry, a Lambert quadrilateral is a kite with three right angles. A quadrilateral is a if and only if any one of the following conditions is true. One diagonal is the bisector of the other diagonal. One diagonal is a line of symmetry, one diagonal bisects a pair of opposite angles. The kites are the quadrilaterals that have an axis of symmetry along one of their diagonals, if crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the antiparallelograms. Every kite is orthodiagonal, meaning that its two diagonals are at angles to each other
Kite (geometry)
–
V4.3.4.3
Kite (geometry)
–
A kite showing its sides equal in length and its inscribed circle.
Kite (geometry)
–
V4.3.4.4
Kite (geometry)
–
V4.3.4.5
29.
Circle
–
A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
Circle
–
The
compass in this 13th-century manuscript is a symbol of God's act of
Creation. Notice also the circular shape of the
halo
Circle
–
A circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre (O) in magenta.
Circle
–
Circular piece of silk with Mongol images
Circle
–
Circles in an old
Arabic astronomical drawing.
30.
Diameter
–
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle, both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is called the diameter. In this sense one speaks of the rather than a diameter, because all diameters of a circle or sphere have the same length. Both quantities can be calculated efficiently using rotating calipers, for a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. For an ellipse, the terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse, for example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis, the word diameter is derived from Greek διάμετρος, diameter of a circle, from διά, across, through and μέτρον, measure. It is often abbreviated DIA, dia, d, or ⌀, the definitions given above are only valid for circles, spheres and convex shapes. However, they are cases of a more general definition that is valid for any kind of n-dimensional convex or non-convex object. The diameter of a subset of a space is the least upper bound of the set of all distances between pairs of points in the subset. So, if A is the subset, the diameter is sup, if the distance function d is viewed here as having codomain R, this implies that the diameter of the empty set equals −∞. Some authors prefer to treat the empty set as a case, assigning it a diameter equal to 0. For any solid object or set of scattered points in n-dimensional Euclidean space, in medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the supremum of the set of all distances between pairs of points in the object. In differential geometry, the diameter is an important global Riemannian invariant, the symbol or variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke. In Unicode it is defined as U+2300 ⌀ Diameter sign, on an Apple Macintosh, the diameter symbol can be entered via the character palette, where it can be found in the Technical Symbols category. The character will not display correctly, however, since many fonts do not include it. In many situations the letter ø is a substitute, which in Unicode is U+00F8 ø
Diameter
–
Circle with
circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.
31.
Circumference
–
The circumference of a closed curve or circular object is the linear distance around its edge. The circumference of a circle is of importance in geometry and trigonometry. Informally circumference may also refer to the edge rather than to the length of the edge. The circumference of a circle is the distance around it, the term is used when measuring physical objects, as well as when considering abstract geometric forms. The circumference of a circle relates to one of the most important mathematical constants in all of mathematics and this constant, pi, is represented by the Greek letter π. The numerical value of π is 3.141592653589793, pi is defined as the ratio of a circles circumference C to its diameter d, π = C d Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference, the use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. The constant ratio of circumference to radius C / r =2 π also has uses in mathematics, engineering. These uses include but are not limited to radians, computer programming, the Greek letter τ is sometimes used to represent this constant, but is not generally accepted as proper notation. The circumference of an ellipse can be expressed in terms of the elliptic integral of the second kind. In graph theory the circumference of a graph refers to the longest cycle contained in that graph, arc length Area Caccioppoli set Isoperimetric inequality Pythagorean theorem Volume Numericana - Circumference of an ellipse Circumference of a circle With interactive applet and animation
Circumference
–
Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circumference = π × diameter = 2 × π × radius.
32.
Area of a circle
–
In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents a constant, approximately equal to 3.14159, one method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a polygon is half its perimeter multiplied by the distance from its center to its sides. Therefore, the area of a disk is the precise phrase for the area enclosed by a circle. Modern mathematics can obtain the area using the methods of calculus or its more sophisticated offspring. However the area of a disk was studied by the Ancient Greeks, eudoxus of Cnidus in the fifth century B. C. had found that the area of a disk is proportional to its radius squared. The circumference is 2πr, and the area of a triangle is half the times the height. A variety of arguments have been advanced historically to establish the equation A = π r 2 of varying degrees of mathematical rigor, the area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the polygon tends to a circle, and this suggests that the area of a disk is half the circumference of its bounding circle times the radius. Following Archimedes, compare the area enclosed by a circle to a triangle whose base has the length of the circles circumference. If the area of the circle is not equal to that of the triangle and we eliminate each of these by contradiction, leaving equality as the only possibility. We use regular polygons in the same way, suppose that the area C enclosed by the circle is greater than the area T = 1⁄2cr of the triangle. Let E denote the excess amount, inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments, if the total area of those gaps, G4, is greater than E, split each arc in half. This makes the square into an inscribed octagon, and produces eight segments with a smaller total gap. Continue splitting until the total gap area, Gn, is less than E, now the area of the inscribed polygon, Pn = C − Gn, must be greater than that of the triangle. E = C − T > G n P n = C − G n > C − E P n > T But this forces a contradiction, as follows. Draw a perpendicular from the center to the midpoint of a side of the polygon, its length, also, let each side of the polygon have length s, then the sum of the sides, ns, is less than the circle circumference
Area of a circle
–
Circle with square and octagon inscribed, showing area gap
33.
Volume
–
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
Volume
–
A
measuring cup can be used to measure volumes of
liquids. This cup measures volume in units of
cups,
fluid ounces, and
millilitres.
34.
Cube
–
Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0. 5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-magnetic and non-sparking qualities and it has excellent metalworking, forming and machining properties. It has many specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, beryllium alloys present a toxic inhalation hazard during manufacture. Beryllium copper is a ductile, weldable, and machinable alloy and it is resistant to non-oxidizing acids, to plastic decomposition products, to abrasive wear, and to galling. It can be heat-treated for increased strength, durability, and electrical conductivity, beryllium copper attains the greatest strength of any copper-based alloy. In solid form and as finished objects, beryllium copper presents no known health hazard, however, inhalation of dust, mist, or fume containing beryllium can cause the serious lung condition, chronic beryllium disease. That disease affects primarily the lungs, restricting the exchange of oxygen between the lungs and the bloodstream, the International Agency for Research on Cancer lists beryllium as a Group 1 Human Carcinogen. The National Toxicology Program also lists beryllium as a carcinogen, beryllium copper is a non-ferrous alloy used in springs, spring wire, load cells, and other parts that must retain their shape under repeated stress and strain. It has high electrical conductivity, and is used in low-current contacts for batteries, beryllium copper is non-sparking but physically tough and nonmagnetic, fulfilling the requirements of ATEX directive for Zones 0,1, and 2. Beryllium copper screwdrivers, pliers, wrenches, cold chisels, knives, and hammers are available for environments with explosive hazards, such oil rigs, coal mines, an alternative metal sometimes used for non-sparking tools is aluminium bronze. Compared to steel tools, beryllium copper tools are more expensive, not as strong, and less durable, beryllium copper is frequently used for percussion instruments for its consistent tone and resonance, especially tambourines and triangles. Beryllium copper has been used for armour piercing bullets, though usage is unusual because bullets made from steel alloys are much less expensive and have similar properties. Beryllium copper is used for measurement-while-drilling tools in the drilling industry. A non-magnetic alloy is required, as magnetometers are used for field-strength data received from the tool, beryllium copper gaskets are used to create an RF-tight, electronic seal on doors used with EMC testing and anechoic chambers. For a time, beryllium copper was used in the manufacture of clubs, particularly wedges. Though some golfers prefer the feel of BeCu club heads, regulatory issues, kiefer Plating of Elkhart, Indiana built some beryllium-copper trumpet bells for the Schilke Music Co. of Chicago. These light-weight bells produce a sound preferred by some musicians, beryllium copper wire is produced in many forms, round, square, flat and shaped, in coils, on spools and in straight lengths. Beryllium copper valve seats and guides are used in high performance engines with coated titanium valves
Cube
–
Example of a non-sparking tool made of beryllium copper
Cube
35.
Four-dimensional space
–
For example, the volume of a rectangular box is found by measuring its length, width, and depth. More than two millennia ago Greek philosophers explored in detail the implications of this uniformity, culminating in Euclids Elements. However, it was not until recent times that a handful of insightful mathematical innovators generalized the concept of dimensions to more than three. The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid 1700s, in 1880 Charles Howard Hinton popularized these insights in an essay titled What is the Fourth Dimension. Which was notable for explaining the concept of a cube by going through a step-by-step generalization of the properties of lines, squares. The simplest form of Hintons method is to draw two ordinary cubes separated by a distance, and then draw lines between their equivalent vertices. This form can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube, the eight lines connecting the vertices of the two cubes in that case represent a single direction in the unseen fourth dimension. Higher dimensional spaces have become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without the use of such spaces, calendar entries for example are usually 4D locations, such as a meeting at time t at the intersection of two streets on some building floor. In list form such a meeting place at the 4D location. Einsteins concept of spacetime uses such a 4D space, though it has a Minkowski structure that is a bit more complicated than Euclidean 4D space, when dimensional locations are given as ordered lists of numbers such as they are called vectors or n-tuples. It is only when such locations are linked together into more complicated shapes that the richness and geometric complexity of 4D. A hint of that complexity can be seen in the animation of one of simplest possible 4D objects. Lagrange wrote in his Mécanique analytique that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, and one of time, the possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843 and this associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R, one of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension. Published in the Dublin University magazine and he coined the terms tesseract, ana and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension. Hintons ideas inspired a fantasy about a Church of the Fourth Dimension featured by Martin Gardner in his January 1962 Mathematical Games column in Scientific American, in 1886 Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams
Four-dimensional space
–
5-cell
Four-dimensional space
–
3D projection of a
tesseract undergoing a
simple rotation in four dimensional space.
36.
Tesseract
–
In geometry, the tesseract is the four-dimensional analog of the cube, the tesseract is to the cube as the cube is to the square. Just as the surface of the consists of six square faces. The tesseract is one of the six convex regular 4-polytopes, the tesseract is also called an 8-cell, C8, octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the family of hypercubes or measure polytopes. In this publication, as well as some of Hintons later work, the tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384, constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64, as an orthotope it can be represented by composite Schläfli symbol × × × or 4, with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the hull of the points. That is, it consists of the points, A tesseract is bounded by eight hyperplanes, each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge, there are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes,24 squares,32 edges, the construction of a hypercube can be imagined the following way, 1-dimensional, Two points A and B can be connected to a line, giving a new line segment AB. 2-dimensional, Two parallel line segments AB and CD can be connected to become a square, 3-dimensional, Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional, Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube and it is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices, the scheme is similar to the construction of a cube from two squares, juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length, the regular complex polytope 42, in C2 has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 42 has 16 vertices, and 8 4-edges and its symmetry is 42, order 32. It also has a lower construction, or 4×4, with symmetry 44
Tesseract
–
Schlegel diagram
37.
Apollonius of Perga
–
Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic and his definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Apollonius worked on other topics, including astronomy. Most of the work has not survived except in references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, for such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states, “Apollonius, the geometrician. Came from Perga in Pamphylia in the times of Ptolemy Euergetes, the ruins of the city yet stand. It was a center of Hellenistic culture, Euergetes, “benefactor, ” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC, times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain, the approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt, never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there, to the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria, philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Iranian empire, the material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” and he intended to verify and emend the books, releasing each one as it was completed
Apollonius of Perga
–
Pages from the 9th century Arabic translation of the Conics
Apollonius of Perga
–
Parabola connection with areas of a square and a rectangle, that inspired Apollonius of Perga to give the parabola its current name.
38.
Harold Scott MacDonald Coxeter
–
Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
Harold Scott MacDonald Coxeter
–
Harold Scott MacDonald Coxeter
39.
Euclid
–
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
Euclid
–
Euclid by
Justus van Gent, 15th century
Euclid
–
One of the oldest surviving fragments of Euclid's Elements, found at
Oxyrhynchus and dated to circa AD 100 (
P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid
–
Statue in honor of Euclid in the
Oxford University Museum of Natural History
40.
Mikhail Leonidovich Gromov
–
Mikhail Leonidovich Gromov, is a French-Russian mathematician known for important contributions in many different areas of mathematics, including geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University, Gromov has won several prizes, including the Abel Prize in 2009 for his revolutionary contributions to geometry. Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union and his father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. Gromov was born during World War II, and his mother, when Gromov was nine years old, his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him. Gromov studied mathematics at Leningrad State University where he obtained a degree in 1965. His thesis advisor was Vladimir Rokhlin, in 1970, invited to give a presentation at the International Congress of Mathematicians in France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings, disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel and he changed his last name to that of his mother. When the request was granted in 1974, he moved directly to New York where a position had been arranged for him at Stony Brook. In 1981 he left Stony Brook to join the faculty of University of Paris VI, at the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences since 1996. He adopted French citizenship in 1992, Gromovs style of geometry often features a coarse or soft viewpoint, analyzing asymptotic or large-scale properties. In the 1980s, Gromov introduced the Gromov–Hausdorff metric, a measure of the difference between two metric spaces. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, Gromov was also the first to study the space of all possible Riemannian structures on a given manifold. Gromov introduced geometric group theory, the study of infinite groups via the geometry of their Cayley graphs, in 1981 he proved Gromovs theorem on groups of polynomial growth, a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof uses the Gromov–Hausdorff metric mentioned above, along with Eliyahu Rips he introduced the notion of hyperbolic groups. Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves and this led to Gromov–Witten invariants which are used in string theory and to his non-squeezing theorem. Gromov is also interested in biology, the structure of the brain and the thinking process. Member of the French Academy of Sciences Gromov, M. Hyperbolic manifolds, groups, riemann surfaces and related topics, Proceedings of the 1978 Stony Brook Conference, pp. 183–213, Ann. of Math
Mikhail Leonidovich Gromov
–
Mikhail Gromov
41.
Felix Klein
–
His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day. Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents, his father, Kleins mother was Sophie Elise Klein. He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, at that time, Julius Plücker held Bonns chair of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plückers interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868, Plücker died in 1868, leaving his book on the foundations of line geometry incomplete. Klein was the person to complete the second part of Plückers Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch. Klein visited Clebsch the following year, along with visits to Berlin, in July 1870, at the outbreak of the Franco-Prussian War, he was in Paris and had to leave the country. For a short time, he served as an orderly in the Prussian army before being appointed lecturer at Göttingen in early 1871. Erlangen appointed Klein professor in 1872, when he was only 23, in this, he was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day. Klein did not build a school at Erlangen where there were few students, in 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. After five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig, there his colleagues included Walther von Dyck, Rohn, Eduard Study and Friedrich Engel. Kleins years at Leipzig,1880 to 1886, fundamentally changed his life, in 1882, his health collapsed, in 1883–1884, he was plagued by depression. Nonetheless his research continued, his work on hyperelliptic sigma functions dates from around this period. Klein accepted a chair at the University of Göttingen in 1886, from then until his 1913 retirement, he sought to re-establish Göttingen as the worlds leading mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own role as the leader of a school of geometry, at Göttingen, he taught a variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory. The research center Klein established at Göttingen served as a model for the best such centers throughout the world and he introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein hired David Hilbert away from Königsberg, this appointment proved fateful, under Kleins editorship, Mathematische Annalen became one of the very best mathematics journals in the world. Founded by Clebsch, only under Kleins management did it first rival then surpass Crelles Journal based out of the University of Berlin, Klein set up a small team of editors who met regularly, making democratic decisions. The journal specialized in analysis, algebraic geometry, and invariant theory
Felix Klein
–
Felix Klein
42.
Hermann Minkowski
–
Hermann Minkowski was a Jewish German mathematician, professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used methods to solve problems in number theory, mathematical physics. Hermann was a brother of the medical researcher, Oskar. In different sources Minkowskis nationality is given as German, Polish, Lithuanian or Lithuanian-German. To escape persecution in Russia the family moved to Königsberg in 1872, Minkowski studied in Königsberg and taught in Bonn, Königsberg and Zurich, and finally in Göttingen from 1902 until his premature death in 1909. He married Auguste Adler in 1897 with whom he had two daughters, the engineer and inventor Reinhold Rudenberg was his son-in-law. Minkowski died suddenly of appendicitis in Göttingen on 12 January 1909 and our science, which we loved above all else, brought us together, it seemed to us a garden full of flowers. He was for me a gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst, however, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us. The main-belt asteroid 12493 Minkowski and M-matrices are named in Minkowskis honor, Minkowski was educated in Germany at the Albertina University of Königsberg, where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. In 1883, while still a student at Königsberg, he was awarded the Mathematics Prize of the French Academy of Sciences for his manuscript on the theory of quadratic forms and he also became a friend of another renowned mathematician, David Hilbert. His brother, Oskar Minkowski, was a physician and researcher. Minkowski taught at the universities of Bonn, Göttingen, Königsberg, at the Eidgenössische Polytechnikum, today the ETH Zurich, he was one of Einsteins teachers. Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, in 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory. He is also the creator of the Minkowski Sausage and the Minkowski cover of a curve, in 1902, he joined the Mathematics Department of Göttingen and became a close colleague of David Hilbert, whom he first met at university in Königsberg. Constantin Carathéodory was one of his students there, henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 53–111. English translation, The Fundamental Equations for Electromagnetic Processes in Moving Bodies, in, The Principle of Relativity, Calcutta, University Press, 1–69 Minkowski, Hermann
Hermann Minkowski
–
Hermann Minkowski
43.
Blaise Pascal
–
Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a prodigy who was educated by his father. Pascal also wrote in defence of the scientific method, in 1642, while still a teenager, he started some pioneering work on calculating machines. After three years of effort and 50 prototypes, he built 20 finished machines over the following 10 years, following Galileo Galilei and Torricelli, in 1647, he rebutted Aristotles followers who insisted that nature abhors a vacuum. Pascals results caused many disputes before being accepted, in 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. Following a religious experience in late 1654, he began writing works on philosophy. His two most famous works date from this period, the Lettres provinciales and the Pensées, the set in the conflict between Jansenists and Jesuits. In that year, he wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids, Pascal had poor health, especially after the age of 18, and he died just two months after his 39th birthday. Pascal was born in Clermont-Ferrand, which is in Frances Auvergne region and he lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal, who also had an interest in science and mathematics, was a local judge, Pascal had two sisters, the younger Jacqueline and the elder Gilberte. In 1631, five years after the death of his wife, the newly arrived family soon hired Louise Delfault, a maid who eventually became an instrumental member of the family. Étienne, who never remarried, decided that he alone would educate his children, for they all showed extraordinary intellectual ability, the young Pascal showed an amazing aptitude for mathematics and science. Particularly of interest to Pascal was a work of Desargues on conic sections and it states that if a hexagon is inscribed in a circle then the three intersection points of opposite sides lie on a line. Pascals work was so precocious that Descartes was convinced that Pascals father had written it, in France at that time offices and positions could be—and were—bought and sold. In 1631 Étienne sold his position as president of the Cour des Aides for 65,665 livres. The money was invested in a government bond which provided, if not a lavish, then certainly a comfortable income which allowed the Pascal family to move to, but in 1638 Richelieu, desperate for money to carry on the Thirty Years War, defaulted on the governments bonds. Suddenly Étienne Pascals worth had dropped from nearly 66,000 livres to less than 7,300 and it was only when Jacqueline performed well in a childrens play with Richelieu in attendance that Étienne was pardoned
Blaise Pascal
–
Painting of Blaise Pascal made by François II Quesnel for Gérard Edelinck in 1691.
Blaise Pascal
–
An early
Pascaline on display at the
Musée des Arts et Métiers, Paris
Blaise Pascal
–
Portrait of Pascal
Blaise Pascal
–
Pascal studying the
cycloid, by
Augustin Pajou, 1785,
Louvre
44.
Pythagoras
–
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
Pythagoras
–
Bust of Pythagoras of Samos in the
Capitoline Museums,
Rome.
Pythagoras
–
Bust of Pythagoras,
Vatican
Pythagoras
–
A scene at the
Chartres Cathedral shows a philosopher, on one of the
archivolts over the right door of the west portal at Chartres, which has been attributed to depict Pythagoras.
Pythagoras
–
Croton on the southern coast of Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling overburdened in Samos.
45.
Before Common Era
–
Common Era or Current Era is a year-numbering system for the Julian and Gregorian calendars that refers to the years since the start of this era, i. e. since AD1. The preceding era is referred to as before the Common or Current Era, the Current Era notation system can be used as a secular alternative to the Dionysian era system, which distinguishes eras as AD and BC. The two notation systems are equivalent, thus 2017 CE corresponds to AD2017 and 400 BCE corresponds to 400 BC. The year-numbering system for the Gregorian calendar is the most widespread civil calendar used in the world today. For decades, it has been the standard, recognized by international institutions such as the United Nations. The expression has been traced back to Latin usage to 1615, as vulgaris aerae, the term Common Era can be found in English as early as 1708, and became more widely used in the mid-19th century by Jewish academics. He attempted to number years from a reference date, an event he referred to as the Incarnation of Jesus. Dionysius labeled the column of the table in which he introduced the new era as Anni Domini Nostri Jesu Christi, numbering years in this manner became more widespread in Europe with its usage by Bede in England in 731. Bede also introduced the practice of dating years before what he supposed was the year of birth of Jesus, in 1422, Portugal became the last Western European country to switch to the system begun by Dionysius. The first use of the Latin term vulgaris aerae discovered so far was in a 1615 book by Johannes Kepler, Kepler uses it again in a 1616 table of ephemerides, and again in 1617. A1635 English edition of that book has the title page in English – so far, a 1701 book edited by John LeClerc includes Before Christ according to the Vulgar Æra,6. A1716 book in English by Dean Humphrey Prideaux says, before the beginning of the vulgar æra, a 1796 book uses the term vulgar era of the nativity. The first so-far-discovered usage of Christian Era is as the Latin phrase aerae christianae on the page of a 1584 theology book. In 1649, the Latin phrase æræ Christianæ appeared in the title of an English almanac, a 1652 ephemeris is the first instance so-far-found for English usage of Christian Era. The English phrase common Era appears at least as early as 1708, a 1759 history book uses common æra in a generic sense, to refer to the common era of the Jews. The first-so-far found usage of the phrase before the era is in a 1770 work that also uses common era and vulgar era as synonyms. The 1797 edition of the Encyclopædia Britannica uses the terms vulgar era, the Catholic Encyclopedia in at least one article reports all three terms being commonly understood by the early 20th century. Thus, the era of the Jews, the common era of the Mahometans, common era of the world
Before Common Era
–
Key concepts
46.
Middle Ages
–
In the history of Europe, the Middle Ages or Medieval Period lasted from the 5th to the 15th century. It began with the fall of the Western Roman Empire and merged into the Renaissance, the Middle Ages is the middle period of the three traditional divisions of Western history, classical antiquity, the medieval period, and the modern period. The medieval period is subdivided into the Early, High. Population decline, counterurbanisation, invasion, and movement of peoples, the large-scale movements of the Migration Period, including various Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. In the seventh century, North Africa and the Middle East—once part of the Byzantine Empire—came under the rule of the Umayyad Caliphate, although there were substantial changes in society and political structures, the break with classical antiquity was not complete. The still-sizeable Byzantine Empire survived in the east and remained a major power, the empires law code, the Corpus Juris Civilis or Code of Justinian, was rediscovered in Northern Italy in 1070 and became widely admired later in the Middle Ages. In the West, most kingdoms incorporated the few extant Roman institutions, monasteries were founded as campaigns to Christianise pagan Europe continued. The Franks, under the Carolingian dynasty, briefly established the Carolingian Empire during the later 8th, the Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation states, reducing crime and violence, intellectual life was marked by scholasticism, a philosophy that emphasised joining faith to reason, and by the founding of universities. Controversy, heresy, and the Western Schism within the Catholic Church paralleled the conflict, civil strife. Cultural and technological developments transformed European society, concluding the Late Middle Ages, the Middle Ages is one of the three major periods in the most enduring scheme for analysing European history, classical civilisation, or Antiquity, the Middle Ages, and the Modern Period. Medieval writers divided history into periods such as the Six Ages or the Four Empires, when referring to their own times, they spoke of them as being modern. In the 1330s, the humanist and poet Petrarch referred to pre-Christian times as antiqua, leonardo Bruni was the first historian to use tripartite periodisation in his History of the Florentine People. Bruni and later argued that Italy had recovered since Petrarchs time. The Middle Ages first appears in Latin in 1469 as media tempestas or middle season, in early usage, there were many variants, including medium aevum, or middle age, first recorded in 1604, and media saecula, or middle ages, first recorded in 1625. The alternative term medieval derives from medium aevum, tripartite periodisation became standard after the German 17th-century historian Christoph Cellarius divided history into three periods, Ancient, Medieval, and Modern. The most commonly given starting point for the Middle Ages is 476, for Europe as a whole,1500 is often considered to be the end of the Middle Ages, but there is no universally agreed upon end date. English historians often use the Battle of Bosworth Field in 1485 to mark the end of the period
Middle Ages
–
The
Cross of Mathilde, a
crux gemmata made for
Mathilde, Abbess of Essen (973–1011), who is shown kneeling before the Virgin and Child in the
enamel plaque. The body of Christ is slightly later. Probably made in
Cologne or
Essen, the cross demonstrates several medieval techniques:
cast figurative sculpture,
filigree, enamelling, gem polishing and setting, and the reuse of Classical
cameos and
engraved gems.
Middle Ages
–
A late Roman statue
depicting the four Tetrarchs, now in
Venice
Middle Ages
–
Coin of
Theodoric
Middle Ages
–
Mosaic showing
Justinian with
the bishop of
Ravenna, bodyguards, and courtiers
47.
Pierre de Fermat
–
He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermats principle for light propagation and his Fermats Last Theorem in number theory, Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a leather merchant. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth, there is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban. He attended the University of Orléans from 1623 and received a bachelor in law in 1626. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apolloniuss De Locis Planis to one of the mathematicians there, there he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France and he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat, fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often little or no proof of his theorems. In some of these letters to his friends he explored many of the ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time and this naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, The basis of Fermats mathematics was the classical Greek treatises combined with Vietas new algebraic methods, Fermats pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, in these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of power functions. With his method, he was able to reduce this evaluation to the sum of geometric series, the resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pells equation, perfect numbers, amicable numbers and it was while researching perfect numbers that he discovered Fermats little theorem. Fermat developed the two-square theorem, and the polygonal number theorem, although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived
Pierre de Fermat
–
Pierre de Fermat
Pierre de Fermat
–
Bust in the Salle des Illustres in
Capitole de Toulouse
Pierre de Fermat
–
Place of burial of Pierre de Fermat in Place Jean Jaurés, Castres. Translation of the plaque: in this place was buried on January 13, 1665, Pierre de Fermat, councilor of the chamber of Edit [Parlement of Toulouse] and mathematician of great renown, celebrated for his theorem, a n + b n ≠ c n for n>2
Pierre de Fermat
–
Holographic will handwritten by Fermat on 4 March 1660 — kept at the Departmental Archives of
Haute-Garonne, in
Toulouse
48.
General relativity
–
General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
General relativity
–
A simulated
black hole of 10
solar masses within the
Milky Way, seen from a distance of 600 kilometers.
General relativity
–
Albert Einstein developed the theories of special and general relativity. Picture from 1921.
General relativity
–
Einstein cross: four images of the same astronomical object, produced by a
gravitational lens
General relativity
–
Artist's impression of the space-borne gravitational wave detector
LISA
49.
Real analysis
–
Real analysis is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. The theorems of real analysis rely intimately upon the structure of the number line. The real number system consists of a set, together with two operations and an order, and is, formally speaking, an ordered quadruple consisting of these objects, there are several ways of formalizing the definition of the real number system. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in detail in the main article. In addition to these notions, the real numbers, equipped with the absolute value function as a metric. Many important theorems in real analysis remain valid when they are restated as statements involving metric spaces and these theorems are frequently topological in nature, and placing them in the more abstract setting of metric spaces may lead to proofs that are shorter, more natural, or more elegant. The real numbers have several important lattice-theoretic properties that are absent in the complex numbers, most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the numbers is total. These order-theoretic properties lead to a number of important results in analysis, such as the monotone convergence theorem, the intermediate value theorem. However, while the results in analysis are stated for real numbers. In particular, many ideas in analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces. Also, mathematicians consider real and imaginary parts of complex sequences, a sequence is a function whose domain is a countable, totally ordered set, usually taken to be the natural numbers or whole numbers. Occasionally, it is convenient to consider bidirectional sequences indexed by the set of all integers. Of interest in analysis, a real-valued sequence, here indexed by the natural numbers, is a map a, N → R, n ↦ a n. Each a = a n is referred to as a term of the sequence, a sequence that tends to a limit is said to be convergent, otherwise it is divergent
Real analysis
–
The first four partial sums of the
Fourier series for a
square wave. Fourier series are an important tool in real analysis.
50.
Number theory
–
Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
Number theory
–
A
Lehmer sieve, which is a primitive
digital computer once used for finding
primes and solving simple
Diophantine equations.
Number theory
–
The Plimpton 322 tablet
Number theory
–
Title page of the 1621 edition of Diophantus' Arithmetica, translated into
Latin by
Claude Gaspard Bachet de Méziriac.
Number theory
–
Leonhard Euler
51.
Discrete geometry
–
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of geometric objects, such as points, lines, planes, circles, spheres, polygons. The subject focuses on the properties of these objects, such as how they intersect one another. Although polyhedra and tessellations had been studied for years by people such as Kepler and Cauchy. Coxeter and Paul Erdős, laid the foundations of discrete geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions, some theories further generalize the idea to include such objects as unbounded polytopes, and abstract polytopes. A sphere packing is an arrangement of non-overlapping spheres within a containing space, the spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, a tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions, topics in this area include, Cauchys theorem Flexible polyhedra Incidence structures generalize planes as can be seen from their axiomatic definitions. Incidence structures also generalize the higher-dimensional analogs and the structures are sometimes called finite geometries. Formally, a structure is a triple C =. Where P is a set of points, L is a set of lines, the elements of I are called flags. If ∈ I, we say that point p lies on line l, a geometric graph is a graph in which the vertices or edges are associated with geometric objects. Examples include Euclidean graphs, the 1-skeleton of a polyhedron or polytope, intersection graphs, simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a complex is an abstract simplicial complex. The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology, lovászs proof used the Borsuk-Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has used in the study of fair division problems. Topics in this include, Sperners lemma Regular maps A discrete group is a group G equipped with the discrete topology
Discrete geometry
–
A collection of
circles and the corresponding
unit disk graph
52.
Combinatorics
–
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics
Combinatorics
–
An example of
change ringing (with six bells), one of the earliest nontrivial results in Graph Theory.
53.
Arab
–
Arabs are an ethnic group inhabiting the Arab world. They primarily live in the Arab states in Western Asia, North Africa, the Horn of Africa, the Arabs are first mentioned in the mid-ninth century BCE as a tribal people dwelling in the central Arabian Peninsula. The Arabs appear to have been under the vassalage of the Neo-Assyrian Empire, tradition holds that Arabs descend from Ishmael, the son of Abraham. The Arabian Desert is the birthplace of Arab, there are other Arab groups as well that spread in the land and existed for millennia. Before the expansion of the Caliphate, Arab referred to any of the largely nomadic Semitic people from the northern to the central Arabian Peninsula and Syrian Desert. Presently, Arab refers to a number of people whose native regions form the Arab world due to spread of Arabs throughout the region during the early Arab conquests of the 7th and 8th centuries. The Arabs forged the Rashidun, Umayyad and the Abbasid caliphates, whose borders reached southern France in the west, China in the east, Anatolia in the north, and this was one of the largest land empires in history. The Great Arab Revolt has had as big an impact on the modern Middle East as the World War I, the war signaled the end of the Ottoman Empire. They are modern states and became significant as distinct political entities after the fall and defeat, following adoption of the Alexandria Protocol in 1944, the Arab League was founded on 22 March 1945. The Charter of the Arab League endorsed the principle of an Arab homeland whilst respecting the sovereignty of its member states. Beyond the boundaries of the League of Arab States, Arabs can also be found in the global diaspora, the ties that bind Arabs are ethnic, linguistic, cultural, historical, identical, nationalist, geographical and political. The Arabs have their own customs, language, architecture, art, literature, music, dance, media, cuisine, dress, society, sports, the total number of Arabs are an estimated 450 million. This makes them the second largest ethnic group after the Han Chinese. Arabs are a group in terms of religious affiliations and practices. In the pre-Islamic era, most Arabs followed polytheistic religions, some tribes had adopted Christianity or Judaism, and a few individuals, the hanifs, apparently observed monotheism. Today, Arabs are mainly adherents of Islam, with sizable Christian minorities, Arab Muslims primarily belong to the Sunni, Shiite, Ibadi, Alawite, Druze and Ismaili denominations. Arab Christians generally follow one of the Eastern Christian Churches, such as the Maronite, Coptic Orthodox, Greek Orthodox, Greek Catholic, or Chaldean churches. Listed among the booty captured by the army of king Shalmaneser III of Assyria in the Battle of Qarqar are 1000 camels of Gi-in-di-buu the ar-ba-a-a or Gindibu belonging to the Arab
Arab
Arab
–
Schoolgirls in
Gaza lining up for class, 2009
Arab
–
Syrian immigrants in New York City, as depicted in 1895
Arab
–
Lebanese–Mexican billionaire
Carlos Slim has been ranked by Forbes as the
second richest person in the world.
54.
Mesopotamia
–
In the Iron Age, it was controlled by the Neo-Assyrian and Neo-Babylonian Empires. The Sumerians and Akkadians dominated Mesopotamia from the beginning of history to the fall of Babylon in 539 BC. It fell to Alexander the Great in 332 BC, and after his death, around 150 BC, Mesopotamia was under the control of the Parthian Empire. Mesopotamia became a battleground between the Romans and Parthians, with parts of Mesopotamia coming under ephemeral Roman control. In AD226, eastern part of it fell to the Sassanid Persians, division of Mesopotamia between Roman and Sassanid Empires lasted until the 7th century Muslim conquest of Persia of the Sasanian Empire and Muslim conquest of the Levant from Byzantines. A number of primarily neo-Assyrian and Christian native Mesopotamian states existed between the 1st century BC and 3rd century AD, including Adiabene, Osroene, and Hatra, Mesopotamia is the site of the earliest developments of the Neolithic Revolution from around 10,000 BC. The regional toponym Mesopotamia comes from the ancient Greek root words μέσος middle and ποταμός river and it is used throughout the Greek Septuagint to translate the Hebrew equivalent Naharaim. In the Anabasis, Mesopotamia was used to designate the land east of the Euphrates in north Syria, the Aramaic term biritum/birit narim corresponded to a similar geographical concept. The neighbouring steppes to the west of the Euphrates and the part of the Zagros Mountains are also often included under the wider term Mesopotamia. A further distinction is made between Northern or Upper Mesopotamia and Southern or Lower Mesopotamia. Upper Mesopotamia, also known as the Jazira, is the area between the Euphrates and the Tigris from their sources down to Baghdad, Lower Mesopotamia is the area from Baghdad to the Persian Gulf and includes Kuwait and parts of western Iran. In modern academic usage, the term Mesopotamia often also has a chronological connotation and it is usually used to designate the area until the Muslim conquests, with names like Syria, Jazirah, and Iraq being used to describe the region after that date. It has been argued that these later euphemisms are Eurocentric terms attributed to the region in the midst of various 19th-century Western encroachments, Mesopotamia encompasses the land between the Euphrates and Tigris rivers, both of which have their headwaters in the Armenian Highlands. Both rivers are fed by tributaries, and the entire river system drains a vast mountainous region. Overland routes in Mesopotamia usually follow the Euphrates because the banks of the Tigris are frequently steep and difficult. The climate of the region is semi-arid with a vast desert expanse in the north which gives way to a 15,000 square kilometres region of marshes, lagoons, mud flats, in the extreme south, the Euphrates and the Tigris unite and empty into the Persian Gulf. In the marshlands to the south of the area, a complex water-borne fishing culture has existed since prehistoric times, periodic breakdowns in the cultural system have occurred for a number of reasons. Alternatively, military vulnerability to invasion from marginal hill tribes or nomadic pastoralists has led to periods of trade collapse and these trends have continued to the present day in Iraq
Mesopotamia
–
Known world of the Mesopotamian, Babylonian, and Assyrian cultures from documentary sources
Mesopotamia
–
Map showing the extent of Mesopotamia
Mesopotamia
–
One of 18
Statues of Gudea, a ruler around 2090 BC
Mesopotamia
–
One of the
Nimrud ivories shows a lion eating a man. Neo-Assyrian period, 9th to 7th centuries BC.
55.
Surveying
–
Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional position of points and the distances and angles between them. A land surveying professional is called a land surveyor, Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages and the law. Surveying has been an element in the development of the environment since the beginning of recorded history. The planning and execution of most forms of construction require it and it is also used in transport, communications, mapping, and the definition of legal boundaries for land ownership. It is an important tool for research in other scientific disciplines. Basic surveyance has occurred since humans built the first large structures, the prehistoric monument at Stonehenge was set out by prehistoric surveyors using peg and rope geometry. In ancient Egypt, a rope stretcher would use simple geometry to re-establish boundaries after the floods of the Nile River. The almost perfect squareness and north-south orientation of the Great Pyramid of Giza, built c.2700 BC, the Groma instrument originated in Mesopotamia. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao Suanjing or The Sea Island Mathematical Manual, the Romans recognized land surveyors as a profession. They established the basic measurements under which the Roman Empire was divided, Roman surveyors were known as Gromatici. In medieval Europe, beating the bounds maintained the boundaries of a village or parish and this was the practice of gathering a group of residents and walking around the parish or village to establish a communal memory of the boundaries. Young boys were included to ensure the memory lasted as long as possible, in England, William the Conqueror commissioned the Domesday Book in 1086. It recorded the names of all the owners, the area of land they owned, the quality of the land. It did not include maps showing exact locations, abel Foullon described a plane table in 1551, but it is thought that the instrument was in use earlier as his description is of a developed instrument. Gunters chain was introduced in 1620 by English mathematician Edmund Gunter and it enabled plots of land to be accurately surveyed and plotted for legal and commercial purposes. Leonard Digges described a Theodolite that measured horizontal angles in his book A geometric practice named Pantometria, joshua Habermel created a theodolite with a compass and tripod in 1576. Johnathon Sission was the first to incorporate a telescope on a theodolite in 1725, in the 18th century, modern techniques and instruments for surveying began to be used. Jesse Ramsden introduced the first precision theodolite in 1787 and it was an instrument for measuring angles in the horizontal and vertical planes
Surveying
–
A surveyor at work with an infrared reflector used for distance measurement.
Surveying
–
Table of Surveying, 1728
Cyclopaedia
Surveying
–
A map of India showing the Great Trigonometrical Survey, produced in 1870
Surveying
–
A German engineer surveying during the
First World War, 1918
56.
Construction
–
Construction is the process of constructing a building or infrastructure. Construction as an industry comprises six to nine percent of the domestic product of developed countries. Construction starts with planning, design, and financing, and continues until the project is built, large-scale construction requires collaboration across multiple disciplines. An architect normally manages the job, and a manager, design engineer. For the successful execution of a project, effective planning is essential, the largest construction projects are referred to as megaprojects. Construction is a term meaning the art and science to form objects, systems, or organizations. Construction is used as a verb, the act of building, and a noun, how a building was built, in general, there are three sectors of construction, buildings, infrastructure and industrial. Building construction is further divided into residential and non-residential. Infrastructure is often called heavy/highway, heavy civil or heavy engineering and it includes large public works, dams, bridges, highways, water/wastewater and utility distribution. Industrial includes refineries, process chemical, power generation, mills, there are other ways to break the industry into sectors or markets. Engineering News-Record is a magazine for the construction industry. Each year, ENR compiles and reports on data about the size of design and they publish a list of the largest companies in the United States and also a list the largest global firms. In 2014, ENR compiled the data in nine market segments and it was divided as transportation, petroleum, buildings, power, industrial, water, manufacturing, sewer/waste, telecom, hazardous waste plus a tenth category for other projects. In their reporting on the Top 400, they used data on transportation, sewer, hazardous waste, the Standard Industrial Classification and the newer North American Industry Classification System have a classification system for companies that perform or otherwise engage in construction. To recognize the differences of companies in this sector, it is divided into three subsectors, building construction, heavy and civil engineering construction, and specialty trade contractors, there are also categories for construction service firms and construction managers. Building construction is the process of adding structure to real property or construction of buildings, the majority of building construction jobs are small renovations, such as addition of a room, or renovation of a bathroom. Often, the owner of the property acts as laborer, paymaster, for this reason, those with experience in the field make detailed plans and maintain careful oversight during the project to ensure a positive outcome. Residential construction practices, technologies, and resources must conform to local building authority regulations, materials readily available in the area generally dictate the construction materials used
Construction
–
In large construction projects, such as this
skyscraper in
Melbourne,
Australia,
cranes are essential.
Construction
–
Military residential unit construction by U.S. Navy personnel in Afghanistan
Construction
–
The National Cement Share Company of
Ethiopia 's new plant in
Dire Dawa.
Construction
–
Framing
57.
Plimpton 322
–
Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G. A and this tablet, believed to have been written about 1800 BC, has a table of four columns and 15 rows of numbers in the cuneiform script of the period. This table lists what are now called Pythagorean triples, i. e. integers a, b, from a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem. Although the tablet was interpreted in the past as a table, more recently published work sees this as anachronistic. For readable popular treatments of this tablet see Robson or, more briefly, Robson is a more detailed and technical discussion of the interpretation of the tablets numbers, with an extensive bibliography. Plimpton 322 is partly broken, approximately 13 cm wide,9 cm tall, according to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa. More specifically, based on formatting similarities with other tablets from Larsa that have explicit dates written on them, Robson points out that Plimpton 322 was written in the same format as other administrative, rather than mathematical, documents of the period. The main content of Plimpton 322 is a table of numbers, with four columns and fifteen rows, the fourth column is just a row number, in order from 1 to 15. The second and third columns are visible in the surviving tablet. Conversion of these numbers from sexagesimal to decimal raises additional ambiguities, the sixty sexigesimal entries are exact, no truncations or rounding off. In each row, the number in the column can be interpreted as the shortest side s of a right triangle. The number in the first column is either the fraction s 2 l 2 or d 2 l 2 =1 + s 2 l 2, scholars still differ, however, on how these numbers were generated. Below is the translation of the tablet. Otto E. Neugebauer argued for an interpretation, pointing out that this table provides a list of Pythagorean triples. For instance, line 11 of the table can be interpreted as describing a triangle with short side 3/4 and hypotenuse 5/4, forming the side, hypotenuse ratio of the familiar right triangle. If p and q are two numbers, one odd and one even, then form a Pythagorean triple. For instance, line 11 can be generated by this formula with p =2 and q =1, as Neugebauer argues, each line of the tablet can be generated by a pair that are both regular numbers, integer divisors of a power of 60. This property of p and q being regular leads to a denominator that is regular, neugebauers explanation is the one followed e. g. by Conway & Guy
Plimpton 322
–
The Plimpton 322 tablet.
58.
Frustum
–
In geometry, a frustum is the portion of a solid that lies between one or two parallel planes cutting it. A right frustum is a truncation of a right pyramid. The term is used in computer graphics to describe the viewing frustum. It is formed by a pyramid, in particular, frustum culling is a method of hidden surface determination. In the aerospace industry, frustum is the term for the fairing between two stages of a multistage rocket, which is shaped like a truncated cone. Each plane section is a floor or base of the frustum and its axis if any, is that of the original cone or pyramid. A frustum is circular if it has circular bases, it is if the axis is perpendicular to both bases, and oblique otherwise. The height of a frustum is the distance between the planes of the two bases. Cones and pyramids can be viewed as degenerate cases of frusta, the pyramidal frusta are a subclass of the prismatoids. Two frusta joined at their bases make a bifrustum, the Egyptians knew the correct formula for obtaining the volume of a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus. V = h 1 a h 12 − h 2 a h 223 = a 3 By factoring the difference of two cubes we get h1−h2 = h, the height of the frustum, and α/3. Distributing α and substituting from its definition, the Heronian mean of areas B1, the alternative formula is therefore V = h 3 Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary number, the square root of negative one. In particular, the volume of a circular cone frustum is V = π h 3 where π is 3.14159265. and R1, R2 are the radii of the two bases. The volume of a frustum whose bases are n-sided regular polygons is V = n h 12 cot π n where a1. The surface area of a frustum whose bases are similar regular n-sided polygons is A = n 4 where a1. On the back of a United States one-dollar bill, a pyramidal frustum appears on the reverse of the Great Seal of the United States, certain ancient Native American mounds also form the frustum of a pyramid. The John Hancock Center in Chicago, Illinois is a frustum whose bases are rectangles, the Washington Monument is a narrow square-based pyramidal frustum topped by a small pyramid. The viewing frustum in 3D computer graphics is a photographic or video cameras usable field of view modeled as a pyramidal frustum
Frustum
59.
Displacement (vector)
–
A displacement is a vector that is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a line from the initial position to the final position of the point. The velocity then is distinct from the speed which is the time rate of change of the distance traveled along a specific path. The velocity may be defined as the time rate of change of the position vector. For motion over an interval of time, the displacement divided by the length of the time interval defines the average velocity. In dealing with the motion of a body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement, for a position vector s that is a function of time t, the derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, vibration sensing and other sciences, by extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the displacement function. Such higher-order terms are required in order to represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering. The fourth order derivative is called jounce
Displacement (vector)
–
Displacement versus distance traveled along a path
60.
Mean speed theorem
–
It essentially says that, a uniformly accelerated body travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body. Oresme essentially provided a geometrical verification for the generalized Merton Rule, clay tablets used in Babylonian astronomy present trapezoid procedures for computing Jupiters position and motion and anticipate the theorem by 14 centuries. The medieval scientists demonstrated this theorem — the foundation of The Law of Falling Bodies — long before Galileo, in principle, the qualities of Greek physics were replaced, at least for motions, by the numerical quantities that have ruled Western science ever since. The work was quickly diffused into France, Italy, and other parts of Europe, the theorem is a special case of the more general kinematics equations for uniform acceleration. Science in the Middle Ages Scholasticism Sylla, Edith The Oxford Calculators, in Kretzmann, Kenny & Pinborg, longeway, John William Heytesbury, in The Stanford Encyclopedia of Philosophy
Mean speed theorem
–
Galileo 's demonstration of the law of the space traversed in case of uniformly varied motion. It's the same demonstration that
Oresme had made centuries earlier.
61.
Greek mathematics
–
Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
Greek mathematics
–
Statue of Euclid in the
Oxford University Museum of Natural History
Greek mathematics
–
An illustration of
Euclid 's proof of the
Pythagorean Theorem
Greek mathematics
–
The
Antikythera mechanism, an ancient mechanical calculator.
62.
Thales of Miletus
–
Thales of Miletus was a pre-Socratic Greek/Phoenician philosopher, mathematician and astronomer from Miletus in Asia Minor. He was one of the Seven Sages of Greece, Thales is recognized for breaking from the use of mythology to explain the world and the universe, and instead explaining natural objects and phenomena by theories and hypothesis, i. e. science. Aristotle reported Thales hypothesis that the principle of nature and the nature of matter was a single material substance. In mathematics, Thales used geometry to calculate the heights of pyramids and he is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to Thales theorem. He is the first known individual to whom a mathematical discovery has been attributed, the ancient source, Apollodorus of Athens, writing during the 2nd century BCE, thought Thales was born about the year 625 BCE. The dates of Thales life are not exactly known, but are roughly established by a few events mentioned in the sources. According to Herodotus Thales predicted the eclipse of May 28,585 BC. Diogenes Laërtius quotes the chronicle of Apollodorus of Athens as saying that Thales died at the age of 78 during the 58th Olympiad and attributes his death to heat stroke while watching the games. Plutarch had earlier told this version, Solon visited Thales and asked him why he remained single, nevertheless, several years later, anxious for family, he adopted his nephew Cybisthus. Thales involved himself in many activities, taking the role of an innovator, some say that he left no writings, others say that he wrote On the Solstice and On the Equinox. Diogenes Laërtius quotes two letters from Thales, one to Pherecydes of Syros, offering to review his book on religion, Thales identifies the Milesians as Athenian colonists. He was aware of the existence of the lodestone, and was the first to be connected to knowledge of this in history, according to Aristotle, Thales thought lodestones had souls, because iron is attracted to them. According to Hieronymus, historically quoted by Diogenes Laertius, Thales found the height of pyramids by comparison between the lengths of the shadows cast by a person and by the pyramids, several anecdotes suggest Thales was not just a philosopher, but also a businessman. A story, with different versions, recounts how Thales achieved riches from an olive harvest by prediction of the weather, in one version, he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. Thales’ political life had mainly to do with the involvement of the Ionians in the defense of Anatolia against the power of the Persians. In neighbouring Lydia, a king had come to power, Croesus and he had conquered most of the states of coastal Anatolia, including the cities of the Ionians. The story is told in Herodotus, the war endured for five years, but in the sixth an eclipse of the Sun spontaneously halted a battle in progress. It seems that Thales had predicted this solar eclipse, the Seven Sages were most likely already in existence, as Croesus was also heavily influenced by Solon of Athens, another sage
Thales of Miletus
–
Thales of Miletus
Thales of Miletus
–
An olive mill and an olive press dating from Roman times in
Capernaum, Israel.
Thales of Miletus
–
Total
eclipse of the
Sun
Thales of Miletus
–
The Ionic Stoa on the Sacred Way in Miletus
63.
Pythagoreans
–
Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Pythagorean ideas exercised an influence on Aristotle, and Plato. According to tradition, pythagoreanism developed at some point into two schools of thought, the mathēmatikoi and the akousmatikoi. There is the inner and outer circle John Burnet noted Lastly, we have one admitted instance of a philosophic guild, that of the Pythagoreans. And it will be found that the hypothesis, if it is to be called by that name, of a regular organisation of scientific activity will alone explain all the facts. The development of doctrine in the hands of Thales, Anaximander, according to Iamblichus in The life of Pythagoras, by Thomas Taylor There were also two forms of philosophy, for the two genera of those that pursued it, the Acusmatici and the Mathematici. The latter are acknowledged to be Pythagoreans by the rest but the Mathematici do not admit that the Acusmatici derived their instructions from Pythagoras, memory was the most valued faculty. All these auditions were of three kinds, some signifying what a thing is, others what it especially is, others what ought or ought not to be done. By musical sounds alone unaccompanied with words they healed the passions of the soul and certain diseases, enchanting in reality and it is probable that from hence this name epode, i. e. enchantment, came to be generally used. Each of these he corrected through the rule of virtue, attempering them through appropriate melodies, therefore its function is none of what are called ‘parts of virtue’, for it is better than all of them and the end produced is always better than the knowledge that produces it. Nor is every virtue of the soul in that way a function, nor is success, for if it is to be productive, different ones will produce different things, as the skill of building produces a house. However, intelligence is a part of virtue and of success, according to historians like Thomas Gale, Thomas Taler, or Cantor, Archytas became the head of the school, about a century after the murder of Pythagoras. According to August Böckh, who cites Nicomachus, Philolaus was the successor of Pythagoras, and according to Cicero, Philolaus was teacher of Archytas of Tarentum. According to the historians from the Stanford Encyclopedia of Philosophy, Philolaus and Eurytus are identified by Aristoxenus as teachers of the last generation of Pythagoreans, a Echecrates is mentioned by Aristoxenus as a student of Philolaus and Eurytus. The mathēmatikoi were supposed to have extended and developed the more mathematical, the mathēmatikoi did think that the akousmatikoi were Pythagorean, but felt that their own group was more representative of Pythagoras. Commentary from Sir William Smith, Dictionary of Greek and Roman Biography, Aristotle states the fundamental maxim of the Pythagoreans in various forms. According to Philolaus, number is the dominant and self-produced bond of the continuance of things. But number has two forms, the even and the odd, and a third, resulting from the mixture of the two, the even-odd and this third species is one itself, for it is both even and odd
Pythagoreans
–
Bust of
Pythagoras,
Musei Capitolini,
Rome.
Pythagoreans
–
Pythagoreans celebrate sunrise by
Fyodor Bronnikov
Pythagoreans
–
Excerpt from
Philolaus Pythagoras book, (Charles Peter Mason, 1870)
64.
Incommensurable magnitudes
–
In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
Incommensurable magnitudes
–
The mathematical constant
pi (π) is an irrational number that is much represented in popular culture.
65.
Syracuse, Italy
–
Syracuse is a historic city in Sicily, the capital of the province of Syracuse. The city is notable for its rich Greek history, culture, amphitheatres, architecture and this 2, 700-year-old city played a key role in ancient times, when it was one of the major powers of the Mediterranean world. Syracuse is located in the southeast corner of the island of Sicily, the city was founded by Ancient Greek Corinthians and Teneans and became a very powerful city-state. Syracuse was allied with Sparta and Corinth and exerted influence over the entirety of Magna Graecia, described by Cicero as the greatest Greek city and the most beautiful of them all, it equaled Athens in size during the fifth century BC. It later became part of the Roman Republic and Byzantine Empire, after this Palermo overtook it in importance, as the capital of the Kingdom of Sicily. Eventually the kingdom would be united with the Kingdom of Naples to form the Two Sicilies until the Italian unification of 1860, in the modern day, the city is listed by UNESCO as a World Heritage Site along with the Necropolis of Pantalica. In the central area, the city itself has a population of around 125,000 people, the inhabitants are known as Siracusans. Syracuse is mentioned in the Bible in the Acts of the Apostles book at 28,12 as Paul stayed there, the patron saint of the city is Saint Lucy, she was born in Syracuse and her feast day, Saint Lucys Day, is celebrated on 13 December. Syracuse was founded in 734 or 733 BC by Greek settlers from Corinth and Tenea, there are many attested variants of the name of the city including Συράκουσαι Syrakousai, Συράκοσαι Syrakosai and Συρακώ Syrako. The nucleus of the ancient city was the island of Ortygia. The settlers found the fertile and the native tribes to be reasonably well-disposed to their presence. The city grew and prospered, and for some time stood as the most powerful Greek city anywhere in the Mediterranean, colonies were founded at Akrai, Kasmenai, Akrillai, Helorus and Kamarina. The descendants of the first colonists, called Gamoroi, held power until they were expelled by the Killichiroi, the former, however, returned to power in 485 BC, thanks to the help of Gelo, ruler of Gela. Gelo himself became the despot of the city, and moved many inhabitants of Gela, Kamarina and Megera to Syracuse, building the new quarters of Tyche, the enlarged power of Syracuse made unavoidable the clash against the Carthaginians, who ruled western Sicily. In the Battle of Himera, Gelo, who had allied with Theron of Agrigento, a temple dedicated to Athena, was erected in the city to commemorate the event. Syracuse grew considerably during this time and its walls encircled 120 hectares in the fifth century, but as early as the 470s BC the inhabitants started building outside the walls. The complete population of its territory approximately numbered 250,000 in 415 BC, Gelo was succeeded by his brother Hiero, who fought against the Etruscans at Cumae in 474 BC. His rule was eulogized by poets like Simonides of Ceos, Bacchylides and Pindar, a democratic regime was introduced by Thrasybulos
Syracuse, Italy
–
Ortygia island, where Syracuse was founded in
ancient Greek times.
Mount Etna is visible in the distance.
Syracuse, Italy
–
A Syracusan
tetradrachm (c. 415–405 BC), sporting
Arethusa and a
quadriga.
Syracuse, Italy
–
Decadrachme from Sicile struck at Syracuse and sign d'Évainète
Syracuse, Italy
–
The siege of Syracuse in a 17th-century engraving.
66.
Parabola
–
A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
Parabola
–
Parabolic compass designed by
Leonardo da Vinci
Parabola
–
Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.
Parabola
–
A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and
air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.
Parabola
–
Parabolic trajectories of water in a fountain.
67.
Pi
–
The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
Pi
–
The constant π is represented in this
mosaic outside the Mathematics Building at the
Technical University of Berlin.
Pi
–
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
Pi
–
Archimedes developed the polygonal approach to approximating π.
Pi
–
Isaac Newton used
infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
68.
Surface of revolution
–
A surface of revolution is a surface in Euclidean space created by rotating a curve around an axis of rotation. Examples of surfaces of revolution generated by a line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. The sections of the surface of revolution made by planes through the axis are called meridional sections, any meridional section can be considered to be the generatrix in the plane determined by it and the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles, some special cases of hyperboloids and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular and this formula is the calculus equivalent of Pappuss centroid theorem. The quantity 2 +2 comes from the Pythagorean theorem and represents a segment of the arc of the curve. The quantity 2πx is the path of this segment, as required by Pappus theorem. Likewise, when the axis of rotation is the x-axis and provided that y is never negative and these come from the above formula. For example, the surface with unit radius is generated by the curve y = sin, x = cos. Its area is therefore A =2 π ∫0 π sin 2 +2 d t =2 π ∫0 π sin d t =4 π. A basic problem in the calculus of variations is finding the curve between two points that produces this surface of revolution. There are only two minimal surfaces of revolution, the plane and the catenoid, geodesics on a surface of revolution are governed by Clairauts relation. A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid, for example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus, the use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to surface area without the use of measuring the length
Surface of revolution
–
A portion of the curve x =2+cos z rotated around the z axis
69.
Indian mathematics
–
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama, the decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, in addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China and this was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved, all mathematical works were orally transmitted until approximately 500 BCE, thereafter, they were transmitted both orally and in manuscript form. A later landmark in Indian mathematics was the development of the series expansions for functions by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a theory of differentiation and integration. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of practical mathematics. The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4,2,1, considered favourable for the stability of a brick structure. They used a system of weights based on the ratios, 1/20, 1/10, 1/5, 1/2,1,2,5,10,20,50,100,200. They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, the inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts, bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. The religious texts of the Vedic Period provide evidence for the use of large numbers, by the time of the Yajurvedasaṃhitā-, numbers as high as 1012 were being included in the texts. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta, With three-fourths Puruṣa went up, the Satapatha Brahmana contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Śulba Sūtras list rules for the construction of fire altars. Most mathematical problems considered in the Śulba Sūtras spring from a single theological requirement, according to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. The diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately and they contain lists of Pythagorean triples, which are particular cases of Diophantine equations
Indian mathematics
Indian mathematics
–
The design of the domestic fire altar in the Śulba Sūtra
70.
Pythagorean triples
–
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written, and an example is. If is a Pythagorean triple, then so is for any integer k. A primitive Pythagorean triple is one in which a, b and c are coprime, a right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. However, right triangles with non-integer sides do not form Pythagorean triples, for instance, the triangle with sides a = b =1 and c = √2 is right, but is not a Pythagorean triple because √2 is not an integer. Moreover,1 and √2 do not have a common multiple because √2 is irrational. There are 16 primitive Pythagorean triples with c ≤100, Note, for example, each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot. The formula states that the integers a = m 2 − n 2, b =2 m n, c = m 2 + n 2 form a Pythagorean triple. The triple generated by Euclids formula is primitive if and only if m and n are coprime, every primitive triple arises from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples and this relationship of a, b and c to m and n from Euclids formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclids formula does not produce all triples—for example and this can be remedied by inserting an additional parameter k to the formula. That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra, many formulas for generating triples with particular properties have been developed since the time of Euclid. A proof of the necessity that a, b, c be expressed by Euclids formula for any primitive Pythagorean triple is as follows, all such triples can be written as where a2 + b2 = c2 and a, b, c are coprime. Thus a, b, c are pairwise coprime, as a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd, from a 2 + b 2 = c 2 we obtain c 2 − a 2 = b 2 and hence = b 2. Since b is rational, we set it equal to m n in lowest terms, thus b = n m, as being the reciprocal of b. As m n is fully reduced, m and n are coprime, and they cannot be both even. If they were odd, the numerator of m 2 − n 22 m n would be a multiple of 4
Pythagorean triples
–
The Pythagorean theorem: a 2 + b 2 = c 2
71.
Diophantine equations
–
In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one, an exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations, in more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis, the solutions are described by the following theorem, This Diophantine equation has a solution if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if is a solution, then the solutions have the form, where k is an arbitrary integer. Proof, If d is this greatest common divisor, Bézouts identity asserts the existence of integers e and f such that ae + bf = d, If c is a multiple of d, then c = dh for some integer h, and is a solution. On the other hand, for pair of integers x and y. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution, we have a + b = ax + by + k = ax + by + k = ax + by, showing that is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u + v =0. As u and v are coprime, Euclids lemma shows that exists a integer k such that x2 − x1 = kv. Therefore, x2 = x1 + kv and y2 = y1 − ku, the system to be solved may thus be rewritten as B = UC. Calling yi the entries of V−1X and di those of D = UC and it follows that the system has a solution if and only if bi, i divides di for i ≤ k and di =0 for i > k. If this condition is fulfilled, the solutions of the system are V. Hermite normal form may also be used for solving systems of linear Diophantine equations, however, Hermite normal form does not directly provide the solutions, to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form is more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, the Hermite normal form is substantially easier to compute than the Smith normal form. Integer linear programming amounts to finding some integer solutions of systems that include also inequations
Diophantine equations
–
Finding all
right triangles with integer side-lengths is equivalent to solving the Diophantine equation.
72.
Cyclic quadrilateral
–
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic, the center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case, the word cyclic is from the Ancient Greek κύκλος which means circle or wheel. All triangles have a circumcircle, but not all quadrilaterals do, an example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle, any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles, a bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential. A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent and this common point is the circumcenter. A convex quadrilateral ABCD is cyclic if and only if its opposite angles are supplementary, the direct theorem was Proposition 22 in Book 3 of Euclids Elements. Equivalently, a quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle. That is, for example, ∠ A C B = ∠ A D B. Ptolemys theorem expresses the product of the lengths of the two e and f of a cyclic quadrilateral as equal to the sum of the products of opposite sides. That is, if this equation is satisfied in a convex quadrilateral, then it is a cyclic quadrilateral. If two lines, one containing segment AC and the other containing segment BD, intersect at P, then the four points A, B, C, D are concyclic if, the intersection P may be internal or external to the circle. In the former case, the quadrilateral is ABCD, and in the latter case. When the intersection is internal, the equality states that the product of the segment lengths into which P divides one diagonal equals that of the other diagonal and this is known as the intersecting chords theorem since the diagonals of the cyclic quadrilateral are chords of the circumcircle. Yet another characterization is that a convex quadrilateral ABCD is cyclic if, the area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmaguptas formula K = where s, the semiperimeter, is s = 1/2. This is a corollary of Bretschneiders formula for the general quadrilateral, If also d =0, the cyclic quadrilateral becomes a triangle and the formula is reduced to Herons formula
Cyclic quadrilateral
–
Examples of cyclic quadrilaterals.
73.
Al-Mahani
–
Abu-Abdullah Muhammad ibn Īsa Māhānī was a Persian Muslim mathematician and astronomer from Mahan, Kermān, Persia. A series of observations of lunar and solar eclipses and planetary conjunctions and he wrote commentaries on Euclid and Archimedes, and improved Ishaq ibn Hunayns translation of Menelaus of Alexandrias Spherics. He tried vainly to solve an Archimedean problem, to divide a sphere by means of a plane into two segments being in a ratio of volume. That problem led to an equation, x 3 + c 2 b = c x 2 which Muslim writers called al-Mahanis equation. List of Iranian scientists H. Suter, Die Mathematiker und Astronomen der Araber 26,1900 and his failure to solve the Archimedean problem is quoted by Omar al-Khayyami). Woepcke, Lalgebra dOmar Alkhayyami 2,96 sq. OConnor, John J. Robertson, Edmund F. Abu Abd Allah Muhammad ibn Isa Al-Mahani, MacTutor History of Mathematics archive, al-Māhānī, Abū Abd Allāh Muḥammad Ibn Īsā
Al-Mahani
–
v
74.
Ratio
–
In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, when the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units, but in many applications, the word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
Ratio
–
The ratio of width to height of
standard-definition television.
75.
Cubic equation
–
In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d, where a is nonzero. Setting f =0 produces an equation of the form. The solutions of this equation are called roots of the polynomial f, If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the equation can be found algebraically. The roots can also be found trigonometrically, alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newtons method. The coefficients do not need to be complex numbers, much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, the problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, in the 3rd century, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. In the 11th century, the Persian poet-mathematician, Omar Khayyám, in an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution, in the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of an equation, x3 + 12x = 6x2 +35. He used what would later be known as the Ruffini-Horner method to approximate the root of a cubic equation. He also developed the concepts of a function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the equation to find algebraic solutions to certain types of cubic equations. Leonardo de Pisa, also known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x =20
Cubic equation
–
Niccolò Fontana Tartaglia
Cubic equation
76.
Ibn al-Haytham
–
Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, and philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and he was the first to explain that vision occurs when light bounces on an object and then is directed to ones eyes. He spent most of his close to the court of the Fatimid Caliphate in Cairo and earned his living authoring various treatises. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called The Physicist and he is also sometimes called al-Baṣrī after his birthplace Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born c.965 in Basra, which was part of the Buyid emirate. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. Legend has it that after deciding the scheme was impractical and fearing the caliphs anger, during this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. Among his students were Sorkhab, a Persian from Semnan who was his student for three years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen. Alhazen made significant contributions to optics, number theory, geometry, astronomy, Alhazens work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text. A Latin translation of the Kitab al-Manazir was made probably in the twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration and he made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the known as Alhazens problem. Alhazen wrote as many as 200 books, although only 55 have survived, some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew, the crater Alhazen on the Moon is named in his honour, as was the asteroid 59239 Alhazen. In honour of Alhazen, the Aga Khan University named its Ophthalmology endowed chair as The Ibn-e-Haitham Associate Professor, Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10, 000-dinar banknote issued in 2003, and on 10-dinar notes from 1982. The 2015 International Year of Light celebrated the 1000th anniversary of the works on optics by Ibn Al-Haytham, Alhazens most famous work is his seven-volume treatise on optics Kitab al-Manazir, written from 1011 to 1021. Optics was translated into Latin by a scholar at the end of the 12th century or the beginning of the 13th century
Ibn al-Haytham
–
Front page of the Opticae Thesaurus, which included the first printed Latin translation of Alhazen's Book of Optics. The illustration incorporates many examples of optical phenomena including perspective effects, the rainbow, mirrors, and refraction.
Ibn al-Haytham
–
Alhazen (Ibn al-Haytham)
Ibn al-Haytham
–
The
theorem of Ibn Haytham
Ibn al-Haytham
–
Alhazen on Iraqi 10 dinars
77.
Saccheri quadrilateral
–
A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base. The first known consideration of the Saccheri quadrilateral was by Omar Khayyam in the late 11th century, for a Saccheri quadrilateral ABCD, the sides AD and BC are equal in length and perpendicular to the base AB. The top CD is the summit or upper base and the angles at C and D are called the summit angles, as it turns out, when the summit angles are right angles, the existence of this quadrilateral is equivalent to the statement expounded by Euclids fifth postulate. When the summit angles are acute, this leads to hyperbolic geometry, and when the summit angles are obtuse. Saccheri himself, however, thought that both the obtuse and acute cases could be shown to be contradictory and he did show that the obtuse case was contradictory, but failed to properly handle the acute case. Saccheri quadrilaterals were first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid. Let ABCD be a Saccheri quadrilateral having AB as base, CD as summit and CA, the following properties are valid in any Saccheri quadrilateral in hyperbolic geometry, The summit angles are equal and acute. The summit is longer than the base, two Saccheri quadrilaterals are congruent if, the base segments and summit angles are congruent the summit segments and summit angles are congruent. The line segment joining the midpoints of the sides is not perpendicular to either side, besides the 2 right angles, these quadrilaterals have acute summit angles. The tilings exhibit a symmetry, and include, Lambert quadrilateral Coxeter. George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag,1975
Saccheri quadrilateral
–
*3322 symmetry
78.
John Wallis
–
John Wallis was an English mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later and he is credited with introducing the symbol ∞ for infinity. He similarly used 1/∞ for an infinitesimal, Wallis was born in Ashford, Kent, the third of five children of Reverend John Wallis and Joanna Chapman. He was initially educated at a school in Ashford but moved to James Movats school in Tenterden in 1625 following an outbreak of plague, as it was intended that he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge. While there, he kept an act on the doctrine of the circulation of the blood and his interests, however, centred on mathematics. He received his Bachelor of Arts degree in 1637 and a Masters in 1640, from 1643 to 1649, he served as a nonvoting scribe at the Westminster Assembly. He was elected to a fellowship at Queens College, Cambridge in 1644, throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to Holbeach at Felsted School. He rendered them great practical assistance in deciphering Royalist dispatches, most ciphers were ad hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. Wallis realised that the latter were far more secure – even describing them as unbreakable and he was also concerned about the use of ciphers by foreign powers, refusing, for example, Gottfried Leibnizs request of 1697 to teach Hanoverian students about cryptography. Returning to London – he had been chaplain at St Gabriel Fenchurch in 1643 – Wallis joined the group of scientists that was later to evolve into the Royal Society. He was finally able to indulge his interests, mastering William Oughtreds Clavis Mathematicae in a few weeks in 1647. He soon began to write his own treatises, dealing with a range of topics. Wallis joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I, in spite of their opposition he was appointed in 1649 to the Savilian Chair of Geometry at Oxford University, where he lived until his death on 28 October 1703. In 1661, he was one of twelve Presbyterian representatives at the Savoy Conference, besides his mathematical works he wrote on theology, logic, English grammar and philosophy, and he was involved in devising a system for teaching deaf mutes. William Holder had earlier taught a man, Alexander Popham, to speak plainly and distinctly. Wallis later claimed credit for this, leading Holder to accuse Wallis of rifling his Neighbours, Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I he introduced the term continued fraction, Wallis rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity. In 1655, Wallis published a treatise on conic sections in which they were defined analytically and this was the earliest book in which these curves are considered and defined as curves of the second degree
John Wallis
–
John Wallis
John Wallis
–
Opera mathematica, 1699
79.
Coordinate system
–
The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space, depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems, another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis, for a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point
Coordinate system
–
The
spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in
Euclidean space: radial distance r, polar angle θ (
theta), and azimuthal angle φ (
phi). The symbol ρ (
rho) is often used instead of r.
80.
Equation
–
In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation, there are two kinds of equations, identity equations and conditional equations. An identity equation is true for all values of the variable, a conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation, each member will contain one or more terms. The equation, A x 2 + B x + C = y has two members, A x 2 + B x + C and y, the left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, an equation is analogous to a scale into which weights are placed. When equal weights of something are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality. In geometry, equations are used to describe geometric figures and this is the starting idea of algebraic geometry, an important area of mathematics. Algebra studies two main families of equations, polynomial equations and, among them the case of linear equations. Polynomial equations have the form P =0, where P is a polynomial, linear equations have the form ax + b =0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques, algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory and these equations are difficult in general, one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions. Differential equations are equations that involve one or more functions and their derivatives and they are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in such as physics, chemistry, biology. The = symbol, which appears in equation, was invented in 1557 by Robert Recorde. An equation is analogous to a scale, balance, or seesaw
Equation
–
A
strange attractor which arises when solving a certain
differential equation.
81.
Girard Desargues
–
Girard Desargues was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour, born in Lyon, Desargues came from a family devoted to service to the French crown. His father was a notary, an investigating commissioner of the Seneschals court in Lyon. Girard Desargues worked as an architect from 1645, prior to that, he had worked as a tutor and may have served as an engineer and technical consultant in the entourage of Richelieu. As an architect, Desargues planned several private and public buildings in Paris, as an engineer, he designed a system for raising water that he installed near Paris. It was based on the use of the at the time unrecognized principle of the epicycloidal wheel and his work was rediscovered and republished in 1864. A collection of his works was published in 1951, and the 1864 compilation remains in print, one notable work, often cited by others in mathematics, is Rough draft for an essay on the results of taking plane sections of a cone. Late in his life, Desargues published a paper with the title of DALG. The most common theory about what this stands for is Des Argues, Lyonnais, rené Taton Sur la naissance de Girard Desargues. Revue dhistoire des sciences et de leurs applications Tome 15 n°2, oConnor, John J. Robertson, Edmund F. Girard Desargues, MacTutor History of Mathematics archive, University of St Andrews. Richard Westfall, Gerard Desargues, The Galileo Project Gerard Desargues, Brouillon Project dune Atteinte aux Evenemens des Rencontres du Cone avec un Plan
Girard Desargues
–
Girard Desargues
82.
Riemann surface
–
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the plane, locally near every point they look like patches of the complex plane. For example, they can look like a sphere or a torus or several sheets glued together, the main point of Riemann surfaces is that holomorphic functions may be defined between them. Every Riemann surface is a real analytic manifold, but it contains more structure which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface if, so the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not. Geometrical facts about Riemann surfaces are as nice as possible, and they provide the intuition and motivation for generalizations to other curves. The Riemann–Roch theorem is an example of this influence. There are several equivalent definitions of a Riemann surface, a Riemann surface X is a complex manifold of complex dimension one. This means that X is a Hausdorff topological space endowed with an atlas, the map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the maps between two overlapping charts are required to be holomorphic. A Riemann surface is a manifold of dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the supplement Riemann signifies that X is endowed with an additional structure which allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same, choosing an equivalence class of metrics on X is the additional datum of the conformal structure. A complex structure gives rise to a structure by choosing the standard Euclidean metric given on the complex plane. Showing that a structure determines a complex structure is more difficult. The complex plane C is the most basic Riemann surface, the map f = z defines a chart for C, and is an atlas for C. The map g = z* also defines a chart on C and is an atlas for C, the charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. In fact, given a Riemann surface X and its atlas A, the conjugate atlas B = is never compatible with A, in an analogous fashion, every non-empty open subset of the complex plane can be viewed as a Riemann surface in a natural way
Riemann surface
–
Riemann surface for the function ƒ (z) = √ z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real part of √ z. For the imaginary part of √ z, rotate the plot 180° around the vertical axis.
83.
Complex analysis
–
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. As a differentiable function of a variable is equal to the sum of its Taylor series. Complex analysis is one of the branches in mathematics, with roots in the 19th century. Important mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has very popular through a new boost from complex dynamics. Another important application of analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is one in which the independent variable and the dependent variable are complex numbers. More precisely, a function is a function whose domain. In other words, the components of the f, u = u and v = v can be interpreted as real-valued functions of the two real variables, x and y. The basic concepts of complex analysis are often introduced by extending the elementary real functions into the complex domain, holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. In the context of analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C. Although superficially similar in form to the derivative of a real function, in particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach z 0 in the complex plane. Consequently, complex differentiability has much stronger consequences than usual differentiability, for instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not. For this reason, holomorphic functions are referred to as analytic functions. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions z ↦ ℜ, z ↦ | z |, an important property that characterizes holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy-Riemann conditions. If f, C → C, defined by f = f = u + i v, here, the differential operator ∂ / ∂ z ¯ is defined as. In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations u x = v y and u y = − v x, where the subscripts indicate partial differentiation
Complex analysis
–
Plot of the function f (x) = (x 2 − 1)(x − 2 − i) 2 / (x 2 + 2 + 2 i). The
hue represents the function
argument, while the
brightness represents the magnitude.
Complex analysis
–
The
Mandelbrot set, a
fractal.
84.
Classical mechanics
–
In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
Classical mechanics
–
Sir
Isaac Newton (1643–1727), an influential figure in the history of physics and whose
three laws of motion form the basis of classical mechanics
Classical mechanics
–
Diagram of orbital motion of a satellite around the earth, showing perpendicular velocity and acceleration (force) vectors.
Classical mechanics
–
Hamilton 's greatest contribution is perhaps the reformulation of
Newtonian mechanics, now called
Hamiltonian mechanics.
85.
Non-Euclidean geometries
–
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
Non-Euclidean geometries
–
On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Non-Euclidean geometries
–
Behavior of lines with a common perpendicular in each of the three types of geometry
86.
Incidence geometry
–
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is an object that involves concepts such as length, angles, continuity, betweenness. An incidence structure is what is obtained when all other concepts are removed, even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry, Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently there are different terminologies to describe these objects, in graph theory they are called hypergraphs, and in combinatorial design theory they are called block designs. Besides the difference in terminology, each area approaches the subject differently and is interested in questions about these objects relevant to that discipline, using geometric language, as is done in incidence geometry, shapes the topics and examples that are normally presented. In the examples selected for this article we use only those with a natural geometric flavor, a special case that has generated much interest deals with finite sets of points in the Euclidean plane and what can be said about the number and types of lines they determine. Some results of this situation can extend to more general settings since only incidence properties are considered, if is a flag, we say that A is incident with l or that l is incident with A, and write A I l. Intuitively, a point and line are in this relation if, given a point B and a line m which do not form a flag, that is, the point is not on the line, the pair is called an anti-flag. There is no concept of distance in an incidence structure. However, a combinatorial metric does exist in the incidence graph. Another way to define a distance again uses a graph-theoretic notion in a related structure, the vertices of the collinearity graph are the points of the incidence structure and two points are joined if there exists a line incident with both points. The distance between two points of the structure can then be defined as their distance in the collinearity graph. When distance is considered in a structure, it is necessary to mention how it is being defined. Incidence structures that are most studied are those that satisfy some additional properties, such as planes, affine planes, generalized polygons, partial geometries. Every line contains at least two distinct points, in a partial linear space it is also true that every pair of distinct lines meet in at most one point. This statement does not have to be assumed as it is readily proved from axiom one above, further constraints are provided by the regularity conditions, RLk, Each line is incident with the same number of points. If finite this number is denoted by k
Incidence geometry
–
Simplest non-trivial linear space
Incidence geometry
–
Projective plane of order 2 the Fano plane
87.
Geodesic
–
In differential geometry, a geodesic is a generalization of the notion of a straight line to curved spaces. The term has been generalized to include measurements in more general mathematical spaces, for example, in graph theory. In the presence of a connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. If this connection is the Levi-Civita connection induced by a Riemannian metric, geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles, the shortest path between two points in a curved space can be found by writing the equation for the length of a curve, and then minimizing this length using the calculus of variations. This has some technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. Equivalently, a different quantity may be defined, termed the energy of the curve, intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy. The resulting shape of the band is a geodesic, in Riemannian geometry geodesics are not the same as shortest curves between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points, and are parameterized with constant velocity, going the long way round on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t → t2 from the interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant. Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry, in general relativity, geodesics describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a rock, an orbiting satellite. More generally, the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free and this article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds. The article geodesic discusses the case of general relativity in greater detail. The most familiar examples are the lines in Euclidean geometry. On a sphere, the images of geodesics are the great circles, the shortest path from point A to point B on a sphere is given by the shorter arc of the great circle passing through A and B. If A and B are antipodal points, then there are infinitely many shortest paths between them, geodesics on an ellipsoid behave in a more complicated way than on a sphere, in particular, they are not closed in general. In metric geometry, a geodesic is a curve which is locally a distance minimizer
Geodesic
–
A geodesic triangle on the sphere. The geodesics are
great circle arcs.
88.
Manifold
–
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of a manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights, two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not, for example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane. When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured, symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold, Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 =1. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable
Manifold
–
The surface of the Earth requires (at least) two charts to include every point. Here the
globe is decomposed into charts around the
North and
South Poles.
Manifold
–
The
real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as
Boy's surface.
89.
Unit circle
–
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π
Unit circle
–
Illustration of a unit circle. The variable t is an
angle measure.
90.
Trigonometry
–
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
Trigonometry
–
Hipparchus, credited with compiling the first
trigonometric table, is known as "the father of trigonometry".
Trigonometry
–
All of the
trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
Trigonometry
–
Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a
marine chronometer, the position of the ship can be determined from such measurements.
91.
Algebraic varieties
–
Algebraic varieties are the central objects of study in algebraic geometry. Classically, a variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. For example, some definitions provide that algebraic variety is irreducible, under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility, the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that a variety may have singular points. Generalizing this result, Hilberts Nullstellensatz provides a correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry, an affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way, the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, let k be an algebraically closed field and let An be an affine n-space over k. The polynomials f in the ring k can be viewed as k-valued functions on An by evaluating f at the points in An, i. e. by choosing values in k for each xi. For each set S of polynomials in k, define the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. This topology is called the Zariski topology.2 Given a subset V of An, let f in k be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in Pn in homogeneous coordinates, however, because f is homogeneous, f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish, Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the ring of V is the quotient of the polynomial ring by this ideal.10 A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every variety is quasi-projective. In classical algebraic geometry, all varieties were by definition quasiprojective varieties and it might not have an embedding into projective space
Algebraic varieties
–
The
twisted cubic is a projective algebraic variety.
92.
Neighborhood (topology)
–
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. It is closely related to the concepts of set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount away from that point without leaving the set. If X is a space and p is a point in X. This is also equivalent to p ∈ X being in the interior of V, note that the neighbourhood V need not be an open set itself. If V is open it is called an open neighbourhood, some mathematicians require that neighbourhoods be open, so it is important to note conventions. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of sets containing each of its points. The collection of all neighbourhoods of a point is called the system at the point. If S is a subset of topological space X then a neighbourhood of S is a set V that includes an open set U containing S, the neighbourhood of a point is just a special case of this definition. In a metric space M =, a set V is a neighbourhood of a point p if there exists an open ball with centre p and radius r >0, such that B r = B = is contained in V. V is called uniform neighbourhood of a set S if there exists a number r such that for all elements p of S, B r = is contained in V. For r >0 the r -neighbourhood S r of a set S is the set of all points in X that are at less than r from S, S r = ⋃ p ∈ S B r. It directly follows that an r -neighbourhood is a neighbourhood. The above definition is useful if the notion of open set is already defined, there is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points. In a uniform space S =, V is called a neighbourhood of P if P is not close to X ∖ V. A deleted neighbourhood of a point p is a neighbourhood of p, for instance, the interval = is a neighbourhood of p =0 in the real line, so the set ∪ = ∖ is a deleted neighbourhood of 0. Note that a neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definition of the limit of a function, bredon, Glen E. Topology and geometry
Neighborhood (topology)
–
A set in the
plane is a neighbourhood of a point if a small disk around is contained in.
93.
Diffeomorphism
–
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is a function that maps one differentiable manifold to another such that both the function and its inverse are smooth. Given two manifolds M and N, a map f, M → N is called a diffeomorphism if it is a bijection and its inverse f−1. If these functions are r times continuously differentiable, f is called a Cr-diffeomorphism, two manifolds M and N are diffeomorphic if there is a diffeomorphism f from M to N. They are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable, F is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth. First remark It is essential for V to be connected for the function f to be globally invertible. g. Second remark Since the differential at a point D f x, T x U → T f V is a map, it has a well-defined inverse if. The matrix representation of Dfx is the n × n matrix of partial derivatives whose entry in the i-th row. This so-called Jacobian matrix is used for explicit computations. Third remark Diffeomorphisms are necessarily between manifolds of the same dimension, imagine f going from dimension n to dimension k. If n < k then Dfx could never be surjective, in both cases, therefore, Dfx fails to be a bijection. Fourth remark If Dfx is a bijection at x then f is said to be a local diffeomorphism. Fifth remark Given a smooth map from dimension n to k, if Df is surjective, f is said to be a submersion. Sixth remark A differentiable bijection is not necessarily a diffeomorphism, F = x3, for example, is not a diffeomorphism from R to itself because its derivative vanishes at 0. This is an example of a homeomorphism that is not a diffeomorphism, seventh remark When f is a map between differentiable manifolds, a diffeomorphic f is a stronger condition than a homeomorphic f. For a diffeomorphism, f and its inverse need to be differentiable, for a homeomorphism, f, every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. F, M → N is called a diffeomorphism if, in coordinate charts, more precisely, Pick any cover of M by compatible coordinate charts and do the same for N. Let φ and ψ be charts on, respectively, M and N, with U and V as, respectively, the map ψfφ−1, U → V is then a diffeomorphism as in the definition above, whenever f ⊂ ψ−1
Diffeomorphism
–
Algebraic structure → Group theory
Group theory
94.
Homeomorphism
–
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic, a self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form a relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes, the open interval is homeomorphic to the real numbers R for any a < b. The unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
Homeomorphism
–
A
trefoil knot is homeomorphic to a circle, but not
isotopic. Continuous mappings are not always realizable as deformations. Here the knot has been thickened to make the image understandable.
Homeomorphism
–
A continuous deformation between a coffee
mug and a donut (
torus) illustrating that they are homeomorphic. But there need not be a
continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse.
95.
Differentiable manifold
–
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas, one may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart, in formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. In other words, where the domains of overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the charts to one another are called transition maps. Differentiability means different things in different contexts including, continuously differentiable, k times differentiable, smooth, furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for theories such as classical mechanics, general relativity. It is possible to develop a calculus for differentiable manifolds and this leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry, the emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen and these ideas found a key application in Einsteins theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces, the widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space. This formalizes the notion of patching together pieces of a space to make a manifold – the manifold produced also contains the data of how it has been patched together, However, different atlases may produce the same manifold, a manifold does not come with a preferred atlas. And, thus, one defines a manifold to be a space as above with an equivalence class of atlases. There are a number of different types of manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following, a differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable
Differentiable manifold
–
A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts the
Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.
96.
Metric space
–
In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set, a metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space, in fact, a metric is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the line segment connecting them. Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, since for any x, y ∈ M, The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for a space if it is clear from the context what metric is used. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations, to be a metric there shouldnt be any one-way roads. The triangle inequality expresses the fact that detours arent shortcuts, many of the examples below can be seen as concrete versions of this general idea. The real numbers with the function d = | y − x | given by the absolute difference. The rational numbers with the distance function also form a metric space. The positive real numbers with distance function d = | log | is a metric space. Any normed vector space is a space by defining d = ∥ y − x ∥. Examples, The Manhattan norm gives rise to the Manhattan distance, the maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from x to y. The British Rail metric on a vector space is given by d = ∥ x ∥ + ∥ y ∥ for distinct points x and y. The name alludes to the tendency of railway journeys to proceed via London irrespective of their final destination, If is a metric space and X is a subset of M, then becomes a metric space by restricting the domain of d to X × X. The discrete metric, where d =0 if x = y and d =1 otherwise, is a simple but important example and this, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, any point is a ball, and therefore every subset is open. A finite metric space is a metric space having a number of points
Metric space
–
Diameter of a set.
97.
Hyperbolic metric
–
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, in these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman. A hyperbolic n -manifold is a complete Riemannian n-manifold of constant sectional curvature -1, every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space H n. As a result, the cover of any closed manifold M of constant negative curvature −1 is H n. Thus, every such M can be written as H n / Γ where Γ is a discrete group of isometries on H n. That is, Γ is a subgroup of S O1, n + R. The manifold has finite volume if and only if Γ is a lattice and its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean n-1-manifold and the closed half-ray. The manifold is of finite volume if and only if its part is compact
Hyperbolic metric
–
The
Pseudosphere. Each half of this shape is a hyperbolic 2-manifold (i.e. surface) with boundary.
Hyperbolic metric
–
A perspective projection of a
dodecahedral tessellation in
H^{3}. This is an example of what an observer might see inside a hyperbolic 3-manifold.
98.
Special relativity
–
In physics, special relativity is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einsteins original pedagogical treatment, it is based on two postulates, The laws of physics are invariant in all inertial systems, the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source. It was originally proposed in 1905 by Albert Einstein in the paper On the Electrodynamics of Moving Bodies, as of today, special relativity is the most accurate model of motion at any speed. Even so, the Newtonian mechanics model is useful as an approximation at small velocities relative to the speed of light. Not until Einstein developed general relativity, to incorporate general frames of reference, a translation that has often been used is restricted relativity, special really means special case. It has replaced the notion of an absolute universal time with the notion of a time that is dependent on reference frame. Rather than an invariant time interval between two events, there is an invariant spacetime interval, a defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other, rather space and time are interwoven into a single continuum known as spacetime. Events that occur at the time for one observer can occur at different times for another. The theory is special in that it applies in the special case where the curvature of spacetime due to gravity is negligible. In order to include gravity, Einstein formulated general relativity in 1915, Special relativity, contrary to some outdated descriptions, is capable of handling accelerations as well as accelerated frames of reference. e. At a sufficiently small scale and in conditions of free fall, a locally Lorentz-invariant frame that abides by special relativity can be defined at sufficiently small scales, even in curved spacetime. Galileo Galilei had already postulated that there is no absolute and well-defined state of rest, Einstein extended this principle so that it accounted for the constant speed of light, a phenomenon that had been recently observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light and the independence of physical laws from the choice of inertial system, the Principle of Invariant Light Speed –. Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. That is, light in vacuum propagates with the c in at least one system of inertial coordinates. Following Einsteins original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations, however, the most common set of postulates remains those employed by Einstein in his original paper
Special relativity
–
Albert Einstein around 1905, the year his "
Annus Mirabilis papers " – which included Zur Elektrodynamik bewegter Körper, the paper founding special relativity – were published.
99.
Ruler
–
A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing, engineering and building to measure distances or to rule straight lines. The ruler is a straightedge which may also contain calibrated lines to measure distances, rulers have long been made from different materials and in a wide range of sizes. Plastics have also used since they were invented, they can be molded with length markings instead of being scribed. Metal is used for more durable rulers for use in the workshop,12 inches or 30 cm in length is useful for a ruler to be kept on a desk to help in drawing. Shorter rulers are convenient for keeping in a pocket, longer rulers, e. g.18 inches are necessary in some cases. Rigid wooden or plastic yardsticks,1 yard long, and meter sticks,1 meter long, are also used, classically, long measuring rods were used for larger projects, now superseded by tape measure or laser rangefinders. Desk rulers are used for three purposes, to measure, to aid in drawing straight lines and as a straight guide for cutting and scoring with a blade. Practical rulers have distance markings along their edges, a line gauge is a type of ruler used in the printing industry. These may be made from a variety of materials, typically metal or clear plastic, units of measurement on a basic line gauge usually include inches, agate, picas, and points. More detailed line gauges may contain sample widths of lines, samples of common type in several point sizes, measuring instruments similar in function to rulers are made portable by folding or retracting into a coil when not in use. When extended for use, they are straight, like a ruler, the illustrations on this page show a 2-meter carpenters rule, which folds down to a length of 24 cm to easily fit in a pocket, and a 5-meter-long tape, which retracts into a small housing. A flexible length measuring instrument which is not necessarily straight in use is the tailors fabric tape measure and it is used to measure around a solid body, e. g. a persons waist measurement, as well as linear measurement, e. g. inside leg. It is rolled up when not in use, taking up little space, a contraction rule is made having larger divisions than standard measures to allow for shrinkage of a metal casting. They may also be known as a shrinkage or shrink rule, a ruler software program can be used to measure pixels on a computer screen or mobile phone. These programs are known as screen rulers. In geometry, a ruler without any marks on it may be used only for drawing lines between points. A straightedge is used to help draw accurate graphs and tables. A ruler and compass construction refers to using an unmarked ruler
Ruler
–
A variety of rulers
Ruler
–
A closeup of a steel rule
Ruler
–
Gilded Bronze Ruler - 1
chi = 23.1 cm. Western Han (206 BCE - CE 8).
Hanzhong City, China
Ruler
–
Bronze ruler. Han dynasty, 206 BCE to CE 220. Excavated in
Zichang County, China
100.
Koch snowflake
–
The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled On a continuous curve without tangents, the progression for the area of the snowflake converges to 8/5 times the area of the original triangle, while the progression for the snowflakes perimeter diverges to infinity. Consequently, the snowflake has an area bounded by an infinitely long line. The Koch snowflake can be constructed by starting with a triangle, then recursively altering each line segment as follows. Draw an equilateral triangle that has the middle segment from step 1 as its base, remove the line segment that is the base of the triangle from step 2. After one iteration of this process, the shape is the outline of a hexagram. The Koch snowflake is the limit approached as the steps are followed over and over again. The Koch curve originally described by Helge von Koch is constructed with one of the three sides of the original triangle. In other words, three Koch curves make a Koch snowflake, the Koch curve has an infinite length because the total length of the curve increases by one third with each iteration. Each iteration creates four times as many segments as in the previous iteration. Hence the length of the curve after n iterations will be n times the original triangle perimeter, which is unbounded as n tends to infinity. As the number of iterations tends to infinity, the limit of the perimeter is, lim n → ∞ P n = lim n → ∞3 ⋅ s ⋅ n = ∞, a ln 4/ln 3-dimensional measure exists, but has not been calculated so far. Only upper and lower bounds have been invented, collapsing the geometric sum gives, A n = a 0 = a 05. The limit of the area is, lim n → ∞ A n = lim n → ∞ a 05 ⋅ =85 ⋅ a 0, so the area of the Koch snowflake is 8/5 of the area of the original triangle. Expressed in terms of the length s of the original triangle this is 2 s 235. The Koch snowflake is self-replicating with six copies around a central point, hence it is an irreptile which is irrep-7. The fractal dimension of the Koch curve is ln 4/ln 3 ≈1.26186 and this is greater than the dimension of a line but less than Peanos space-filling curve. The Koch curve is continuous everywhere but differentiable nowhere and it is possible to tessellate the plane by copies of Koch snowflakes in two different sizes
Koch snowflake
–
Closeup of Haines sphereflake
Koch snowflake
–
The first four
iterations of the Koch snowflake
Koch snowflake
–
Koch curve in 3D
101.
Hilbert space
–
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis —and ergodic theory, john von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis, geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist
Hilbert space
–
David Hilbert
Hilbert space
–
The state of a
vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct
overtones is given by the projection of the point onto the coordinate axes in the space.
102.
General topology
–
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, another name for general topology is point-set topology. The fundamental concepts in topology are continuity, compactness, and connectedness, Continuous functions, intuitively. Compact sets are those that can be covered by finitely many sets of small size. Connected sets are sets that cannot be divided into two pieces that are far apart, the words nearby, arbitrarily small, and far apart can all be made precise by using open sets. If we change the definition of open set, we change what continuous functions, compact sets, each choice of definition for open set is called a topology. A set with a topology is called a topological space, metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces, General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, let X be a set and let τ be a family of subsets of X. The notation Xτ may be used to denote a set X endowed with the particular topology τ, the members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ, a subset of X may be open, closed, both, or neither. The empty set and X itself are both closed and open. A base B for a topological space X with topology T is a collection of sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T, every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of spaces, the product can be given the product topology. For example, in products, a basis for the product topology consists of all products of open sets. For infinite products, there is the requirement that in a basic open set, all. In other words, the quotient topology is the finest topology on Y for which f is continuous, a common example of a quotient topology is when an equivalence relation is defined on the topological space X
General topology
–
This subspace of R ² is path-connected, because a path can be drawn between any two points in the space.
103.
Invariance of domain
–
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states, If U is a subset of Rn and f, U → Rn is an injective continuous map, then V = f is open. The theorem and its proof are due to L. E. J. Brouwer, the proof uses tools of algebraic topology, notably the Brouwer fixed point theorem. The conclusion of the theorem can equivalently be formulated as, f is an open map, furthermore, the theorem says that if two subsets U and V of Rn are homeomorphic, and U is open, then V must be open as well. Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space and it is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f, → R2 with f = and this map is injective and continuous, the domain is an open subset of R, but the image is not open in R2. A more extreme example is g, → R2 with g = because here g is injective and continuous, the theorem is also not generally true in infinite dimensions. Consider for instance the Banach space l∞ of all bounded real sequences, define f, l∞ → l∞ as the shift f =. Then f is injective and continuous, the domain is open in l∞, an important consequence of the domain invariance theorem is that Rn cannot be homeomorphic to Rm if m ≠ n. Indeed, no non-empty open subset of Rn can be homeomorphic to any subset of Rm in this case. There are also generalizations to certain types of maps from a Banach space to itself. Open mapping theorem for other conditions that ensure that a continuous map is open. Mill, J. van, Domain invariance, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Invariance of domain
–
A map which is not a homeomorphism onto its image: g: (−1.1, 1) → R 2 with g (t) = (t 2 − 1, t 3 − t)
104.
Geometric topology
–
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. This was the origin of simple homotopy theory, manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in terms, embeddings in codimension 3. Low-dimensional topology is concerned with questions in dimensions up to 4, thus the topological classification of 4-manifolds is in principle easy, and the key questions are, does a topological manifold admit a differentiable structure, and if so, how many. Notably, the case of dimension 4 is the last open case of the generalized Poincaré conjecture. The distinction is because surgery theory works in dimension 5 and above, in dimension 4 and below, surgery theory does not work, and other phenomena occur. Indeed, one approach to discussing low-dimensional manifolds is to ask what would surgery theory predict to be true, were it to work, – and then understand low-dimensional phenomena as deviations from this. The precise reason for the difference at dimension 5 is because the Whitney embedding theorem, in surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2, the Whitney trick works. The key consequence of this is Smales h-cobordism theorem, which works in dimension 5 and above, the limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4. A manifold is orientable if it has a consistent choice of orientation, in this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Thus an i-handle is the analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory, the modification of handle structures is closely linked to Cerf theory. Local flatness is a property of a submanifold in a manifold of larger dimension. In the category of manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Suppose a d dimensional manifold N is embedded into an n dimensional manifold M and that is, there exists a homeomorphism U → R n such that the image of U ∩ N coincides with R d. Brown and Mazur received the Veblen Prize for their independent proofs of this theorem, low-dimensional topology includes, Surface s 3-manifolds 4-manifolds each have their own theory, where there are some connections. Knot theory is the study of mathematical knots, while inspired by knots which appear in daily life in shoelaces and rope, a mathematicians knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, to gain further insight, mathematicians have generalized the knot concept in several ways
Geometric topology
–
A
Seifert surface bounded by a set of
Borromean rings. Seifert surfaces for
links are a useful tool in geometric topology.
105.
William Kingdon Clifford
–
William Kingdon Clifford FRS was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, the operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular, have been of increasing importance to mathematical physics, geometry. Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry, in his philosophical writings he coined the expression mind-stuff. Born at Exeter, William Clifford showed great promise at school and he went on to Kings College London and Trinity College, Cambridge, where he was elected fellow in 1868, after being second wrangler in 1867 and second Smiths prizeman. Being second was a fate he shared with others who became famous mathematicians, including William Thomson, in 1870, he was part of an expedition to Italy to observe the solar eclipse of December 22,1870. During that voyage he survived a shipwreck along the Sicilian coast, in 1871, he was appointed professor of mathematics and mechanics at University College London, and in 1874 became a fellow of the Royal Society. He was also a member of the London Mathematical Society and the Metaphysical Society, on 7 April 1875 Clifford married Lucy Lane. In 1876, Clifford suffered a breakdown, probably brought on by overwork and he taught and administered by day, and wrote by night. A half-year holiday in Algeria and Spain allowed him to resume his duties for 18 months and he went to the island of Madeira to recover, but died there of tuberculosis after a few months, leaving a widow with two children. Born eleven days later, Albert Einstein would go on to develop the theory of gravity that Clifford had suggested nine years earlier. Clifford enjoyed entertaining children and wrote a collection of fairy stories, Clifford and his wife are buried in Londons Highgate Cemetery just north of the grave of Karl Marx, and near the graves of George Eliot and Herbert Spencer. Clifford was above all and before all a geometer, the discovery of non-Euclidean geometry opened new possibilities in geometry in Cliffords era. The field of differential geometry was born, with the concept of curvature broadly applied to space itself as well as to curved lines and surfaces. Clifford was very impressed by Bernhard Riemann’s 1854 essay On the hypotheses which lie at the bases of geometry. In 1870 he reported to the Cambridge Philosophical Society on the curved space concepts of Riemann, Cliffords translation of Riemanns paper was published in Nature in 1873. His report at Cambridge, On the Space-Theory of Matter, was published in 1876, Clifford elaborated elliptic space geometry as a non-Euclidean metric space. Equidistant curves in space are now said to be Clifford parallels
William Kingdon Clifford
–
William Kingdon Clifford (1845–1879)
William Kingdon Clifford
–
Clifford by
John Collier
William Kingdon Clifford
–
William Kingdon Clifford
William Kingdon Clifford
–
Marker for W. K. Clifford and his wife in Highgate Cemetery (c. 1986)
106.
Sophus Lie
–
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of symmetry and applied it to the study of geometry. His first mathematical work, Repräsentation der Imaginären der Plangeometrie, was published, in 1869, by the Academy of Sciences in Christiania and that same year he received a scholarship and traveled to Berlin, where he stayed from September to February 1870. There, he met Felix Klein and they became close friends, when he left Berlin, Lie traveled to Paris, where he was joined by Klein two months later. There, they met Camille Jordan and Gaston Darboux, but on 19 July 1870 the Franco-Prussian War began and Klein had to leave France very quickly. Lie left for Fontainebleau where after a while he was arrested under suspicion of being a German spy and he was released from prison after a month, thanks to the intervention of Darboux. Lie obtained his PhD at the University of Christiania in 1871 with a thesis entitled On a class of geometric transformations and it would be described by Darboux as “one of the most handsome discoveries of modern Geometry”. The next year, the Norwegian Parliament established a professorship for him. That same year, Lie visited Klein, who was then at Erlangen, at the end of 1872, Sophus Lie proposed to Anna Birch, then eighteen years old, and they were married in 1874. The couple had three children, Marie, Dagny and Herman, in 1884, Friedrich Engel arrived at Christiania to help him, with the support of Klein and Adolph Mayer. Engel would help Lie to write his most important treatise, Theorie der Transformationsgruppen, decades later, Engel would also be one of the two editors of Lies collected works. In 1886 Lie became professor at Leipzig, replacing Klein, who had moved to Göttingen, in November 1889, Lie suffered a mental breakdown and had to be hospitalized until June 1890. After that, he returned to his post, but over the years his anaemia progressed to the point where he decided to return to his homeland, consequently, in 1898 he tendered his resignation in May, and left for home in September the same year. He died the year,1899. Sophus Lie died at the age of 56, due to pernicious anemia, the generators are subject to a linearized version of the group law, now called the commutator bracket, and have the structure of what is today called a Lie algebra. Hermann Weyl used Lies work on theory in his papers from 1922 and 1923. Lie, Sophus, Theorie der Transformationsgruppen I, Leipzig, B. G. Teubner, written with the help of Friedrich Engel. Lie, Sophus, Theorie der Transformationsgruppen II, Leipzig, B. G. Teubner, written with the help of Friedrich Engel
Sophus Lie
–
Sophus Lie
107.
Geometric group theory
–
Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory, currently combinatorial group theory as an area is largely subsumed by geometric group theory. Other precursors of geometric group theory include small cancellation theory and Bass–Serre theory, small cancellation theory was introduced by Martin Grindlinger in the 1960s and further developed by Roger Lyndon and Paul Schupp. It studies van Kampen diagrams, corresponding to finite group presentations, via combinatorial curvature conditions, Bass–Serre theory, introduced in the 1977 book of Serre, derives structural algebraic information about groups by studying group actions on simplicial trees. The emergence of geometric group theory as an area of mathematics is usually traced to the late 1980s. The work of Gromov had an effect on the study of discrete groups. Notable themes and developments in geometric group theory in 1990s and 2000s include, a particularly influential broad theme in the area is Gromovs program of classifying finitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry and this program involves, The study of properties that are invariant under quasi-isometry. Theorems which use quasi-isometry invariants to prove results about groups, for example, Gromovs polynomial growth theorem, Stallings ends theorem. Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some group or metric space. This direction was initiated by the work of Schwartz on quasi-isometric rigidity of rank-one lattices, the theory of word-hyperbolic and relatively hyperbolic groups. A particularly important development here is the work of Sela in 1990s resulting in the solution of the problem for word-hyperbolic groups. The notion of a relatively hyperbolic groups was introduced by Gromov in 1987 and refined by Farb and Bowditch. The study of hyperbolic groups gained prominence in the 2000s. Interactions with mathematical logic and the study of theory of free groups. Particularly important progress occurred on the famous Tarski conjectures, due to the work of Sela as well as of Kharlampovich, the study of limit groups and introduction of the language and machinery of non-commutative algebraic geometry gained prominence. Interactions with computer science, complexity theory and the theory of formal languages, the study of isoperimetric inequalities, Dehn functions and their generalizations for finitely presented group
Geometric group theory
–
The
Cayley graph of a
free group with two generators. This is a
hyperbolic group whose
Gromov boundary is a
Cantor set. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs.
108.
Theorem
–
In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. In some cases, a picture alone may be sufficient to prove a theorem, because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being trivial, or difficult, or deep and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
Theorem
–
A
planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The
four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
109.
Vector space
–
A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars
Vector space
–
Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2 w.
110.
Euclidean space
–
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
Euclidean space
–
A
sphere, the most perfect spatial shape according to
Pythagoreans, also is an important concept in modern understanding of Euclidean spaces
111.
Riemann
–
Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, through his pioneering contributions to differential geometry, Bernhard Riemann laid the foundations of the mathematics of general relativity. Riemann was born on September 17,1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of Germany today and his father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood, Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional skills, such as calculation abilities, from an early age but suffered from timidity. During 1840, Riemann went to Hanover to live with his grandmother, after the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations. In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his familys finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, however, once there, he began studying mathematics under Carl Friedrich Gauss. Gauss recommended that Riemann give up his work and enter the mathematical field, after getting his fathers approval. During his time of study, Jacobi, Lejeune Dirichlet, Steiner and he stayed in Berlin for two years and returned to Göttingen in 1849. Riemann held his first lectures in 1854, which founded the field of Riemannian geometry, in 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary, in 1859, following Lejeune Dirichlets death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than three or four in order to describe physical reality. In 1862 he married Elise Koch and had a daughter, Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his journey to Italy in Selasca where he was buried in the cemetery in Biganzolo
Riemann
–
Bernhard Riemann in 1863.
Riemann
–
Riemann's tombstone in Biganzolo
112.
Einstein
–
Albert Einstein was a German-born theoretical physicist. He developed the theory of relativity, one of the two pillars of modern physics, Einsteins work is also known for its influence on the philosophy of science. Einstein is best known in popular culture for his mass–energy equivalence formula E = mc2, near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field. This led him to develop his theory of relativity during his time at the Swiss Patent Office in Bern. Briefly before, he aquired the Swiss citizenship in 1901, which he kept for his whole life and he continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of particle theory and the motion of molecules. He also investigated the properties of light which laid the foundation of the photon theory of light. In 1917, Einstein applied the theory of relativity to model the large-scale structure of the universe. He was visiting the United States when Adolf Hitler came to power in 1933 and, being Jewish, did not go back to Germany and he settled in the United States, becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project, Einstein supported defending the Allied forces, but generally denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, Einstein signed the Russell–Einstein Manifesto, Einstein was affiliated with the Institute for Advanced Study in Princeton, New Jersey, until his death in 1955. Einstein published more than 300 scientific papers along with over 150 non-scientific works, on 5 December 2014, universities and archives announced the release of Einsteins papers, comprising more than 30,000 unique documents. Einsteins intellectual achievements and originality have made the word Einstein synonymous with genius, Albert Einstein was born in Ulm, in the Kingdom of Württemberg in the German Empire, on 14 March 1879. His parents were Hermann Einstein, a salesman and engineer, the Einsteins were non-observant Ashkenazi Jews, and Albert attended a Catholic elementary school in Munich from the age of 5 for three years. At the age of 8, he was transferred to the Luitpold Gymnasium, the loss forced the sale of the Munich factory. In search of business, the Einstein family moved to Italy, first to Milan, when the family moved to Pavia, Einstein stayed in Munich to finish his studies at the Luitpold Gymnasium. His father intended for him to electrical engineering, but Einstein clashed with authorities and resented the schools regimen. He later wrote that the spirit of learning and creative thought was lost in strict rote learning, at the end of December 1894, he travelled to Italy to join his family in Pavia, convincing the school to let him go by using a doctors note. During his time in Italy he wrote an essay with the title On the Investigation of the State of the Ether in a Magnetic Field
Einstein
–
Albert Einstein in 1921
Einstein
–
Einstein at the age of 3 in 1882
Einstein
–
Albert Einstein in 1893 (age 14)
Einstein
–
Einstein's matriculation certificate at the age of 17, showing his final grades from the Argovian cantonal school (Aargauische Kantonsschule, on a scale of 1–6, with 6 being the highest possible mark)
113.
General relativity theory
–
General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
General relativity theory
–
A simulated
black hole of 10
solar masses within the
Milky Way, seen from a distance of 600 kilometers.
General relativity theory
–
Albert Einstein developed the theories of special and general relativity. Picture from 1921.
General relativity theory
–
Einstein cross: four images of the same astronomical object, produced by a
gravitational lens
General relativity theory
–
Artist's impression of the space-borne gravitational wave detector
LISA
114.
Computer graphics
–
Computer graphics are pictures and films created using computers. Usually, the term refers to computer-generated image data created with help from specialized hardware and software. It is a vast and recent area in computer science, the phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes referred to as CGI. The overall methodology depends heavily on the sciences of geometry, optics. Computer graphics is responsible for displaying art and image data effectively and meaningfully to the user and it is also used for processing image data received from the physical world. Computer graphic development has had a significant impact on many types of media and has revolutionized animation, movies, advertising, video games, the term computer graphics has been used a broad sense to describe almost everything on computers that is not text or sound. Such imagery is found in and on television, newspapers, weather reports, a well-constructed graph can present complex statistics in a form that is easier to understand and interpret. In the media such graphs are used to illustrate papers, reports, thesis, many tools have been developed to visualize data. Computer generated imagery can be categorized into different types, two dimensional, three dimensional, and animated graphics. As technology has improved, 3D computer graphics have become more common, Computer graphics has emerged as a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. Screens could display art since the Lumiere brothers use of mattes to create effects for the earliest films dating from 1895. New kinds of displays were needed to process the wealth of information resulting from such projects, early projects like the Whirlwind and SAGE Projects introduced the CRT as a viable display and interaction interface and introduced the light pen as an input device. Douglas T. Ross of the Whirlwind SAGE system performed an experiment in 1954 in which a small program he wrote captured the movement of his finger. Electronics pioneer Hewlett-Packard went public in 1957 after incorporating the decade prior, and established ties with Stanford University through its founders. This began the transformation of the southern San Francisco Bay Area into the worlds leading computer technology hub - now known as Silicon Valley. The field of computer graphics developed with the emergence of computer graphics hardware, further advances in computing led to greater advancements in interactive computer graphics. In 1959, the TX-2 computer was developed at MITs Lincoln Laboratory, the TX-2 integrated a number of new man-machine interfaces
Computer graphics
–
A
Blender 2.45 screenshot, displaying the
3D test model Suzanne.
Computer graphics
–
Spacewar! running on the
Computer History Museum 's
PDP-1
Computer graphics
–
Dire Straits '
1985 music video for their hit song
Money For Nothing - the "I Want My
MTV " song – became known as an early example of fully three-dimensional,
animated computer-generated imagery.
Computer graphics
–
Quarxs, series poster,
Maurice Benayoun,
François Schuiten, 1992
115.
Coxeter group
–
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators
Coxeter group
116.
Discrete group
–
For example, the integers, Z, form a discrete subgroup of the reals, R, but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology, any group can be given the discrete topology. Since every map from a space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups. Hence, there is an isomorphism between the category of groups and the category of discrete groups, discrete groups can therefore be identified with their underlying groups. There are some occasions when a group or Lie group is usefully endowed with the discrete topology. This happens for example in the theory of the Bohr compactification, a discrete isometry group is an isometry group such that for every point of the metric space the set of images of the point under the isometries is a discrete set. A discrete symmetry group is a group that is a discrete isometry group. Since topological groups are homogeneous, one need look at a single point to determine if the topological group is discrete. In particular, a group is discrete if and only if the singleton containing the identity is an open set. A discrete group is the thing as a zero-dimensional Lie group. The identity component of a group is just the trivial subgroup while the group of components is isomorphic to the group itself. Since the only Hausdorff topology on a set is the discrete one. It follows that every subgroup of a Hausdorff group is discrete. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian, other properties, every discrete group is totally disconnected every subgroup of a discrete group is discrete. Every quotient of a group is discrete. The product of a number of discrete groups is discrete. A discrete group is compact if and only if it is finite, every discrete group is locally compact. Every discrete subgroup of a Hausdorff group is closed, every discrete subgroup of a compact Hausdorff group is finite
Discrete group
–
Algebraic structure → Group theory
Group theory
117.
Universe
–
The Universe is all of time and space and its contents. It includes planets, moons, minor planets, stars, galaxies, the contents of intergalactic space, the size of the entire Universe is unknown. The earliest scientific models of the Universe were developed by ancient Greek and Indian philosophers and were geocentric, over the centuries, more precise astronomical observations led Nicolaus Copernicus to develop the heliocentric model with the Sun at the center of the Solar System. In developing the law of gravitation, Sir Isaac Newton built upon Copernicuss work as well as observations by Tycho Brahe. Further observational improvements led to the realization that our Solar System is located in the Milky Way galaxy and it is assumed that galaxies are distributed uniformly and the same in all directions, meaning that the Universe has neither an edge nor a center. Discoveries in the early 20th century have suggested that the Universe had a beginning, the majority of mass in the Universe appears to exist in an unknown form called dark matter. The Big Bang theory is the prevailing cosmological description of the development of the Universe, under this theory, space and time emerged together 13. 799±0.021 billion years ago with a fixed amount of energy and matter that has become less dense as the Universe has expanded. After the initial expansion, the Universe cooled, allowing the first subatomic particles to form, giant clouds later merged through gravity to form galaxies, stars, and everything else seen today. Some physicists have suggested various multiverse hypotheses, in which the Universe might be one among many universes that likewise exist, the Universe can be defined as everything that exists, everything that has existed, and everything that will exist. According to our current understanding, the Universe consists of spacetime, forms of energy, the Universe encompasses all of life, all of history, and some philosophers and scientists suggest that it even encompasses ideas such as mathematics and logic. The word universe derives from the Old French word univers, which in turn derives from the Latin word universum, the Latin word was used by Cicero and later Latin authors in many of the same senses as the modern English word is used. Another synonym was ὁ κόσμος ho kósmos, synonyms are also found in Latin authors and survive in modern languages, e. g. the German words Das All, Weltall, and Natur for Universe. The same synonyms are found in English, such as everything, the cosmos, the world, the prevailing model for the evolution of the Universe is the Big Bang theory. The Big Bang model states that the earliest state of the Universe was extremely hot and dense, the model is based on general relativity and on simplifying assumptions such as homogeneity and isotropy of space. The Big Bang model accounts for such as the correlation of distance and redshift of galaxies, the ratio of the number of hydrogen to helium atoms. The initial hot, dense state is called the Planck epoch, after the Planck epoch and inflation came the quark, hadron, and lepton epochs. Together, these epochs encompassed less than 10 seconds of time following the Big Bang, the observed abundance of the elements can be explained by combining the overall expansion of space with nuclear and atomic physics. As the Universe expands, the density of electromagnetic radiation decreases more quickly than does that of matter because the energy of a photon decreases with its wavelength
Universe
–
The
Hubble ultra deep field image shows some of the most remote galaxies that can be seen with present technology
Universe
–
In this diagram, time passes from left to right, so at any given time, the Universe is represented by a disk-shaped "slice" of the diagram.
Universe
–
This diagram shows Earth's location in the Universe.
118.
Curvature
–
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. This article deals primarily with extrinsic curvature and its canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius everywhere. Smaller circles bend more sharply, and hence have higher curvature, the curvature of a smooth curve is defined as the curvature of its osculating circle at each point. Curvature is normally a scalar quantity, but one may define a curvature vector that takes into account the direction of the bend in addition to its magnitude. The curvature of more objects is described by more complex objects from linear algebra. This article sketches the mathematical framework which describes the curvature of a curve embedded in a plane, the curvature of C at a point is a measure of how sensitive its tangent line is to moving the point to other nearby points. There are a number of equivalent ways that this idea can be made precise and it is natural to define the curvature of a straight line to be constantly zero. The curvature of a circle of radius R should be large if R is small and small if R is large, thus the curvature of a circle is defined to be the reciprocal of the radius, κ =1 R. Given any curve C and a point P on it, there is a circle or line which most closely approximates the curve near P. The curvature of C at P is then defined to be the curvature of that circle or line, the radius of curvature is defined as the reciprocal of the curvature. Another way to understand the curvature is physical, suppose that a particle moves along the curve with unit speed. Taking the time s as the parameter for C, this provides a natural parametrization for the curve, the unit tangent vector T also depends on time. The curvature is then the magnitude of the rate of change of T. Symbolically and this is the magnitude of the acceleration of the particle and the vector dT/ds is the acceleration vector. Geometrically, the curvature κ measures how fast the unit tangent vector to the curve rotates. If a curve close to the same direction, the unit tangent vector changes very little and the curvature is small, where the curve undergoes a tight turn. These two approaches to the curvature are related geometrically by the following observation, in the first definition, the curvature of a circle is equal to the ratio of the angle of an arc to its length. e. For such a curve, there exists a reparametrization with respect to arc length s. This is a parametrization of C such that ∥ γ ′ ∥2 = x ′2 + y ′2 =1, the velocity vector T is the unit tangent vector
Curvature
119.
Morse theory
–
In another context, a Morse function can also mean an anharmonic oscillator, see Morse potential. In mathematics, specifically in topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds, before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics and these techniques were used in Raoul Botts proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory, consider, for purposes of illustration, a mountainous landscape M. If f is the function M → R sending each point to its elevation, each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. Contour lines may also have points of order, but these are unstable. Double points in contour lines occur at points, or passes. Saddle points are points where the surrounding landscape curves up in one direction, imagine flooding this landscape with water. Then, the covered by water when the water reaches an elevation of a is f−1(−∞, a]. Consider how the topology of this changes as the water rises. It appears, intuitively, that it does not change except when a passes the height of a point, that is. In other words, it does not change except when the water either starts filling a basin, covers a saddle, to each of these three types of critical points – basins, passes, and peaks – one associates a number called the index. Intuitively speaking, the index of a point b is the number of independent directions around b in which f decreases. Therefore, the indices of basins, passes, and peaks are 0,1, rigorously, the index of a critical point is the dimension of the negative-definite submatrix of the hessian matrix calculated at that point. In case of smooth maps, the matrix turns out to be a symmetric matrix. Starting from the bottom of the torus, let p, q, r, and s be the four points of index 0,1,1
Morse theory
–
A saddle point