1.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
Geometry
–
Visual checking of the
Pythagorean theorem for the (3, 4, 5)
triangle as in the
Chou Pei Suan Ching 500–200 BC.
Geometry
–
An illustration of
Desargues' theorem, an important result in
Euclidean and
projective geometry
Geometry
–
Geometry lessons in the 20th century
Geometry
–
A
European and an
Arab practicing geometry in the 15th century.
2.
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
3.
History of geometry
–
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of mathematics, the other being the study of numbers. Classic geometry was focused in compass and straightedge constructions, geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, the earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, and ancient Babylonia from around 3000 BC. Among these were some surprisingly sophisticated principles, and a mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 B. C. Problem 30 of the Ahmes papyrus uses these methods to calculate the area of a circle and this assumes that π is 4×², with an error of slightly over 0.63 percent. Problem 48 involved using a square with side 9 units and this square was cut into a 3x3 grid. The diagonal of the squares were used to make an irregular octagon with an area of 63 units. This gave a value for π of 3.111. The two problems together indicate a range of values for π between 3.11 and 3.16. Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula, the Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3, the Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3, the Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a used for measuring the travel of the Sun, therefore. There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did, the Indian Vedic period had a tradition of geometry, mostly expressed in the construction of elaborate altars. Early Indian texts on this include the Satapatha Brahmana and the Śulba Sūtras. According to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, the diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately
History of geometry
–
Part of the " Tab.Geometry. " (Table of Geometry) from the 1728
Cyclopaedia.
History of geometry
–
Rigveda manuscript in
Devanagari.
History of geometry
–
Statue of Euclid in the
Oxford University Museum of Natural History.
History of geometry
–
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's
Elements, (c. 1310)
4.
Euclidean geometry
–
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. 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. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
Euclidean geometry
–
Detail from
Raphael 's
The School of Athens featuring a Greek mathematician – perhaps representing
Euclid or
Archimedes – using a
compass to draw a geometric construction.
Euclidean geometry
–
A surveyor uses a
level
Euclidean geometry
–
Sphere packing applies to a stack of
oranges.
Euclidean geometry
–
Geometry is used in art and architecture.
5.
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.
6.
Synthetic geometry
–
Synthetic geometry is the study of geometry without the use of coordinates or formulas. It relies on the method and the tools directly related to them. Only after the introduction of methods was there a reason to introduce the term synthetic geometry to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, geometry, as presented by Euclid in the elements, is the quintessential example of the use of the synthetic method. It was the method of Isaac Newton for the solution of geometric problems. Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective geometry, for example the geometer Jakob Steiner hated analytic geometry, and always gave preference to synthetic methods. The process of logical synthesis begins with some arbitrary but definite starting point and this starting point is the introduction of primitive notions or primitives and axioms about these primitives, Primitives are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are such as points, lines and planes. Axioms are statements about these primitives, for example, any two points are incident with just one line. Axioms are assumed true, and not proven and they are the building blocks of geometric concepts, since they specify the properties that the primitives have. From a given set of axioms, synthesis proceeds as a carefully constructed logical argument, when a significant result is proved rigorously, it becomes a theorem. There is no fixed set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of geometry in the singular, historically, Euclids parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry, other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and betweeness are also optional, for example, following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development. One of the early French analysts summarized synthetic geometry this way, for example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory than is found by starting with a vector space of dimension three
Synthetic geometry
–
Projecting a
sphere to a
plane.
7.
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".
8.
Projective geometry
–
Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry, the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
Projective geometry
–
Growth measure and the polar vortices. Based on the work of Lawrence Edwards
Projective geometry
–
Projecting a
sphere to a
plane.
Projective geometry
–
Forms
9.
Dimension
–
In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
Dimension
10.
Compass-and-straightedge construction
–
The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, the ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in cases were unable to do so. Gauss showed that some polygons are constructible but that most are not, some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems, in terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its not drawing a circle, the straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and it would appear that the modern compass is a more powerful instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, the ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon
Compass-and-straightedge construction
–
A compass
Compass-and-straightedge construction
–
Creating a regular
hexagon with a ruler and compass
11.
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.
12.
Curve
–
In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
Curve
–
Megalithic art from Newgrange showing an early interest in curves
Curve
–
A
parabola, a simple example of a curve
13.
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.
14.
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.
15.
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
16.
Line segment
–
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
Line segment
–
historical image – create a line segment (1699)
17.
Length
–
In geometric measurements, length is the most extended dimension of an object. In the International System of Quantities, length is any quantity with dimension distance, in other contexts length is the measured dimension of an object. For example, it is possible to cut a length of a wire which is shorter than wire thickness. Length may be distinguished from height, which is vertical extent, and width or breadth, length is a measure of one dimension, whereas area is a measure of two dimensions and volume is a measure of three dimensions. In most systems of measurement, the unit of length is a base unit, measurement has been important ever since humans settled from nomadic lifestyles and started using building materials, occupying land and trading with neighbours. As society has become more technologically oriented, much higher accuracies of measurement are required in a diverse set of fields. One of the oldest units of measurement used in the ancient world was the cubit which was the length of the arm from the tip of the finger to the elbow. This could then be subdivided into shorter units like the foot, hand or finger, the cubit could vary considerably due to the different sizes of people. After Albert Einsteins special relativity, length can no longer be thought of being constant in all reference frames. Thus a ruler that is one meter long in one frame of reference will not be one meter long in a frame that is travelling at a velocity relative to the first frame. This means length of an object is variable depending on the observer, in the physical sciences and engineering, when one speaks of units of length, the word length is synonymous with distance. There are several units that are used to measure length, in the International System of Units, the basic unit of length is the metre and is now defined in terms of the speed of light. The centimetre and the kilometre, derived from the metre, are commonly used units. In U. S. customary units, English or Imperial system of units, commonly used units of length are the inch, the foot, the yard, and the mile. Units used to denote distances in the vastness of space, as in astronomy, are longer than those typically used on Earth and include the astronomical unit, the light-year. Dimension Distance Orders of magnitude Reciprocal length Smoot Unit of length
Length
–
Base quantity
18.
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
19.
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
20.
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.
21.
Rhomboid
–
Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled. A parallelogram with sides of length is a rhombus but not a rhomboid. A parallelogram with right angled corners is a rectangle but not a rhomboid, the term rhomboid is now more often used for a parallelepiped, a solid figure with six faces in which each face is a parallelogram and pairs of opposite faces lie in parallel planes. Some crystals are formed in three-dimensional rhomboids and this solid is also sometimes called a rhombic prism. The term occurs frequently in science terminology referring to both its two- and three-dimensional meaning, and let quadrilaterals other than these be called trapezia. Heath suggests that rhomboid was a term already in use. The rhomboid has no line of symmetry, but it has symmetry of order 2. In biology, rhomboid may describe a geometric rhomboid or a bilaterally-symmetrical kite-shaped or diamond-shaped outline, in a type of arthritis called pseudogout, crystals of calcium pyrophosphate dihydrate accumulate in the joint, causing inflammation. Aspiration of the joint fluid reveals rhomboid-shaped crystals under a microscope
Rhomboid
–
These shapes are rhomboids
22.
Trapezoid
–
The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, the first recorded use of the Greek word translated trapezoid was by Marinus Proclus in his Commentary on the first book of Euclids Elements. This article uses the term trapezoid in the sense that is current in the United States, in many other languages using a word derived from the Greek for this figure, the form closest to trapezium is used. A right trapezoid has two adjacent right angles, right trapezoids are used in the trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge, while an obtuse trapezoid has one acute, an acute trapezoid is also an isosceles trapezoid, if its sides have the same length, and the base angles have the same measure. An obtuse trapezoid with two pairs of sides is a parallelogram. A parallelogram has central 2-fold rotational symmetry, a Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the plane has 3 right angles. A tangential trapezoid is a trapezoid that has an incircle, there is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having one pair of parallel sides. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides, the latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined and this article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals, under the inclusive definition, all parallelograms are trapezoids. Rectangles have mirror symmetry on mid-edges, rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. Four lengths a, c, b, d can constitute the sides of a non-parallelogram trapezoid with a and b parallel only when | d − c | < | b − a | < d + c. The quadrilateral is a parallelogram when d − c = b − a =0, the angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal. The diagonals cut each other in mutually the same ratio, the diagonals cut the quadrilateral into four triangles of which one opposite pair are similar. The diagonals cut the quadrilateral into four triangles of which one pair have equal areas
Trapezoid
–
The
Temple of Dendur in the
Metropolitan Museum of Art in
New York City
Trapezoid
–
Trapezoid
Trapezoid
–
Example of a trapeziform
pronotum outlined on a spurge bug
23.
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.
24.
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.
25.
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
26.
Cylinder (geometry)
–
In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
Cylinder (geometry)
–
Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section
Cylinder (geometry)
–
A right circular cylinder with radius r and height h.
Cylinder (geometry)
–
In
projective geometry, a cylinder is simply a cone whose
apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
27.
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.
28.
Aryabhata
–
Aryabhata or Aryabhata I was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya and the Arya-siddhanta, furthermore, in most instances Aryabhatta would not fit the metre either. Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga and this corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra, Bhāskara I describes Aryabhata as āśmakīya, one belonging to the Aśmaka country. During the Buddhas time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India. It has been claimed that the aśmaka where Aryabhata originated may be the present day Kodungallur which was the capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr, however, K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence. Aryabhata mentions Lanka on several occasions in the Aryabhatiya, but his Lanka is an abstraction and it is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I, identify Kusumapura as Pāṭaliputra, Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. Aryabhata is the author of treatises on mathematics and astronomy. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and it also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, a third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, direct details of Aryabhatas work are known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra and it is also occasionally referred to as Arya-shatas-aShTa, because there are 108 verses in the text. It is written in the terse style typical of sutra literature
Aryabhata
–
Statue of Aryabhata on the grounds of
IUCAA,
Pune. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.
Aryabhata
–
India's first satellite named after Aryabhata
29.
Alhazen
–
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
Alhazen
–
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.
Alhazen
–
Alhazen (Ibn al-Haytham)
Alhazen
–
The
theorem of Ibn Haytham
Alhazen
–
Alhazen on Iraqi 10 dinars
30.
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.
31.
Archimedes
–
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
Archimedes
–
Archimedes Thoughtful by
Fetti (1620)
Archimedes
–
Cicero Discovering the Tomb of Archimedes by
Benjamin West (1805)
Archimedes
–
Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by
Giulio Parigi.
Archimedes
–
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. A
sphere and
cylinder were placed on the tomb of Archimedes at his request. (see also:
Equiareal map)
32.
Brahmagupta
–
Brahmagupta was an Indian mathematician and astronomer. He is the author of two works on mathematics and astronomy, the Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka. According to his commentators, Brahmagupta was a native of Bhinmal, Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in verse in Sanskrit. As no proofs are given, it is not known how Brahmaguptas results were derived, Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala during the reign of the Chapa dynasty ruler Vyagrahamukha and he was the son of Jishnugupta. He was a Shaivite by religion, even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a part of his life. Prithudaka Svamin, a commentator, called him Bhillamalacharya, the teacher from Bhillamala. Sociologist G. S. Ghurye believed that he might have been from the Multan region or the Abu region and it was also a center of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, in the year 628, at an age of 30, he composed Brāhmasphuṭasiddhānta which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he has incorported a great deal of originality to his revision, the book consists of 24 chapters with 1008 verses in the ārya meter. Later, Brahmagupta moved to Ujjain, which was also a centre for astronomy. At the mature age of 67, he composed his next well known work Khanda-khādyaka and he is believed to have died in Ujjain. Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, the division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmaguptas case, the disagreements stemmed largely from the choice of astronomical parameters, the historian of science George Sarton called him one of the greatest scientists of his race and the greatest of his time. Brahmaguptas mathematical advances were carried on to further extent by Bhāskara II, a descendant in Ujjain. Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations, lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka
Brahmagupta
33.
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
34.
Leonhard Euler
–
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
Leonhard Euler
–
Portrait by
Jakob Emanuel Handmann (1756)
Leonhard Euler
–
1957
Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Leonhard Euler
–
Stamp of the former
German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his
polyhedral formula, nowadays written as " v − e + f = 2".
Leonhard Euler
–
Euler's grave at the
Alexander Nevsky Monastery
35.
Carl Friedrich Gauss
–
Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
Carl Friedrich Gauss
–
Carl Friedrich Gauß (1777–1855), painted by
Christian Albrecht Jensen
Carl Friedrich Gauss
–
Statue of Gauss at his birthplace,
Brunswick
Carl Friedrich Gauss
–
Title page of Gauss's
Disquisitiones Arithmeticae
Carl Friedrich Gauss
–
Gauss's portrait published in
Astronomische Nachrichten 1828
36.
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
37.
David Hilbert
–
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895, in 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust
David Hilbert
–
David Hilbert (1912)
David Hilbert
–
The Mathematical Institute in Göttingen. Its new building, constructed with funds from the
Rockefeller Foundation, was opened by Hilbert and Courant in 1930.
David Hilbert
–
Hilbert's tomb: Wir müssen wissen Wir werden wissen
38.
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
39.
Nikolai Lobachevsky
–
Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry. William Kingdon Clifford called Lobachevsky the Copernicus of Geometry due to the character of his work. He was one of three children and his father, a clerk in a land surveying office, died when he was seven, and his mother moved to Kazan. Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807 and then received a scholarship to Kazan University, at Kazan University, Lobachevsky was influenced by professor Johann Christian Martin Bartels, a former teacher and friend of German mathematician Carl Friedrich Gauss. Lobachevsky received a degree in physics and mathematics in 1811. He served in administrative positions and became the rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva and they had a large number of children. He was dismissed from the university in 1846, ostensibly due to his health, by the early 1850s, he was nearly blind. He died in poverty in 1856, Lobachevskys main achievement is the development of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclids fifth postulate from other axioms, Euclids fifth is a rule in Euclidean geometry which states that for any given line and point not on the line, there is one parallel line through the point not intersecting the line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true and this idea was first reported on February 23,1826 to the session of the department of physics and mathematics, and this research was printed in the UMA in 1829–1830. The non-Euclidean geometry that Lobachevsky developed is referred to as hyperbolic geometry and he developed the angle of parallelism which depends on the distance the point is off the given line. In hyperbolic geometry the sum of angles in a triangle must be less than 180 degrees. Non-Euclidean geometry stimulated the development of geometry which has many applications. Hyperbolic geometry is referred to as Lobachevskian geometry or Bolyai–Lobachevskian geometry. Some mathematicians and historians have claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss. Gauss himself appreciated Lobachevskys published works very highly, but they never had personal correspondence between them prior to the publication, Lobachevskys magnum opus Geometriya was completed in 1823, but was not published in its exact original form until 1909, long after he had died. Lobachevsky was also the author of New Foundations of Geometry and he also wrote Geometrical Investigations on the Theory of Parallels and Pangeometry
Nikolai Lobachevsky
–
Portrait by Lev Kryukov (c. 1843)
Nikolai Lobachevsky
–
Annual celebration of Lobachevsky's birthday by participants of
Volga 's student Mathematical Olympiad
40.
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
41.
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.
42.
Nasir al-Din al-Tusi
–
Khawaja Muhammad ibn Muhammad ibn al-Hasan al-Tūsī, better known as Nasīr al-Dīn Tūsī, was a Persian polymath, architect, philosopher, physician, scientist, theologian and Marja Taqleed. He was of the Twelver Shī‘ah Islamic belief, the Muslim scholar Ibn Khaldun considered Tusi to be the greatest of the later Persian scholars. Nasir al-Din Tusi was born in the city of Tus in medieval Khorasan in the year 1201, in Hamadan and Tus he studied the Quran, Hadith, Shia jurisprudence, logic, philosophy, mathematics, medicine and astronomy. He was apparently born into a Shī‘ah family and lost his father at a young age, at a young age he moved to Nishapur to study philosophy under Farid al-Din Damad and mathematics under Muhammad Hasib. He met also Farid al-Din Attar, the legendary Sufi master who was killed by Mongol invaders. In Mosul he studied mathematics and astronomy with Kamal al-Din Yunus and he was captured after the invasion of the Alamut castle by the Mongol forces. Tusi has about 150 works, of which 25 are in Persian and the remaining are in Arabic, here are some of his major works, Kitāb al-Shakl al-qattāʴ Book on the complete quadrilateral. A five volume summary of trigonometry, al-Tadhkirah fiilm al-hayah – A memoir on the science of astronomy. Many commentaries were written about this work called Sharh al-Tadhkirah - Commentaries were written by Abd al-Ali ibn Muhammad ibn al-Husayn al-Birjandi, akhlaq-i Nasiri – A work on ethics. Al-Risalah al-Asturlabiyah – A Treatise on astrolabe, Zij-i ilkhani – A major astronomical treatise, completed in 1272. Sharh al-isharat Awsaf al-Ashraf a short work in Persian Tajrīd al-iʿtiqād – A commentary on Shia doctrines. During his stay in Nishapur, Tusi established a reputation as an exceptional scholar, tusi’s prose writing, which number over 150 works, represent one of the largest collections by a single Islamic author. Writing in both Arabic and Persian, Nasir al-Din Tusi dealt with religious topics and non-religious or secular subjects. His works include the definitive Arabic versions of the works of Euclid, Archimedes, Ptolemy, Autolycus, Tusi convinced Hulegu Khan to construct an observatory for establishing accurate astronomical tables for better astrological predictions. Beginning in 1259, the Rasad Khaneh observatory was constructed in Azarbaijan, south of the river Aras, and to the west of Maragheh, the capital of the Ilkhanate Empire. Based on the observations in this for the time being most advanced observatory and this book contains astronomical tables for calculating the positions of the planets and the names of the stars. His model for the system is believed to be the most advanced of his time. Between Ptolemy and Copernicus, he is considered by many to be one of the most eminent astronomers of his time, for his planetary models, he invented a geometrical technique called a Tusi-couple, which generates linear motion from the sum of two circular motions
Nasir al-Din al-Tusi
–
Persian Muslim scholar Nasīr al-Dīn Tūsī
Nasir al-Din al-Tusi
–
A Treatise on
Astrolabe by Tusi, Isfahan 1505
Nasir al-Din al-Tusi
–
Tusi couple from Vat. Arabic ms 319
Nasir al-Din al-Tusi
–
The Astronomical Observatory of Nasir al-
Dīn Tusi.
43.
Oswald Veblen
–
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905, while this was considered the first rigorous proof. Veblen was born in Decorah, Iowa and his parents were Andrew Anderson Veblen and Kirsti Veblen. Veblens uncle was Thorstein Veblen, noted economist and sociologist and he went to school in Iowa City. He did his studies at the University of Iowa, where he received an A. B. in 1898, and Harvard University. For his graduate studies, he went to study mathematics at the University of Chicago and his dissertation, A System of Axioms for Geometry was written under the supervision of E. H. Moore. During World War I, Veblen served first as a captain, Veblen taught mathematics at Princeton University from 1905 to 1932. In 1926, he was named Henry B, in 1932, he helped organize the Institute for Advanced Study in Princeton, resigning his professorship to become the first professor at the Institute that same year. He kept his professorship at the Institute until he was made emeritus in 1950, during his years in Princeton, Veblen and his wife Elizabeth accumulated land along the Princeton Ridge. In 1957 they donated 82 acres to establish the Herrontown Woods Arboretum, one of the largest nature preserves in Princeton, Veblen died in Brooklin, Maine, in 1960 at age 80. After his death the American Mathematical Society created an award in his name and it is awarded every three years, and is the most prestigious award in recognition of outstanding research in geometry. During his career, Veblen made important contributions in topology and in projective and differential geometries and he introduced the Veblen axioms for projective geometry and proved the Veblen–Young theorem. He introduced the Veblen functions of ordinals and used an extension of them to define the small and he also published a paper in 1912 on the four-color conjecture. In 1908, he married Elizabeth Richardson, the sister of British physicist Owen Willans Richardson, obituary and Bibliography of Oswald Veblen Works by Oswald Veblen at Project Gutenberg Works by or about Oswald Veblen at Internet Archive Projective relativity theory, transl. by D. H. Delphenich
Oswald Veblen
–
Oswald Veblen (photo ca. 1915)
44.
Yang Hui
–
Yang Hui, courtesy name Qianguang, was a late-Song dynasty Chinese mathematician from Qiantang. Yang worked on magic squares, magic circles and the binomial theorem and this triangle was the same as Pascals Triangle, discovered by Yangs predecessor Jia Xian. Yang was also a contemporary to the famous mathematician Qin Jiushao. In his book known as Rújī Shìsuǒ or Piling-up Powers and Unlocking Coefficients, Jia described the method used as li cheng shi suo. It appeared again in a publication of Zhu Shijies book Jade Mirror of the Four Unknowns of 1303 AD, around 1275 AD, Yang finally had two published mathematical books, which were known as the Xugu Zhaiqi Suanfa and the Suanfa Tongbian Benmo. In his writing, he criticized the earlier works of Li Chunfeng and Liu Yi. In his written work, Yang provided theoretical proof for the proposition that the complements of the parallelograms which are about the diameter of any given parallelogram are equal to one another. This was the idea expressed in the Greek mathematician Euclids forty-third proposition of his first book, only Yang used the case of a rectangle. There were also a number of other problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system. However, the first books of Euclid to be translated into Chinese was by the effort of the Italian Jesuit Matteo Ricci. History of mathematics List of mathematicians Chinese mathematics Needham, Joseph, science and Civilization in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Encyclopedia of China, 1st ed. Yang Hui at MacTutor
Yang Hui
–
1433 Korean edition of Yang Hui suan fa
Yang Hui
–
Yang Hui triangle (Pascal's triangle) using
rod numerals, as depicted in a publication of
Zhu Shijie in 1303 AD.
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.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
47.
Right angle
–
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two adjacent parts of a straight line. More precisely, if a ray is placed so that its endpoint is on a line, as a rotation, a right angle corresponds to a quarter turn. The presence of an angle in a triangle is the defining factor for right triangles. The term is a calque of Latin angulus rectus, here rectus means upright, in Unicode, the symbol for a right angle is U+221F ∟ Right angle. It should not be confused with the similarly shaped symbol U+231E ⌞ Bottom left corner, related symbols are U+22BE ⊾ Right angle with arc, U+299C ⦜ Right angle variant with square, and U+299D ⦝ Measured right angle with dot. The symbol for an angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland. Right angles are fundamental in Euclids Elements and they are defined in Book 1, definition 10, which also defines perpendicular lines. Euclid uses right angles in definitions 11 and 12 to define acute angles, two angles are called complementary if their sum is a right angle. Book 1 Postulate 4 states that all angles are equal. Euclids commentator Proclus gave a proof of this using the previous postulates. Saccheri gave a proof as well but using a more explicit assumption, in Hilberts axiomatization of geometry this statement is given as a theorem, but only after much groundwork. A right angle may be expressed in different units, 1/4 turn, 90° π/2 radians 100 grad 8 points 6 hours Throughout history carpenters and masons have known a quick way to confirm if an angle is a true right angle. It is based on the most widely known Pythagorean triple and so called the Rule of 3-4-5 and this measurement can be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem, Thales theorem states that an angle inscribed in a semicircle is a right angle. Two application examples in which the angle and the Thales theorem are included. Cartesian coordinate system Orthogonality Perpendicular Rectangle Types of angles Wentworth, G. A, Euclid, commentary and trans. by T. L. Heath Elements Vol.1 Google Books
Right angle
–
A right angle is equal to 90 degrees.
48.
Cathetus
–
In a right triangle, a cathetus, commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a side about the right angle, the side opposite the right angle is the hypotenuse. In the context of the hypotenuse, the catheti are sometimes referred to simply as the two sides. If the catheti of a triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor and major cathetus. In a right triangle, the length of a cathetus is the mean of the length of the adjacent segment cut by the altitude to the hypotenuse. By the Pythagorean theorem, the sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse, geographic Information Systems, An Introduction, 3rd ed. New York, Wiley, p.271,2002, Cathetus at Encyclopaedia of Mathematics Weisstein, Eric W. Cathetus
Cathetus
–
A right-angled triangle where c 1 and c 2 are the catheti and h is the hypotenuse
49.
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.
50.
Ancient Greece
–
Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 12th-9th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and this was followed by the period of Classical Greece, an era that began with the Greco-Persian Wars, lasting from the 5th to 4th centuries BC. Due to the conquests by Alexander the Great of Macedonia, Hellenistic civilization flourished from Central Asia to the end of the Mediterranean Sea. Classical Greek culture, especially philosophy, had a influence on ancient Rome. For this reason Classical Greece is generally considered to be the culture which provided the foundation of modern Western culture and is considered the cradle of Western civilization. Classical Antiquity in the Mediterranean region is considered to have begun in the 8th century BC. Classical Antiquity in Greece is preceded by the Greek Dark Ages and this period is succeeded, around the 8th century BC, by the Orientalizing Period during which a strong influence of Syro-Hittite, Jewish, Assyrian, Phoenician and Egyptian cultures becomes apparent. The end of the Dark Ages is also dated to 776 BC. The Archaic period gives way to the Classical period around 500 BC, Ancient Periods Astronomical year numbering Dates are approximate, consult particular article for details The history of Greece during Classical Antiquity may be subdivided into five major periods. The earliest of these is the Archaic period, in which artists made larger free-standing sculptures in stiff, the Archaic period is often taken to end with the overthrow of the last tyrant of Athens and the start of Athenian Democracy in 508 BC. It was followed by the Classical period, characterized by a style which was considered by observers to be exemplary, i. e. classical, as shown in the Parthenon. This period saw the Greco-Persian Wars and the Rise of Macedon, following the Classical period was the Hellenistic period, during which Greek culture and power expanded into the Near and Middle East. This period begins with the death of Alexander and ends with the Roman conquest, Herodotus is widely known as the father of history, his Histories are eponymous of the entire field. Herodotus was succeeded by authors such as Thucydides, Xenophon, Demosthenes, Plato, most of these authors were either Athenian or pro-Athenian, which is why far more is known about the history and politics of Athens than those of many other cities. Their scope is limited by a focus on political, military and diplomatic history, ignoring economic. In the 8th century BC, Greece began to emerge from the Dark Ages which followed the fall of the Mycenaean civilization, literacy had been lost and Mycenaean script forgotten, but the Greeks adopted the Phoenician alphabet, modifying it to create the Greek alphabet. The Lelantine War is the earliest documented war of the ancient Greek period and it was fought between the important poleis of Chalcis and Eretria over the fertile Lelantine plain of Euboea. Both cities seem to have suffered a decline as result of the long war, a mercantile class arose in the first half of the 7th century BC, shown by the introduction of coinage in about 680 BC
Ancient Greece
–
The
Parthenon, a temple dedicated to
Athena, located on the
Acropolis in
Athens, is one of the most representative symbols of the culture and sophistication of the ancient Greeks.
Ancient Greece
–
Dipylon Vase of the late Geometric period, or the beginning of the Archaic period,
c. 750 BC.
Ancient Greece
–
Political geography of ancient Greece in the Archaic and Classical periods
51.
Mathematical proof
–
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
Mathematical proof
–
One of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.
Mathematical proof
–
Visual proof for the (3, 4, 5) triangle as in the
Chou Pei Suan Ching 500–200 BC.
52.
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
53.
Q.E.D.
–
Q. E. D. is an initialism of the Latin phrase quod erat demonstrandum, meaning what was to be demonstrated, or, less formally, thus it has been demonstrated. The abbreviation thus signals the completion of the proof, the phrase quod erat demonstrandum is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι. Translating from the Latin into English yields, what was to be demonstrated, however, since the verb δείκνυμι also means to show or to prove, a better translation from the Greek would read, The very thing it was required to have shown. The phrase was used by many early Greek mathematicians, including Euclid, in the European Renaissance, scholars often wrote in Latin, and phrases such as Q. E. D. were often used to conclude proofs. Perhaps the most famous use of Q. E. D. in an argument is found in the Ethics of Baruch Spinoza. Written in Latin, it is considered by many to be Spinozas magnum opus, the style and system of the book is, as Spinoza says, demonstrated in geometrical order, with axioms and definitions followed by propositions. For Spinoza, this is an improvement over René Descartess writing style in the Meditations. There is another Latin phrase with a different meaning. Quod erat faciendum, originating from the Greek geometers closing ὅπερ ἔδει ποιῆσαι, Euclid used this phrase to close propositions which were not proofs of theorems, but constructions. For example, Euclids first proposition showing how to construct an equilateral triangle given one side is concluded this way, the phrase is usually shortened to QEF. WWWWW or W5 - an abbreviation of Which Was What Was Wanted - has also been used and this is often considered to be more tongue-in-cheek than the usual Halmos symbol or Q. E. D. Due to the paramount importance of proofs in mathematics, mathematicians since the time of Euclid have developed conventions to demarcate the beginning and/or end of proofs, in English language texts, the formal statements of theorems, lemmas, and propositions are typically set in italics. The beginning of a proof usually follows immediately thereafter and is indicated by the word Proof in boldface or italics, on the other hand, several symbolic conventions exist to indicate the end of a proof. While some authors use the classical abbreviation Q. E. D. this practice is increasingly viewed as archaic or even pretentious. Paul Halmos pioneered the use of a square at the end of a proof as a Q. E. D sign. Halmos claimed to have adopted this symbol from magazine typography in which simple geometric shapes had been used to indicate the end of an article and this symbol was later called the tombstone or Halmos symbol or even a halmos by mathematicians. The Halmos symbol is often drawn on chalkboard to signal the end of a proof during a lecture. The tombstone symbol appears in TeX as the character ◼ and ◻, in the AMS Theorem Environment for LaTeX, the hollow square is the default end-of-proof symbol
Q.E.D.
–
Spinoza 's original text of
Ethics, Part 1. Q.E.D. is used at the end of DEMONSTRATIO of PROPOSITIO III. in the right page.
54.
Law of cosines
–
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. Though the notion of the cosine was not yet developed in his time, Euclids Elements, dating back to the 3rd century BC, the cases of obtuse triangles and acute triangles are treated separately, in Propositions 12 and 13 of Book 2. Using notation as in Fig.2, Euclids statement can be represented by the formula A B2 = C A2 + C B2 +2 and this formula may be transformed into the law of cosines by noting that CH = cos = − cos γ. Proposition 13 contains an analogous statement for acute triangles. In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi, the theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form, the theorem is used in triangulation, for solving a triangle or circle, i. e. These formulas produce high round-off errors in floating point calculations if the triangle is very acute and it is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for a the quadratic equation a2 − 2ab cos γ + b2 − c2 =0 and this equation can have 2,1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if b sin γ < c < b, only one positive solution if c = b sin γ and these different cases are also explained by the side-side-angle congruence ambiguity. Consider a triangle with sides of length a, b, c and this triangle can be placed on the Cartesian coordinate system by plotting the following points, as shown in Fig.4, A =, B =, and C =. By the distance formula, we have c =2 +2, an advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. right vs. obtuse. Drop the perpendicular onto the c to get c = a cos β + b cos α. Multiply through by c to get c 2 = a c cos β + b c cos α. By considering the other perpendiculars obtain a 2 = a c cos β + a b cos γ, b 2 = b c cos α + a b cos γ. Adding the latter two equations gives a 2 + b 2 = a c cos β + b c cos α +2 a b cos γ and this proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in any right triangle
Law of cosines
–
Figure 1 – A triangle. The angles α (or A), β (or B), and γ (or C) are respectively opposite the sides a, b, and c.
55.
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.
56.
Cosine
–
In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
Cosine
–
Trigonometric functions in the complex plane
Cosine
–
Trigonometry
Cosine
Cosine
57.
James A. Garfield
–
James Abram Garfield was the 20th President of the United States, serving from March 4,1881, until his assassination later that year. He is the only sitting House member to be elected president, Garfield was raised in humble circumstances on an Ohio farm by his widowed mother. He worked at various jobs, including on a canal boat, beginning at age 17, he attended several Ohio schools, then studied at Williams College in Williamstown, Massachusetts, from which he graduated in 1856. A year later, Garfield entered politics as a Republican and he married Lucretia Rudolph in 1858, and served as a member of the Ohio State Senate. Garfield opposed Confederate secession, served as a general in the Union Army during the American Civil War, and fought in the battles of Middle Creek, Shiloh. He was first elected to Congress in 1862 to represent Ohios 19th District, throughout Garfields extended congressional service after the Civil War, he firmly supported the gold standard and gained a reputation as a skilled orator. Garfield initially agreed with Radical Republican views regarding Reconstruction, but later favored an approach for civil rights enforcement for freedmen. At the 1880 Republican National Convention, Senator-elect Garfield attended as campaign manager for Secretary of the Treasury John Sherman, and gave the presidential nomination speech for him. When neither Sherman nor his rivals – Ulysses S. Grant and James G. Blaine – could get votes to secure the nomination. In the 1880 presidential election, Garfield conducted a front porch campaign. Garfield made notable diplomatic and judiciary appointments, including a U. S. Supreme Court justice, Garfield advocated agricultural technology, an educated electorate, and civil rights for African Americans. He also proposed substantial civil service reform, eventually passed by Congress in 1883 and signed into law by his successor, Chester A. Arthur, Presidents due to the short length of his presidency. James Garfield was born the youngest of five children on November 19,1831, in a log cabin in Orange Township, now Moreland Hills, Orange Township was located in the Western Reserve, and like many who settled there, Garfields ancestors were from New England. James father Abram had been born in Worcester, New York and he instead wed her sister Eliza, who had been born in New Hampshire. James was named for a brother, dead in infancy. In early 1833, Abram and Eliza Garfield joined the Church of Christ, Abram Garfield died later that year, his son was raised in poverty in a household led by the strong-willed Eliza. James was her child, and the two remained close for the rest of her life. Eliza Garfield remarried in 1842, but soon left her husband, Warren Belden
James A. Garfield
–
Brady -
Handy photograph of Garfield, taken between 1870 and 1880
James A. Garfield
–
Birthplace site of James Garfield
James A. Garfield
–
Garfield at age 16
James A. Garfield
–
Lucretia Garfield in the 1870s
58.
U.S. Representative
–
The United States House of Representatives is the lower chamber of the United States Congress which, along with the Senate, composes the legislature of the United States. The composition and powers of the House are established by Article One of the United States Constitution, since its inception in 1789, all representatives are elected popularly. The total number of voting representatives is fixed by law at 435, the House is charged with the passage of federal legislation, known as bills, which, after concurrence by the Senate, are sent to the President for consideration. The presiding officer is the Speaker of the House, who is elected by the members thereof and is traditionally the leader of the controlling party. He or she and other leaders are chosen by the Democratic Caucus or the Republican Conferences. The House meets in the wing of the United States Capitol. Under the Articles of Confederation, the Congress of the Confederation was a body in which each state was equally represented. All states except Rhode Island agreed to send delegates, the issue of how to structure Congress was one of the most divisive among the founders during the Convention. The House is referred to as the house, with the Senate being the upper house. Both houses approval is necessary for the passage of legislation, the Virginia Plan drew the support of delegates from large states such as Virginia, Massachusetts, and Pennsylvania, as it called for representation based on population. The smaller states, however, favored the New Jersey Plan, the Constitution was ratified by the requisite number of states in 1788, but its implementation was set for March 4,1789. The House began work on April 1,1789, when it achieved a quorum for the first time, during the first half of the 19th century, the House was frequently in conflict with the Senate over regionally divisive issues, including slavery. The North was much more populous than the South, and therefore dominated the House of Representatives, However, the North held no such advantage in the Senate, where the equal representation of states prevailed. Regional conflict was most pronounced over the issue of slavery, One example of a provision repeatedly supported by the House but blocked by the Senate was the Wilmot Proviso, which sought to ban slavery in the land gained during the Mexican–American War. Conflict over slavery and other issues persisted until the Civil War, the war culminated in the Souths defeat and in the abolition of slavery. Because all southern senators except Andrew Johnson resigned their seats at the beginning of the war, the years of Reconstruction that followed witnessed large majorities for the Republican Party, which many Americans associated with the Unions victory in the Civil War and the ending of slavery. The Reconstruction period ended in about 1877, the ensuing era, the Democratic and the Republican Party held majorities in the House at various times. The late 19th and early 20th centuries also saw an increase in the power of the Speaker of the House
U.S. Representative
–
United States House of Representatives
U.S. Representative
–
Seal of the House
U.S. Representative
–
Republican
Thomas Brackett Reed, occasionally ridiculed as "Czar Reed", was a U.S. Representative from
Maine, and
Speaker of the House from 1889 to 1891 and from 1895 to 1899.
U.S. Representative
–
House Speaker
Nancy Pelosi, Majority Leader
Steny Hoyer, and Education and Labor Committee Chairman
George Miller confer with President
Barack Obama at the
Oval Office in 2009.
59.
Calculus
–
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
Calculus
–
Isaac Newton developed the use of calculus in his
laws of motion and
gravitation.
Calculus
–
Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
–
Maria Gaetana Agnesi
Calculus
–
The
logarithmic spiral of the
Nautilus shell is a classical image used to depict the growth and change related to calculus
60.
Differential equation
–
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
Differential equation
–
Navier–Stokes differential equations used to simulate airflow around an obstruction.
61.
Pythagorean triple
–
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 triple
–
The Pythagorean theorem: a 2 + b 2 = c 2
62.
Spiral of Theodorus
–
In geometry, the spiral of Theodorus is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene, the spiral is started with an isosceles right triangle, with each leg having unit length. The process then repeats, the i th triangle in the sequence is a triangle with side lengths √i and 1. For example, the 16th triangle has sides measuring 4,1 and hypotenuse of √17 Although all of Theodorus work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells of his work. It is assumed that Theodorus had proved that all of the roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus. Plato does not attribute the irrationality of the root of 2 to Theodorus. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories, each of the triangles hypotenuses hi gives the square root of the corresponding natural number, with h1 = √2. Plato, tutored by Theodorus, questioned why Theodorus stopped at √17, the reason is commonly believed to be that the √17 hypotenuse belongs to the last triangle that does not overlap the figure. In 1958, Erich Teuffel proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued, also, if the sides of unit length are extended into a line, they will never pass through any of the other vertices of the total figure. Theodorus stopped his spiral at the triangle with a hypotenuse of √17, if the spiral is continued to infinitely many triangles, many more interesting characteristics are found. If φn is the angle of the nth triangle, then, therefore, the growth of the angle φn of the next triangle n is, φ n = arctan . The sum of the angles of the first k triangles is called the total angle φ for the kth triangle. It grows proportionally to the root of k, with a bounded correction term c2. The growth of the radius of the spiral at a certain triangle n is Δ r = n +1 − n, the Spiral of Theodorus approximates the Archimedean spiral. An alternative derivation is given in, some have suggested a different interpolant which connects the spiral and an alternative inner spiral, as in
Spiral of Theodorus
–
The spiral of Theodorus up to the triangle with a hypotenuse of √17
63.
Hippasus
–
Hippasus of Metapontum, was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, the discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer. Some modern scholars though have suggested that he discovered the irrationality of √2, little is known about the life of Hippasus. He may have lived in the late 5th century BC, about a century after the time of Pythagoras, Metapontum in Italy is usually referred to as his birthplace, although according to Iamblichus some claim Metapontum to be his birthplace, while others the nearby city of Croton. Hippasus is recorded under the city of Sybaris in Iamblichus list of each citys Pythagoreans, 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. Diogenes Laërtius tells us that Hippasus believed that there is a time which the changes in the universe take to complete. According to one statement, Hippasus left no writings, according to another he was the author of the Mystic Discourse, written to bring Pythagoras into disrepute. A scholium on Platos Phaedo notes him as an experimenter in music theory, claiming that he made use of bronze disks to discover the fundamental musical ratios,4,3,3,2. Hippasus is sometimes credited with the discovery of the existence of irrational numbers, Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is confused, pappus merely says that the knowledge of irrational numbers originated in the Pythagorean school, and that the member who first divulged the secret perished by drowning. Iamblichus gives a series of inconsistent reports, Iamblichus clearly states that the drowning at sea was a punishment from the gods for impious behaviour. These stories are taken together to ascribe the discovery of irrationals to Hippasus. In principle, the stories can be combined, since it is possible to discover irrational numbers when constructing dodecahedrons, irrationality, by infinite reciprocal subtraction, can be easily seen in the Golden ratio of the regular pentagon. Some modern scholars prefer to credit Hippasus with the discovery of the irrationality of √2. Plato in his Theaetetus, describes how Theodorus of Cyrene proved the irrationality of √3, √5, etc. up to √17, in the hands of modern writers this combination of vague ancient reports and modern guesswork has sometimes evolved into a much more emphatic and colourful tale
Hippasus
–
Hippasus of Metapontum
64.
Complex number
–
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
Complex number
–
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an
Argand diagram, representing the
complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the
imaginary unit which satisfies i 2 = −1.
65.
Absolute value
–
In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English, the notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot be directly generalised for a complex number
Absolute value
–
The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its
complex conjugate z have the same absolute value.
66.
Complex plane
–
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the part of a complex number represented by a displacement along the x-axis. The concept of the plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, in particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is known as the Argand plane. 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 this customary notation the number z corresponds to the point in the Cartesian plane. In the Cartesian plane the point can also be represented in coordinates as = =. In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2, and some care must be taken to define the real arctangent function for points when x ≤0. Here |z| is the value or modulus of the complex number z, θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π. Notice that without the constraint on the range of θ, the argument of z is multi-valued, because the exponential function is periodic. Thus, if θ is one value of arg, the values are given by arg = θ + 2nπ. The theory of contour integration comprises a part of complex analysis. In this context the direction of travel around a curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the direction is counterclockwise. Almost all of complex analysis is concerned with complex functions – that is, here it is customary to speak of the domain of f as lying in the z-plane, while referring to the range or image of f as a set of points in the w-plane. In symbols we write z = x + i y, f = w = u + i v and it can be useful to think of the complex plane as if it occupied the surface of a sphere. We can establish a correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows
Complex plane
–
Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z.
67.
Cartesian coordinates
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. 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. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
Cartesian coordinates
–
The
right hand rule.
Cartesian coordinates
–
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinates
–
3D Cartesian Coordinate Handedness
68.
Euclidean space
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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
69.
Polar coordinates
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The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, the concepts of angle and radius were already used by ancient peoples of the first millennium BC. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle, the Greek work, however, did not extend to a full coordinate system. From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca —and its distance—from any location on the Earth, from the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. There are various accounts of the introduction of polar coordinates as part of a coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidges Origin of Polar Coordinates, grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs, in Method of Fluxions, Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the Seventh Manner, For Spirals, and nine other coordinate systems. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole, Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoullis work extended to finding the radius of curvature of curves expressed in these coordinates, the actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacocks 1816 translation of Lacroixs Differential and Integral Calculus, alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them. The radial coordinate is often denoted by r or ρ, the angular coordinate is specified as ϕ by ISO standard 31-11. Angles in polar notation are generally expressed in degrees or radians. Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics, in many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis. In mathematical literature, the axis is often drawn horizontal. Adding any number of turns to the angular coordinate does not change the corresponding direction. Also, a radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the point can be expressed with an infinite number of different polar coordinates or
Polar coordinates
–
Hipparchus
Polar coordinates
–
Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).
Polar coordinates
–
A
planimeter, which mechanically computes polar integrals
70.
Curvilinear coordinates
–
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point and this means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space are Cartesian, cylindrical and spherical polar coordinates. A Cartesian coordinate surface in space is a coordinate plane. In the same space, the coordinate surface r =1 in spherical coordinates is the surface of a unit sphere. The formalism of curvilinear coordinates provides a unified and general description of the coordinate systems. Curvilinear coordinates are used to define the location or distribution of physical quantities which may be, for example, scalars, vectors. Such expressions then become valid for any curvilinear coordinate system, depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a problem with spherical symmetry defined in R3 is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Equations with boundary conditions that follow coordinate surfaces for a particular coordinate system may be easier to solve in that system. One would for instance describe the motion of a particle in a box in Cartesian coordinates. Spherical coordinates are one of the most used curvilinear coordinate systems in fields as Earth sciences, cartography, and physics. A point P in 3d space can be defined using Cartesian coordinates, by r = x e x + y e y + z e z and it can also be defined by its curvilinear coordinates if this triplet of numbers defines a single point in an unambiguous way. The coordinate axes are determined by the tangents to the curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates. Applying the same derivatives to the curvilinear system locally at point P defines the basis vectors. Such a basis, whose vectors change their direction and/or magnitude from point to point is called a local basis, all bases associated with curvilinear coordinates are necessarily local. Basis vectors that are the same at all points are global bases, note, for this article e is reserved for the standard basis and h or b is for the curvilinear basis
Curvilinear coordinates
–
Curvilinear,
affine, and
Cartesian coordinates in two-dimensional space
71.
Pythagorean trigonometric identity
–
The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the formulae, it is one of the basic relations between the sine and cosine functions. The identity is given by the formula, sin 2 θ + cos 2 θ =1 and this relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. If the length of the hypotenuse of a triangle is 1. Therefore, this trigonometric identity follows from the Pythagorean theorem, thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ. Alternatively, the identities found at Trigonometric symmetry, shifts, by the periodicity identities we can say if the formula is true for −π < θ ≤ π then it is true for all real θ. Next we prove the range π/2 < θ ≤ π, to do this we let t = θ − π/2, t will now be in the range 0 < t ≤ π/2. All that remains is to prove it for −π < θ <0, the identities 1 + tan 2 θ = sec 2 θ and 1 + cot 2 θ = csc 2 θ are also called Pythagorean trigonometric identities. If one leg of a triangle has length 1, then the tangent of the angle adjacent to that leg is the length of the other leg. Tan θ = b a, and, sec θ = c a, in this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 has cotangent equal to the length of the other leg, in that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem. The following table gives the identities with the factor or divisor that relates them to the main identity, the unit circle centered at the origin in the Euclidean plane is defined by the equation, x 2 + y 2 =1. Consequently, from the equation for the circle, cos 2 θ + sin 2 θ =1. In the figure, the point P has a negative x-coordinate, and is given by x = cosθ. Point P has a positive y-coordinate, and sinθ = sin >0, as θ increases from zero to the full circle θ = 2π, the sine and cosine change signs in the various quadrants to keep x and y with the correct signs. The figure shows how the sign of the function varies as the angle changes quadrant. Because the x- and y-axes are perpendicular, this Pythagorean identity is equivalent to the Pythagorean theorem for triangles with hypotenuse of length 1. See unit circle for a short explanation, the trigonometric functions may also be defined using power series, namely, sin x = ∑ n =0 ∞ n
Pythagorean trigonometric identity
–
Similar right triangles showing sine and cosine of angle θ
72.
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.
73.
Cross product
–
In mathematics and vector algebra, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. Given two linearly independent vectors a and b, the product, a × b, is a vector that is perpendicular to both a and b and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and it should not be confused with dot product. If two vectors have the direction or if either one has zero length, then their cross product is zero. The cross product is anticommutative and is distributive over addition, the space R3 together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space, but if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. If one adds the further requirement that the product be uniquely defined, the cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used, if the vectors a and b are parallel, by the above formula, the cross product of a and b is the zero vector 0. Then, the n is coming out of the thumb. Using this rule implies that the cross-product is anti-commutative, i. e. b × a = −. By pointing the forefinger toward b first, and then pointing the finger toward a. Using the cross product requires the handedness of the system to be taken into account. If a left-handed coordinate system is used, the direction of the n is given by the left-hand rule. This, however, creates a problem because transforming from one arbitrary reference system to another, the problem is clarified by realizing that the cross product of two vectors is not a vector, but rather a pseudovector. See cross product and handedness for more detail, in 1881, Josiah Willard Gibbs, and independently Oliver Heaviside, introduced both the dot product and the cross product using a period and an x, respectively, to denote them. These alternative names are widely used in the literature. Both the cross notation and the cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of a and b. Conversely, a dot product a ⋅ b involves multiplications between corresponding components of a and b, as explained below, the cross product can be expressed in the form of a determinant of a special 3 ×3 matrix
Cross product
–
The cross-product in respect to a right-handed coordinate system
74.
Seven-dimensional cross product
–
In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in R7 a vector a × b also in R7, like the cross product in three dimensions, the seven-dimensional product is anticommutative and a × b is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the cross product is unique up to a sign. The seven-dimensional cross product has the relationship to the octonions as the three-dimensional product does to the quaternions. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results, the product can be given by a multiplication table, such as the one here. This table, due to Cayley, gives the product of basis vectors ei, for example, from the table e 1 × e 2 = e 3 = − e 2 × e 1 The table can be used to calculate the product of any two vectors. This can be repeated for the six components. There are 480 such tables, one for each of the products satisfying the definition, the top left 3 ×3 corner of this table gives the cross product in three dimensions. The first property states that the product is perpendicular to its arguments, a third statement of the magnitude condition is | x × y | = | x | | y | if =0. Given the properties of bilinearity, orthogonality and magnitude, a cross product exists only in three and seven dimensions. This can be shown by postulating the properties required for the cross product, in zero dimensions there is only the zero vector, while in one dimension all vectors are parallel, so in both these cases the product must be identically zero. The restriction to 0,1,3 and 7 dimensions is related to Hurwitzs theorem, in contrast the three-dimensional cross product, which is unique, there are many possible binary cross products in seven dimensions. Unlike in three dimensions, x × y = a × b does not imply that a and b lie in the plane as x and y. One possible multiplication table is described in the Example section, unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product. Once we have established a multiplication table, it is applied to general vectors x and y by expressing x and y in terms of the basis. More compactly this rule can be written as e i × e i +1 = e i +3 with i =1.7 modulo 7, together with anticommutativity this generates the product. This rule directly produces the two immediately adjacent to the diagonal of zeros in the table. Also, from an identity in the subsection on consequences, e i × = − e i +1 = e i × e i +3, which produces diagonals further out, and so on
Seven-dimensional cross product
–
Fano planes for the two multiplication tables used here.
75.
Hippocrates of Chios
–
Hippocrates of Chios was an ancient Greek mathematician, geometer, and astronomer, who lived c.470 – c.410 BCE. He was born on the isle of Chios, where he originally was a merchant, after some misadventures he went to Athens, possibly for litigation. There he grew into a leading mathematician, on Chios, Hippocrates may have been a pupil of the mathematician and astronomer Oenopides of Chios. The reductio ad absurdum argument has been traced to him, only a single, and famous, fragment of Hippocrates Elements is existent, embedded in the work of Simplicius. In this fragment the area is calculated of some so-called Hippocratic lunes — see Lune of Hippocrates. This was part of a programme to achieve the quadrature of the circle. The strategy apparently was to divide a circle into a number of crescent-shaped parts, if it were possible to calculate the area of each of those parts, then the area of the circle as a whole would be known too. Only much later was it proven that this approach had no chance of success, the number π is the ratio of the circumference to the diameter of a circle, and also the ratio of the area to the square of the radius. In the century after Hippocrates at least four other mathematicians wrote their own Elements, steadily improving terminology, in this way Hippocrates pioneering work laid the foundation for Euclids Elements that was to remain the standard geometry textbook for many centuries. Hippocrates is believed to have originated the use of letters to refer to the points and figures in a proposition, e. g. triangle ABC for a triangle with vertices at points A, B. Two other contributions by Hippocrates in the field of mathematics are noteworthy and he found a way to tackle the problem of duplication of the cube, that is, the problem of how to construct a cube root. Like the quadrature of the circle this was another of the three great mathematical problems of Antiquity. Hippocrates also invented the technique of reduction, that is, to transform specific mathematical problems into a general problem that is more easy to solve. The solution to the general problem then automatically gives a solution to the original problem. In the field of astronomy Hippocrates tried to explain the phenomena of comets, ivor Bulmer-Thomas, Hippocrates of Chios, in, Dictionary of Scientific Biography, Charles Coulston Gillispie, ed. pp. 410–418. Björnbo, Hippokrates, in, Paulys Realencyclopädie der Classischen Altertumswissenschaft, G. Wissowa, oConnor, John J. Robertson, Edmund F. Hippocrates of Chios, MacTutor History of Mathematics archive, University of St Andrews. The Quadrature of the Circle and Hippocrates Lunes at Convergence
Hippocrates of Chios
–
The
Lune of Hippocrates. Partial solution of the "
Squaring the circle " task, suggested by Hippocrates. The area of the shaded figure is equal to the area of the triangle ABC. This is not a complete solution of the task (the complete solution is proven to be impossible with
compass and straightedge).
76.
Inner product space
–
In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a quantity known as the inner product of the vectors. Inner products allow the introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors, inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. An inner product induces a associated norm, thus an inner product space is also a normed vector space. A complete space with a product is called a Hilbert space. An space with a product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C, formally, an inner product space is a vector space V over the field F together with an inner product, i. e. Some authors, especially in physics and matrix algebra, prefer to define the inner product, then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product ⟨ x, y ⟩ as ⟨ y | x ⟩, respectively y † x. Here the kets and columns are identified with the vectors of V and this reverse order is now occasionally followed in the more abstract literature, taking ⟨ x, y ⟩ to be conjugate linear in x rather than y. A few instead find a ground by recognizing both ⟨ ⋅, ⋅ ⟩ and ⟨ ⋅ | ⋅ ⟩ as distinct notations differing only in which argument is conjugate linear. There are various reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered subfield in order for non-negativity to make sense, the basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, however in these cases when it is a proper subfield even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as used in quantum computation, are automatically metrically complete. In some cases we need to consider non-negative semi-definite sesquilinear forms and this means that ⟨ x, x ⟩ is only required to be non-negative
Inner product space
–
Geometric interpretation of the angle between two vectors defined using an inner product
77.
Cartesian coordinate system
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. 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. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
Cartesian coordinate system
–
The
right hand rule.
Cartesian coordinate system
–
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinate system
–
3D Cartesian Coordinate Handedness
78.
Spherical law of cosines
–
In spherical trigonometry, the law of cosines is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a triangle on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere. Since this is a sphere, the lengths a, b. As a special case, for C = π/2, then cos C =0, if the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the formulation of the law of haversines is preferable. It can be obtained from consideration of a spherical triangle dual to the given one, a proof of the law of cosines can be constructed as follows. Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. Then, the angle C is given by, cos C = t a ⋅ t b = cos c − cos a cos b sin a sin b from which the law of cosines immediately follows. To the diagram above, add a plane tangent to the sphere at u and we then have two plane triangles with a side in common, the triangle containing u, y and z and the one containing O, y and z. Sides of the first triangle are tan a and tan b, so − tan a tan b cos C =1 − sec a sec b cos c Multiply both sides by cos a cos b and rearrange. The angles and distances do not change if the sphere is rotated, so we can rotate the sphere so that u is at the north pole, with this rotation, the spherical coordinates for v are = and the spherical coordinates for w are =. The Cartesian coordinates for v are = and the Cartesian coordinates for w are =
Spherical law of cosines
–
Spherical triangle solved by the law of cosines.
79.
Big O notation
–
Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, in computer science, big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows. Big O notation characterizes functions according to their rates, different functions with the same growth rate may be represented using the same O notation. The letter O is used because the rate of a function is also referred to as order of the function. A description of a function in terms of big O notation usually only provides a bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, Big O notation is also used in many other fields to provide similar estimates. Let f and g be two functions defined on some subset of the real numbers. That is, f = O if and only if there exists a real number M. In many contexts, the assumption that we are interested in the rate as the variable x goes to infinity is left unstated. If f is a product of several factors, any constants can be omitted, for example, let f = 6x4 − 2x3 +5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms, 6x4, −2x3, and 5, of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the rule, 6x4 is a product of 6. Omitting this factor results in the simplified form x4, thus, we say that f is a big-oh of. Mathematically, we can write f = O, one may confirm this calculation using the formal definition, let f = 6x4 − 2x3 +5 and g = x4. Applying the formal definition from above, the statement that f = O is equivalent to its expansion, | f | ≤ M | x 4 | for some choice of x0 and M. To prove this, let x0 =1 and M =13, Big O notation has two main areas of application. In mathematics, it is used to describe how closely a finite series approximates a given function. In computer science, it is useful in the analysis of algorithms, in both applications, the function g appearing within the O is typically chosen to be as simple as possible, omitting constant factors and lower order terms
Big O notation
–
Example of Big O notation: f (x) ∈ O(g (x)) as there exists c > 0 (e.g., c = 1) and x 0 (e.g., x 0 = 5) such that f (x) < c g (x) whenever x > x 0.
80.
Loss of significance
–
Loss of significance is an undesirable effect in calculations using finite-precision arithmetic. It occurs when an operation on two numbers increases relative error substantially more than it increases absolute error, for example in subtracting two nearly equal numbers, the effect is that the number of significant digits in the result is reduced unacceptably. Ways to avoid this effect are studied in numerical analysis, the effect can be demonstrated with decimal numbers. It is very different when measured in order of precision, the first is accurate to 6981099999999999999♠10×10−20, while the second is only accurate to 6991100000000000000♠10×10−10. In the second case, the answer seems to have one significant digit, the way to indicate this and represent the answer to 10 significant figures is, 6990100000000000000♠1. Furthermore, it usually only postpones the problem, What if the data is accurate to ten digits. One of the most important parts of analysis is to avoid or minimize loss of significance in calculations. If the underlying problem is well-posed, there should be an algorithm for solving it. Let x and y be positive normalized floating point numbers, in the subtraction x − y, r significant bits are lost where q ≤ r ≤ p 2 − p ≤1 − y x ≤2 − q for some positive integers p and q. For example, consider the equation, a x 2 + b x + c =0. This formula may not always produce an accurate result, for example, when c is very small, loss of significance can occur in either of the root calculations, depending on the sign of b. The case a =1, b =200, c = −0.000015 will serve to illustrate the problem, x 2 +200 x −0.000015 =0. We have b 2 −4 a c =2002 +4 ×1 ×0.000015 =200.00000015 … In real arithmetic, in 10-digit floating-point arithmetic, /2 = −200.00000005, /2 =0.00000005. Notice that the solution of greater magnitude is accurate to ten digits, because of the subtraction that occurs in the quadratic equation, it does not constitute a stable algorithm to calculate the two roots. A careful floating point computer implementation combines several strategies to produce a robust result, here sgn denotes the sign function, where sgn is 1 if b is positive and −1 if b is negative. This avoids cancellation problems between b and the root of the discriminant by ensuring that only numbers of the same sign are added. To illustrate the instability of the quadratic formula versus this variant formula. The discriminant b 2 −4 a c needs to be computed in arithmetic of twice the precision of the result to avoid this and this can be in the form of a fused multiply-add operation
Loss of significance
–
Example of LOS in case of computing 2 forms of the same function
81.
Hyperbolic function
–
In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The inverse hyperbolic functions are the hyperbolic sine arsinh and so on. Just as the form a circle with a unit radius. The hyperbolic functions take a real argument called a hyperbolic angle, the size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a triangle covering this sector. Laplaces equations are important in areas of physics, including electromagnetic theory, heat transfer, fluid dynamics. In complex analysis, the hyperbolic functions arise as the parts of sine and cosine. When considered defined by a variable, the hyperbolic functions are rational functions of exponentials. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati, Riccati used Sc. and Cc. to refer to circular functions and Sh. and Ch. to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today, the abbreviations sh and ch are still used in some other languages, like French and Russian. The hyperbolic functions are, Hyperbolic sine, sinh x = e x − e − x 2 = e 2 x −12 e x =1 − e −2 x 2 e − x. Hyperbolic cosine, cosh x = e x + e − x 2 = e 2 x +12 e x =1 + e −2 x 2 e − x, the complex forms in the definitions above derive from Eulers formula. One also has sech 2 x =1 − tanh 2 x csch 2 x = coth 2 x −1 for the other functions, sinh = sinh 2 = sgn cosh −12 where sgn is the sign function. All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x and e − x, and it is possible to express the above functions as Taylor series, sinh x = x + x 33. + ⋯ = ∑ n =0 ∞ x 2 n +1, the function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh, the function cosh x has a Taylor series expression with only even exponents for x. Thus it is a function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the series expression of the exponential function
Hyperbolic function
–
Hyperbolic functions in the complex plane
Hyperbolic function
–
A ray through the
unit hyperbola in the point, where is twice the area between the ray, the hyperbola, and the -axis. For points on the hyperbola below the -axis, the area is considered negative (see
animated version with comparison with the trigonometric (circular) functions).
Hyperbolic function
Hyperbolic function
82.
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.
83.
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
84.
Algebra
–
Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
Algebra
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A page from
Al-Khwārizmī 's
al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Algebra
–
Italian mathematician
Girolamo Cardano published the solutions to the
cubic and
quartic equations in his 1545 book
Ars magna.
85.
Middle Kingdom of Egypt
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Some scholars also include the Thirteenth Dynasty of Egypt wholly into this period as well, in which case the Middle Kingdom would finish c. 1650, while others only include it until Merneferre Ay c.1700 BC, during the Middle Kingdom period, Osiris became the most important deity in popular religion. The period comprises two phases, the 11th Dynasty, which ruled from Thebes and the 12th Dynasty onwards which was centered on el-Lisht, after the collapse of the Old Kingdom, Egypt entered a period of weak Pharaonic power and decentralization called the First Intermediate Period. Towards the end of period, two rival dynasties, known in Egyptology as the Tenth and Eleventh, fought for power over the entire country. The Theban 11th Dynasty only ruled southern Egypt from the first cataract to the Tenth Nome of Upper Egypt, to the north, Lower Egypt was ruled by the rival 10th Dynasty from Herakleopolis. The struggle was to be concluded by Mentuhotep II, who ascended the Theban throne in 2055 B. C, during Mentuhotep IIs fourteenth regnal year, he took advantage of a revolt in the Thinite Nome to launch an attack on Herakleopolis, which met little resistance. After toppling the last rulers of the 10th Dynasty, Mentuhotep began consolidating his power over all Egypt, for this reason, Mentuhotep II is regarded as the founder of the Middle Kingdom. Mentuhotep II commanded military campaigns south as far as the Second Cataract in Nubia and he also restored Egyptian hegemony over the Sinai region, which had been lost to Egypt since the end of the Old Kingdom. He also sent the first expedition to Punt during the Middle Kingdom, by means of ships constructed at the end of Wadi Hammamat, Mentuhotep III was succeeded by Mentuhotep IV, whose name significantly is omitted from all ancient Egyptian king lists. The Turin Papyrus claims that after Mentuhotep III came seven kingless years, despite this absence, his reign is attested from a few inscriptions in Wadi Hammamat that record expeditions to the Red Sea coast and to quarry stone for the royal monuments. The leader of expedition was his vizier Amenemhat, who is widely assumed to be the future pharaoh Amenemhet I. Mentuhotep IVs absence from the king lists has prompted the theory that Amenemhet I usurped his throne, while there are no contemporary accounts of this struggle, certain circumstantial evidence may point to the existence of a civil war at the end of the 11th dynasty. Inscriptions left by one Nehry, the Haty-a of Hermopolis, suggest that he was attacked at a place called Shedyet-sha by the forces of the reigning king, but his forces prevailed. Khnumhotep I, an official under Amenemhet I, claims to have participated in a flotilla of 20 ships to pacify Upper Egypt, donald Redford has suggested these events should be interpreted as evidence of open war between two dynastic claimants. What is certain is that, however he came to power, from the 12th dynasty onwards, pharaohs often kept well-trained standing armies, which included Nubian contingents. These formed the basis of larger forces which were raised for defence against invasion, however, the Middle Kingdom was basically defensive in its military strategy, with fortifications built at the First Cataract of the Nile, in the Delta and across the Sinai Isthmus. Early in his reign, Amenemhet I was compelled to campaign in the Delta region, in addition, he strengthened defenses between Egypt and Asia, building the Walls of the Ruler in the East Delta region. Perhaps in response to this perpetual unrest, Amenemhat I built a new capital for Egypt in the north, known as Amenemhet Itj Tawy, or Amenemhet, the location of this capital is unknown, but is presumably near the citys necropolis, the present-day el-Lisht
Middle Kingdom of Egypt
–
An
Osiride statue of the first pharaoh of the Middle Kingdom, Mentuhotep II
Middle Kingdom of Egypt
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The head of a statue of Senusret I.
Middle Kingdom of Egypt
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Statue head of Senusret III
86.
India
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India, officially the Republic of India, is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and it is bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast. It shares land borders with Pakistan to the west, China, Nepal, and Bhutan to the northeast, in the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives. Indias Andaman and Nicobar Islands share a border with Thailand. The Indian subcontinent was home to the urban Indus Valley Civilisation of the 3rd millennium BCE, in the following millennium, the oldest scriptures associated with Hinduism began to be composed. Social stratification, based on caste, emerged in the first millennium BCE, early political consolidations took place under the Maurya and Gupta empires, the later peninsular Middle Kingdoms influenced cultures as far as southeast Asia. In the medieval era, Judaism, Zoroastrianism, Christianity, and Islam arrived, much of the north fell to the Delhi sultanate, the south was united under the Vijayanagara Empire. The economy expanded in the 17th century in the Mughal empire, in the mid-18th century, the subcontinent came under British East India Company rule, and in the mid-19th under British crown rule. A nationalist movement emerged in the late 19th century, which later, under Mahatma Gandhi, was noted for nonviolent resistance, in 2015, the Indian economy was the worlds seventh largest by nominal GDP and third largest by purchasing power parity. Following market-based economic reforms in 1991, India became one of the major economies and is considered a newly industrialised country. However, it continues to face the challenges of poverty, corruption, malnutrition, a nuclear weapons state and regional power, it has the third largest standing army in the world and ranks sixth in military expenditure among nations. India is a constitutional republic governed under a parliamentary system. It is a pluralistic, multilingual and multi-ethnic society and is home to a diversity of wildlife in a variety of protected habitats. The name India is derived from Indus, which originates from the Old Persian word Hindu, the latter term stems from the Sanskrit word Sindhu, which was the historical local appellation for the Indus River. The ancient Greeks referred to the Indians as Indoi, which translates as The people of the Indus, the geographical term Bharat, which is recognised by the Constitution of India as an official name for the country, is used by many Indian languages in its variations. Scholars believe it to be named after the Vedic tribe of Bharatas in the second millennium B. C. E and it is also traditionally associated with the rule of the legendary emperor Bharata. Gaṇarājya is the Sanskrit/Hindi term for republic dating back to the ancient times, hindustan is a Persian name for India dating back to the 3rd century B. C. E. It was introduced into India by the Mughals and widely used since then and its meaning varied, referring to a region that encompassed northern India and Pakistan or India in its entirety
India
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Flag
India
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The granite
tower of
Brihadeeswarar Temple in
Thanjavur was completed in 1010 CE by
Raja Raja Chola I.
India
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Writing the will and testament of the Mughal king court in Persian, 1590–1595
87.
Isosceles
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In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the isosceles triangle theorem, the two angles opposite the sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles, in an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle, the vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used. A triangle with two equal sides has exactly one axis of symmetry, which goes through the vertex angle. Thus the axis of symmetry coincides with the bisector of the vertex angle, the median drawn to the base, the altitude drawn from the vertex angle. Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle, in Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. The Euler line of any triangle goes through the orthocenter, its centroid. In an isosceles triangle with two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows, if the vertex angle is acute, then the orthocenter, the centroid, and the circumcenter all fall inside the triangle. In an isosceles triangle the incenter lies on the Euler line, the Steiner inellipse of any triangle is the unique ellipse that is internally tangent to the triangles three sides at their midpoints. For any isosceles triangle with area T and perimeter p, we have 2 p b 3 − p 2 b 2 +16 T2 =0. By substituting the height, the formula for the area of a triangle can be derived from the general formula one-half the base times the height. This is what Herons formula reduces to in the isosceles case, if the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is, T =2 = a 2 sin cos . This is derived by drawing a line from the base of the triangle. The bases of two right triangles are both equal to the hypotenuse times the sine of the bisected angle by definition of the term sine. For the same reason, the heights of these triangles are equal to the times the cosine of the bisected angle
Isosceles
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Isosceles triangle with vertical axis of symmetry
88.
Zhoubi Suanjing
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The Zhoubi Suanjing, or Chou Pei Suan Ching, is one of the oldest Chinese mathematical texts. Zhou refers to the ancient dynasty Zhou c, 1046-771 BCE, Bi means thigh and according to the book, it refers to the gnomon of the sundial. The book is dedicated to astronomical observation and calculation, Suan Jing or classic of arithmetic were appended in later time to honor the achievement of the book in mathematics. This book dates from the period of the Zhou Dynasty, yet its compilation and addition of materials continued into the Han Dynasty and it is an anonymous collection of 246 problems encountered by the Duke of Zhou and his astronomer and mathematician, Shang Gao. Each question has stated their numerical answer and corresponding arithmetic algorithm and this book contains one of the first recorded proofs of the Pythagorean Theorem. Commentators such as Liu Hui, Zu Geng, Li Chunfeng and Yang Hui have expanded on this text, tsinghua Bamboo Slips Boyer, Carl B. A History of Mathematics, John Wiley & Sons, Inc, full text of the Zhoubi Suanjing, including diagrams - Chinese Text Project. Full text of the Zhoubi Suanjing, at Project Gutenberg Christopher Cullen, astronomy and Mathematics in Ancient China, The Zhou Bi Suan Jing, Cambridge University Press,2007
Zhoubi Suanjing
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Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing
Zhoubi Suanjing
–
History
89.
China
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China, officially the Peoples Republic of China, is a unitary sovereign state in East Asia and the worlds most populous country, with a population of over 1.381 billion. The state is governed by the Communist Party of China and its capital is Beijing, the countrys major urban areas include Shanghai, Guangzhou, Beijing, Chongqing, Shenzhen, Tianjin and Hong Kong. China is a power and a major regional power within Asia. Chinas landscape is vast and diverse, ranging from forest steppes, the Himalaya, Karakoram, Pamir and Tian Shan mountain ranges separate China from much of South and Central Asia. The Yangtze and Yellow Rivers, the third and sixth longest in the world, respectively, Chinas coastline along the Pacific Ocean is 14,500 kilometers long and is bounded by the Bohai, Yellow, East China and South China seas. China emerged as one of the worlds earliest civilizations in the basin of the Yellow River in the North China Plain. For millennia, Chinas political system was based on hereditary monarchies known as dynasties, in 1912, the Republic of China replaced the last dynasty and ruled the Chinese mainland until 1949, when it was defeated by the communist Peoples Liberation Army in the Chinese Civil War. The Communist Party established the Peoples Republic of China in Beijing on 1 October 1949, both the ROC and PRC continue to claim to be the legitimate government of all China, though the latter has more recognition in the world and controls more territory. China had the largest economy in the world for much of the last two years, during which it has seen cycles of prosperity and decline. Since the introduction of reforms in 1978, China has become one of the worlds fastest-growing major economies. As of 2016, it is the worlds second-largest economy by nominal GDP, China is also the worlds largest exporter and second-largest importer of goods. China is a nuclear weapons state and has the worlds largest standing army. The PRC is a member of the United Nations, as it replaced the ROC as a permanent member of the U. N. Security Council in 1971. China is also a member of numerous formal and informal multilateral organizations, including the WTO, APEC, BRICS, the Shanghai Cooperation Organization, the BCIM, the English name China is first attested in Richard Edens 1555 translation of the 1516 journal of the Portuguese explorer Duarte Barbosa. The demonym, that is, the name for the people, Portuguese China is thought to derive from Persian Chīn, and perhaps ultimately from Sanskrit Cīna. Cīna was first used in early Hindu scripture, including the Mahābhārata, there are, however, other suggestions for the derivation of China. The official name of the state is the Peoples Republic of China. The shorter form is China Zhōngguó, from zhōng and guó and it was then applied to the area around Luoyi during the Eastern Zhou and then to Chinas Central Plain before being used as an occasional synonym for the state under the Qing
China
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Yinxu, ruins of an ancient
palace dating from the
Shang Dynasty (14th century BCE)
China
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Flag
China
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Some of the thousands of life-size
Terracotta Warriors of the
Qin Dynasty, c. 210 BCE
China
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The
Great Wall of China was built by several dynasties over two thousand years to protect the sedentary agricultural regions of the
Chinese interior from incursions by
nomadic pastoralists of the northern
steppes.
90.
Han Dynasty
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The Han dynasty was the second imperial dynasty of China, preceded by the Qin dynasty and succeeded by the Three Kingdoms period. Spanning over four centuries, the Han period is considered an age in Chinese history. To this day, Chinas majority ethnic group refers to itself as the Han people and it was founded by the rebel leader Liu Bang, known posthumously as Emperor Gaozu of Han, and briefly interrupted by the Xin dynasty of the former regent Wang Mang. This interregnum separates the Han dynasty into two periods, the Western Han or Former Han and the Eastern Han or Later Han, the emperor was at the pinnacle of Han society. He presided over the Han government but shared power with both the nobility and appointed ministers who came largely from the gentry class. The Han Empire was divided into areas controlled by the central government using an innovation inherited from the Qin known as commanderies. These kingdoms gradually lost all vestiges of their independence, particularly following the Rebellion of the Seven States, from the reign of Emperor Wu onward, the Chinese court officially sponsored Confucianism in education and court politics, synthesized with the cosmology of later scholars such as Dong Zhongshu. This policy endured until the fall of the Qing dynasty in 1911 AD, the Han dynasty was an age of economic prosperity and saw a significant growth of the money economy first established during the Zhou dynasty. The coinage issued by the government mint in 119 BC remained the standard coinage of China until the Tang dynasty. The period saw a number of limited institutional innovations, the Xiongnu, a nomadic steppe confederation, defeated the Han in 200 BC and forced the Han to submit as a de facto inferior partner, but continued their raids on the Han borders. Emperor Wu of Han launched several campaigns against them. The ultimate Han victory in these wars eventually forced the Xiongnu to accept vassal status as Han tributaries, the territories north of Hans borders were quickly overrun by the nomadic Xianbei confederation. Imperial authority was seriously challenged by large Daoist religious societies which instigated the Yellow Turban Rebellion. When Cao Pi, King of Wei, usurped the throne from Emperor Xian, following Liu Bangs victory in the Chu–Han Contention, the resulting Han dynasty was named after the Hanzhong fief. Chinas first imperial dynasty was the Qin dynasty, the Qin unified the Chinese Warring States by conquest, but their empire became unstable after the death of the first emperor Qin Shi Huangdi. Within four years, the authority had collapsed in the face of rebellion. Although Xiang Yu proved to be a commander, Liu Bang defeated him at Battle of Gaixia. Liu Bang assumed the title emperor at the urging of his followers and is known posthumously as Emperor Gaozu, Changan was chosen as the new capital of the reunified empire under Han
Han Dynasty
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History of China
Han Dynasty
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Han dynasty in 1 AD.
Han Dynasty
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A
silk banner from
Mawangdui,
Changsha,
Hunan province. It was draped over the coffin of
Lady Dai (d. 168 BC), wife of the Marquess Li Cang (利蒼) (d. 186 BC), chancellor for the Kingdom of Changsha.
Han Dynasty
–
A
gilded bronze
oil lamp in the shape of a kneeling female servant, dated 2nd century BC, found in the tomb of
Dou Wan, wife of the Han prince
Liu Sheng; its sliding shutter allows for adjustments in the direction and brightness in light while it also traps smoke within the body.
91.
The Nine Chapters on the Mathematical Art
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The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving texts from China. Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution, and these were commented on by Liu Hui in the 3rd century. The method of chapter 7 was not found in Europe until the 13th century, there is also the mathematical proof given in the treatise for the Pythagorean theorem. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea and its influence on mathematical thought in China persisted until the Qing Dynasty era. Liu Hui wrote a detailed commentary on this book in 263. Lius commentary is of great mathematical interest in its own right, the Nine Chapters is an anonymous work, and its origins are not clear. This is no longer the case, the Suàn shù shū or writings on reckoning is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with writings in 1983 when archaeologists opened a tomb in Hubei province. It is among the corpus of known as the Zhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have closed in 186 BCE. While its relationship to the Nine Chapters is still under discussion by scholars, the Zhoubi Suanjing, a mathematics and astronomy text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by Cai Yong. Contents of The Nine Chapters are as follows, 方田 Fangtian - Bounding fields, areas of fields of various shapes, manipulation of vulgar fractions. Liu Huis commentary includes a method for calculation of π and the value of 3.14159. 粟米 Sumi - Millet and rice, exchange of commodities at different rates, pricing. Distribution of commodities and money at proportional rates, deriving arithmetic and geometric sums, division by mixed numbers, extraction of square and cube roots, diameter of sphere, perimeter and diameter of circle. 商功 Shanggong - Figuring for construction, volumes of solids of various shapes. 盈不足 Yingbuzu - Excess and deficit, linear problems solved using the principle known later in the West as the rule of false position
The Nine Chapters on the Mathematical Art
–
A page of The Nine Chapters on the Mathematical Art
The Nine Chapters on the Mathematical Art
–
History
92.
Cicero
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Marcus Tullius Cicero was a Roman philosopher, politician, lawyer, orator, political theorist, consul, and constitutionalist. He came from a wealthy family of the Roman equestrian order. According to Michael Grant, the influence of Cicero upon the history of European literature, Cicero introduced the Romans to the chief schools of Greek philosophy and created a Latin philosophical vocabulary distinguishing himself as a translator and philosopher. Though he was an orator and successful lawyer, Cicero believed his political career was his most important achievement. During the chaotic latter half of the 1st century BC marked by civil wars, following Julius Caesars death, Cicero became an enemy of Mark Antony in the ensuing power struggle, attacking him in a series of speeches. His severed hands and head were then, as a revenge of Mark Antony. Petrarchs rediscovery of Ciceros letters is often credited for initiating the 14th-century Renaissance in public affairs, humanism, according to Polish historian Tadeusz Zieliński, the Renaissance was above all things a revival of Cicero, and only after him and through him of the rest of Classical antiquity. Cicero was born in 106 BC in Arpinum, a hill town 100 kilometers southeast of Rome and his father was a well-to-do member of the equestrian order and possessed good connections in Rome. However, being a semi-invalid, he could not enter public life, although little is known about Ciceros mother, Helvia, it was common for the wives of important Roman citizens to be responsible for the management of the household. Ciceros brother Quintus wrote in a letter that she was a thrifty housewife, Ciceros cognomen, or personal surname, comes from the Latin for chickpea, cicer. Plutarch explains that the name was given to one of Ciceros ancestors who had a cleft in the tip of his nose resembling a chickpea. However, it is likely that Ciceros ancestors prospered through the cultivation. Romans often chose down-to-earth personal surnames, the family names of Fabius, Lentulus, and Piso come from the Latin names of beans, lentils. Plutarch writes that Cicero was urged to change this name when he entered politics. During this period in Roman history, cultured meant being able to speak both Latin and Greek, Cicero used his knowledge of Greek to translate many of the theoretical concepts of Greek philosophy into Latin, thus translating Greek philosophical works for a larger audience. It was precisely his broad education that tied him to the traditional Roman elite, according to Plutarch, Cicero was an extremely talented student, whose learning attracted attention from all over Rome, affording him the opportunity to study Roman law under Quintus Mucius Scaevola. Ciceros fellow students were Gaius Marius Minor, Servius Sulpicius Rufus, the latter two became Ciceros friends for life, and Pomponius would become, in Ciceros own words, as a second brother, with both maintaining a lifelong correspondence. Cicero wanted to pursue a career in politics along the steps of the Cursus honorum
Cicero
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A first century AD bust of Cicero in the
Capitoline Museums, Rome
Cicero
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The Young Cicero Reading by
Vincenzo Foppa (fresco, 1464), now at the
Wallace Collection
Cicero
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Cicero Denounces Catiline,
fresco by
Cesare Maccari, 1882–88
Cicero
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Cicero's death (France, 15th century)
93.
John Aubrey
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John Aubrey FRS, was an English antiquary, natural philosopher and writer. He is perhaps best known as the author of the Brief Lives and he was a pioneer archaeologist, who recorded numerous megalithic and other field monuments in southern England, and who is particularly noted as the discoverer of the Avebury henge monument. The Aubrey holes at Stonehenge are named after him, although there is doubt as to whether the holes that he observed are those that currently bear the name. He was also a pioneer folklorist, collecting together a miscellany of material on customs, traditions and he set out to compile county histories of both Wiltshire and Surrey, although both projects remained unfinished. His Interpretation of Villare Anglicanum was the first attempt to compile a full-length study of English place-names and he had wider interests in applied mathematics and astronomy, and was friendly with many of the greatest scientists of the day. For much of the 19th and 20th centuries, thanks largely to the popularity of Brief Lives, Aubrey was regarded as more than an entertaining but quirky, eccentric. Only in the 1970s did the full breadth and innovation of his begin to be more widely appreciated. He published little in his lifetime, and many of his most important manuscripts remain unpublished, or published only in partial and unsatisfactory form. Aubrey was born at Easton Piers or Percy, near Kington St Michael, Wiltshire, to a long-established and his grandfather, Isaac Lyte, lived at Lytes Cary Manor, Somerset, now owned by the National Trust. Richard Aubrey, his father, owned lands in Wiltshire and Herefordshire, for many years an only child, he was educated at home with a private tutor, he was melancholy in his solitude. His father was not intellectual, preferring field sports to learning, Aubrey read such books as came his way, including Bacons Essays, and studied geometry in secret. He was educated at the Malmesbury grammar school under Robert Latimer and he then studied at the grammar school at Blandford Forum, Dorset. He entered Trinity College, Oxford, in 1642, but his studies were interrupted by the English Civil War and his earliest antiquarian work dates from this period in Oxford. In 1646 he became a student of the Middle Temple and he spent a pleasant time at Trinity in 1647, making friends among his Oxford contemporaries, and collecting books. He was to show Avebury to Charles II at the Kings request in 1663 and his father died in 1652, leaving Aubrey large estates, but with them some complicated debts. He claimed that his memory was not tenacious by 17th-century standards, but from the early 1640s he kept notes of observations in natural philosophy, his friends ideas. He also began to write Lives of scientists in the 1650s, in 1659 he was recruited to contribute to a collaborative county history of Wiltshire, leading to his unfinished collections on the antiquities and the natural history of the county. His erstwhile friend and fellow-antiquary Anthony Wood predicted that he would one day break his neck while running downstairs in haste to interview some retreating guest or other and he drank the Kings health in Interregnum Herefordshire, but with equal enthusiasm attended meetings in London of the republican Rota Club
John Aubrey
–
John Aubrey
John Aubrey
–
Part of the southern inner ring at
Avebury
John Aubrey
–
An early photograph of Stonehenge taken July 1877
94.
Thomas Hobbes
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Thomas Hobbes, in some older texts Thomas Hobbes of Malmesbury, was an English philosopher who is considered one of the founders of modern political philosophy. Hobbes is best known for his 1651 book Leviathan, which established the social theory that has served as the foundation for most later Western political philosophy. Thomas Hobbes was born at Westport, now part of Malmesbury in Wiltshire, England, born prematurely when his mother heard of the coming invasion of the Spanish Armada, Hobbes later reported that my mother gave birth to twins, myself and fear. His childhood is almost completely unknown, and his mothers name is unknown and his father, Thomas Sr. was the vicar of Charlton and Westport. Thomas Hobbes, the younger, had a brother Edmund, about two years older, and a sister, Thomas Sr. was involved in a fight with the local clergy outside his church, forcing him to leave London and abandon the family. The family was left in the care of Thomas Sr. s older brother, Francis, Hobbes was a good pupil, and around 1603 he went up to Magdalen Hall, the predecessor college to Hertford College, Oxford. The principal John Wilkinson was a Puritan, and he had influence on Hobbes. At university, Hobbes appears to have followed his own curriculum and he did not complete his B. A. Hobbes became a companion to the younger William and they both took part in a grand tour of Europe in 1610. Hobbes was exposed to European scientific and critical methods during the tour and it has been argued that three of the discourses in the 1620 publication known as Horea Subsecivae, Observations and Discourses, also represent the work of Hobbes from this period. Although he associated with figures like Ben Jonson and briefly worked as Francis Bacons amanuensis. His employer Cavendish, then the Earl of Devonshire, died of the plague in June 1628, the widowed countess dismissed Hobbes but he soon found work, again as a tutor, this time to Gervase Clifton, the son of Sir Gervase Clifton, 1st Baronet. This task, chiefly spent in Paris, ended in 1631 when he found work with the Cavendish family, tutoring William. Over the next seven years, as well as tutoring, he expanded his own knowledge of philosophy and he visited Florence in 1636 and was later a regular debater in philosophic groups in Paris, held together by Marin Mersenne. Hobbess first area of study was an interest in the doctrine of motion. Despite his interest in this phenomenon, he disdained experimental work as in physics and he went on to conceive the system of thought to the elaboration of which he would devote his life. He then singled out Man from the realm of Nature and plants, finally he considered, in his crowning treatise, how Men were moved to enter into society, and argued how this must be regulated if Men were not to fall back into brutishness and misery. Thus he proposed to unite the separate phenomena of Body, Man, Hobbes came home, in 1637, to a country riven with discontent which disrupted him from the orderly execution of his philosophic plan. However, by the end of the Short Parliament in 1640, he had written a treatise called The Elements of Law, Natural
Thomas Hobbes
–
Thomas Hobbes
Thomas Hobbes
Thomas Hobbes
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Frontispiece from De Cive (1642)
Thomas Hobbes
–
Frontispiece of Leviathan
95.
Hans Christian Andersen
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Hans Christian Andersen (/ˈhɑːnz ˈkrɪstʃən ˈændərsən/, Danish, often referred to in Scandinavia as H. C. Although a prolific writer of plays, travelogues, novels, and poems, Andersens popularity is not limited to children, his stories, called eventyr in Danish, express themes that transcend age and nationality. Some of his most famous fairy tales include The Emperors New Clothes, The Little Mermaid, The Nightingale, The Snow Queen, The Ugly Duckling, Thumbelina and his stories have inspired ballets, animated and live-action films and plays. Hans Christian Andersen was born in the town of Odense, Denmark, Andersens father, also Hans, considered himself related to nobility. His paternal grandmother had told his father that their family had in the past belonged to a social class. A persistent theory suggests that Andersen was a son of King Christian VIII. Andersens father, who had received an education, introduced Andersen to literature. Andersens mother, Anne Marie Andersdatter, was uneducated and worked as a washerwoman following his fathers death in 1816, she remarried in 1818. Andersen was sent to a school for poor children where he received a basic education and was forced to support himself, working as an apprentice for a weaver and, later. At 14, he moved to Copenhagen to seek employment as an actor, having an excellent soprano voice, he was accepted into the Royal Danish Theatre, but his voice soon changed. A colleague at the theatre told him that he considered Andersen a poet, taking the suggestion seriously, Andersen began to focus on writing. Jonas Collin, director of the Royal Danish Theatre, felt a great affection for Andersen and sent him to a school in Slagelse. Andersen had already published his first story, The Ghost at Palnatokes Grave, though not a keen pupil, he also attended school at Elsinore until 1827. He later said his years in school were the darkest and most bitter of his life, at one school, he lived at his schoolmasters home. There he was abused and was told that it was to improve his character and he later said the faculty had discouraged him from writing in general, causing him to enter a state of depression. A very early fairy tale by Andersen, called The Tallow Candle, was discovered in a Danish archive in October 2012, the story, written in the 1820s, was about a candle who did not feel appreciated. It was written while Andersen was still in school and dedicated to a benefactor, in 1829, Andersen enjoyed considerable success with the short story A Journey on Foot from Holmens Canal to the East Point of Amager. Its protagonist meets characters ranging from Saint Peter to a talking cat, Andersen followed this success with a theatrical piece, Love on St. Nicholas Church Tower, and a short volume of poems
Hans Christian Andersen
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Photograph taken by
Thora Hallager, 1869
Hans Christian Andersen
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Andersen's childhood home in Odense
Hans Christian Andersen
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Paper chimney sweep
cut by Andersen
Hans Christian Andersen
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Painting of Andersen, 1836, by
Christian Albrecht Jensen
96.
Scarecrow (Oz)
–
The Scarecrow is a character in the fictional Land of Oz created by American author L. Frank Baum and illustrator W. W. In his first appearance, the Scarecrow reveals that he lacks a brain, in reality, he is only two days old and merely ignorant. In Baums classic 1900 novel The Wonderful Wizard of Oz, the living scarecrow encounters Dorothy Gale in a field in the Munchkin Country while she is on her way to the Emerald City. He tells her about his creation and of how he at first scared away the crows, before an older one realised he was a straw man, the old crow then told the Scarecrow of the importance of brains. The mindless Scarecrow joins Dorothy in the hope that The Wizard will give him a brain and they are later joined by the Tin Woodman and the Cowardly Lion. When the group goes to the West, he kills the Witchs crows by twisting their necks and he is taken apart by the Flying Monkeys and his clothes thrown up a tree, but when his clothes are filled with straw he is back again. After Dorothy and her friends have completed their mission to kill the Wicked Witch of the West, before he leaves Oz in a balloon, the Wizard appoints the Scarecrow to rule the Emerald City in his stead. He accompanies Dorothy and the others to the palace of the Good Witch of the South Glinda, and she uses the Golden Cap to summon the Winged Monkeys, who take the Scarecrow back to the Emerald City. His desire for a brain notably contrasts with the Tin Woodmans desire for a heart, indeed, both believe they have neither. This occasions philosophical debate between the two friends as to why their own choices are superior, neither convinces the other, and Dorothy, symbolically, because they remain with Dorothy throughout her quest, she is provided with both and need not select. In the musical of Gregory Maguires interpretation of the Oz franchise, this version of the Scarecrow was Fiyero Tigelaar, Fiyero attends school with Elphaba and Glinda the Good, while they were all still young. Fiyero takes a special interest in Elphaba, however, he is highly sought after by Elphabas roommate, Glinda. Fiyero and Elphaba share a secret romance, and when she leaves for the Emerald City, he gives her a long, many years later, after Elphaba goes into hiding, we are shown that Glinda and Fiyero are to be wed. However this is not because of love, at least not on Fiyeros part, once she reappears in the Emerald City, they escape together, much to Glindas discontent. Elphaba then goes to the site of her sisters death, here she is ambushed by guards, and about to be taken away, when Fiyero saves her, but he is merely sacrificing himself to save her. Elphaba in a fit of rage and heartbreak reads a spell to keep Fiyero safe, with the words of the spell including pleas to let him feel no pain and never die however they try to destroy him. In Gregory Maguires novel Wicked, The Life and Times of the Wicked Witch of the West The Scarecrow is a companion of Dorothy and she sends minions after him to tear him apart to see if Fiyero is inside, but he is not. In the novel he is NOT Fiyero, but this is probably where they got the idea for the stage version, the Scarecrow appears briefly in this novel, first giving Liir advice
Scarecrow (Oz)
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Denslow's drawing of scarecrow hung up on pole and helpless, from first edition of book, 1900
Scarecrow (Oz)
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July 1896
Puck cartoon shows farmer hung up on pole and helpless; was this Denslow's inspiration? The hat says "silverite"; the locomotive is gold
Scarecrow (Oz)
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Cover of
The Scarecrow of Oz (1915) by
L. Frank Baum; illustration by
John R. Neill
97.
Wizard (Oz)
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Oscar Zoroaster Phadrig Isaac Norman Henkle Emmannuel Ambroise Diggs is a fictional character in the Land of Oz created by American author L. Frank Baum. The character was further popularized by a play and several movies, most famously the classic 1939 movie. The Wizard is one of the characters in The Wonderful Wizard of Oz, unseen for most of the novel, he is the ruler of the Land of Oz and highly venerated by his subjects. Believing he is the man capable of solving their problems, Dorothy and her friends travel to the Emerald City. Oz is very reluctant to meet them, but eventually each is granted an audience, one by one. In each of these occasions, the Wizard appears in a different form, once as a giant head, once as a fairy, once as a ball of fire. When, at last, he grants an audience to all of them at once, working as a magician for a circus, he wrote OZ on the side of his hot air balloon for promotional purposes. One day his balloon sailed into the Land of Oz, as Oz had no leadership at the time, he became Supreme Ruler of the kingdom, and did his best to sustain the myth. He leaves Oz at the end of the novel, again in a hot air balloon, after the Wizards departure, the Scarecrow is briefly enthroned, until Princess Ozma is freed from the witch Mombi at the end of The Marvelous Land of Oz. In The Marvelous Land of Oz, the Wizard is described as having usurped the throne of King Pastoria and handed over the baby princess to Mombi. This did not please the readers, and in Ozma of Oz, although the character did not appear, the Wizard returns in the novel Dorothy and the Wizard in Oz. With Dorothy and the boy Zeb, he falls through a crack in the earth, in their journey, he acts as their guide. Oz explains that his name is Oscar Zoroaster Phadrig Isaac Norman Henkle Emmannuel Ambroise Diggs. To shorten this name, he used only his initials, but since they spell out the word pinhead, he shortened his name further, Ozma then permits him to live in Oz permanently. He becomes an apprentice to Glinda, Ozma decrees that, besides herself, only The Wizard and Glinda are allowed to use magic unless the other magic users have permits. In later books, he proves himself quite an inventor, providing devices that aid in various characters’ journeys and he introduces to Oz the use of mobile phones in Tik-Tok of Oz. Some of his most elaborate devices are the Ozpril and the Oztober, balloon-powered Ozoplanes in Ozoplaning with the Wizard of Oz, the Wizard has appeared in nearly every silent Oz film, portrayed by different actors each time. His face was also used as the projected image of the Wizard
Wizard (Oz)
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Oscar Diggs aka the Wizard--illustration by
William Wallace Denslow (1900)
98.
San Marino
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Its size is just over 61 km2, with a population of 33,562. Its capital is the City of San Marino and its largest city is Dogana, San Marino has the smallest population of all the members of the Council of Europe. The country takes its name from Marinus, a stonemason originating from the Roman colony on the island of Rab, in 257 CE Marinus participated in the reconstruction of Riminis city walls after their destruction by Liburnian pirates. San Marino is governed by the Constitution of San Marino, a series of six books written in Latin in the late 16th century, the country is considered to have the earliest written governing documents still in effect. The countrys economy mainly relies on finance, industry, services and it is one of the wealthiest countries in the world in terms of GDP, with a figure comparable to the most developed European regions. San Marino is considered to have a stable economy, with one of the lowest unemployment rates in Europe, no national debt. It is the country with more vehicles than people. Saint Marinus left the island of Arba in present-day Croatia with his lifelong friend Leo, and went to the city of Rimini as a stonemason. After the Diocletianic Persecution following his Christian sermons, he escaped to the nearby Monte Titano, the official date of the founding of what is now known as the Republic is 3 September 301. In 1631, its independence was recognized by the Papacy, the offer was declined by the Regents, fearing future retaliation from other states revanchism. During the later phase of the Italian unification process in the 19th century, in recognition of this support, Giuseppe Garibaldi accepted the wish of San Marino not to be incorporated into the new Italian state. The government of San Marino made United States President Abraham Lincoln an honorary citizen and he wrote in reply, saying that the republic proved that government founded on republican principles is capable of being so administered as to be secure and enduring. Italy tried to establish a detachment of Carabinieri in the republic. Two groups of ten volunteers joined Italian forces in the fighting on the Italian front, the first as combatants, the existence of this hospital later caused Austria-Hungary to suspend diplomatic relations with San Marino. From 1923 to 1943, San Marino was under the rule of the Sammarinese Fascist Party. During World War II, San Marino remained neutral, although it was reported in an article from The New York Times that it had declared war on the United Kingdom on 17 September 1940. The Sammarinese government later transmitted a message to the British government stating that they had not declared war on the United Kingdom, Three days after the fall of Benito Mussolini in Italy, PFS rule collapsed and the new government declared neutrality in the conflict. The Fascists regained power on 1 April 1944 but kept neutrality intact, despite that, on 26 June 1944 San Marino was bombed by the Royal Air Force, in the belief that San Marino had been overrun by German forces and was being used to amass stores and ammunition
San Marino
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The San Marino constitution of 1600
San Marino
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Flag
San Marino
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The front passes
Mount Titano in September 1944.
San Marino
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Mount Titano
99.
Sierra Leone
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Sierra Leone, officially the Republic of Sierra Leone, is a country in West Africa. It is bordered by Guinea to the north-east, Liberia to the south-east, Sierra Leone has a tropical climate, with a diverse environment ranging from savannah to rainforests. Sierra Leone has an area of 71,740 km2. Sierra Leone is divided into four regions, the Northern Province, Eastern Province, Southern Province and the Western Area. Freetown, located in the Western Area, is the capital, largest city and its economic, Bo is the second largest city, and is located in the Southern Province, about 160 miles from Freetown. Kenema, located in the Eastern Province, is the third largest city and is about 185 miles from Freetown, Koidu Town, located in the Eastern Province, is the fourth largest city, and is about 275 miles from Freetown. Makeni, located in the Northern Province, is the fifth largest of Sierra Leone five major cities, Sierra Leone is a constitutional republic with a directly elected president and a unicameral legislature. The current constitution of Sierra Leone was adopted in 1991 during the presidency of Joseph Saidu Momoh, since independent to present, Sierra Leone politics has been dominated by two major political parties, the Sierra Leone Peoples party and the All Peoples congress. The current president of SIerra Leone is Ernest Bai Koroma, a member of the APC party, the previous Sierra Leone president was Ahmad Tejan Kabbah, a member of the SLPP party, who was elected president in 1996 and won reelection for his final term in 2002. From 1991 to 2002, the Sierra Leone civil war was fought and this proxy war left more than 50,000 people dead, much of the countrys infrastructure destroyed, and over two million people displaced as refugees in neighbouring countries. In January 2002, then Sierra Leones president Ahmad Tejan Kabbah, fulfilled his promise by ending the civil war, with help by the British Government, ECOWAS. More recently, the 2014 Ebola outbreak overburdened the weak healthcare infrastructure and it created a humanitarian crisis situation and a negative spiral of weaker economic growth. The country has a low life expectancy at 57.8 years. About sixteen ethnic groups inhabit Sierra Leone, each with its own language, the two largest and most influential are the Temne and the Mende people. The Temne are predominantly found in the north of the country, the Krio language unites all the different ethnic groups in the country, especially in their trade and social interaction with each other. Sierra Leone is a predominantly Muslim country, though with an influential Christian minority, Sierra Leone is regarded as one of the most religiously tolerant nations in the world. Muslims and Christians collaborate and interact with each other peacefully, religious violence is extremely rare in the country. In politics, the majority of Sierra Leoneans vote for a candidate without regard to whether the candidate is a Muslim or a Christian
Sierra Leone
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Fragments of
prehistoric pottery from
Kamabai Rock Shelter
Sierra Leone
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Flag
Sierra Leone
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An 1835 illustration of liberated Africans arriving in Sierra Leone.
Sierra Leone
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The colony of Freetown in 1856
100.
Postage stamps
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Postage stamp may also refer to a formatting artifact in the display of film or video, Windowbox. A postage stamp is a piece of paper that is purchased and displayed on an item of mail as evidence of payment of postage. Typically, stamps are printed on special paper, show a national designation and a denomination on the front. They are sometimes a source of net profit to the issuing agency, stamps are usually rectangular, but triangles or other shapes are occasionally used. The stamp is affixed to an envelope or other postal cover the customer wishes to send, the item is then processed by the postal system, where a postmark, sometimes known as a cancellation mark, is usually applied in overlapping manner to stamp and cover. This procedure marks the stamp as used to prevent its reuse, in modern usage, postmarks generally indicate the date and point of origin of the mailing. The mailed item is delivered to the address the customer has applied to the envelope or parcel. Postage stamps have facilitated the delivery of mail since the 1840s, before then, ink and hand-stamps, usually made from wood or cork, were often used to frank the mail and confirm the payment of postage. The first adhesive postage stamp, commonly referred to as the Penny Black, was issued in the United Kingdom in 1840, there are varying accounts of the inventor or inventors of the stamp. The postage stamp resolved this issue in a simple and elegant manner, concurrently with the first stamps, the UK offered wrappers for mail. S. Postal service for priority or express mailing, the postage stamp afforded convenience for both the mailer and postal officials, more effectively recovered costs for the postal service, and ultimately resulted in a better, faster postal system. With the conveniences stamps offered, their use resulted in greatly increased mailings during the 19th and 20th centuries, as postage stamps with their engraved imagery began to appear on a widespread basis, historians and collectors began to take notice. The study of stamps and their use is referred to as philately. Stamp collecting can be both a hobby and a form of study and reference, as government-issued postage stamps. The postage for the item was prepaid by the use of a hand-stamp to frank the mailed item. Though this stamp was applied to a letter instead of a piece of paper it is considered by many historians as the worlds first postage stamp. Rowland Hill The Englishman Sir Rowland Hill began interest in postal reform in 1835, in 1836, a Member of Parliament, Robert Wallace, provided Hill with numerous books and documents, which Hill described as a half hundred weight of material. Hill commenced a study of these documents, leading him to the 1837 publication of a pamphlet entitled Post Office Reform its Importance
Postage stamps
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The main components of a stamp: 1.
Image 2.
Perforations 3.
Denomination 4. Country name
Postage stamps
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Rowland Hill
Postage stamps
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The
Penny Black, the world’s first postage stamp.
Postage stamps
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Rows of perforations in a sheet of postage stamps.
101.
Lp space
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In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, in penalized regression, L1 penalty and L2 penalty refer to penalizing either the L1 norm of a solutions vector of parameter values, or its L2 norm. Techniques which use an L1 penalty, like LASSO, encourage solutions where many parameters are zero, techniques which use an L2 penalty, like ridge regression, encourage solutions where most parameter values are small. Elastic net regularization uses a penalty term that is a combination of the L1 norm, the Fourier transform for the real line, maps Lp to Lq, where 1 ≤ p ≤2 and 1/p + 1/q =1. This is a consequence of the Riesz–Thorin interpolation theorem, and is precise with the Hausdorff–Young inequality. By contrast, if p >2, the Fourier transform does not map into Lq, Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L2 and ℓ2 are both Hilbert spaces, in fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ2, where E is a set with an appropriate cardinality. The length of a vector x = in the real vector space Rn is usually given by the Euclidean norm. The Euclidean distance between two points x and y is the length ||x − y||2 of the line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space, the class of p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science. For a real number p ≥1, the p-norm or Lp-norm of x is defined by ∥ x ∥ p =1 p, of course the absolute value bars are unnecessary when p is a rational number and, in reduced form, has an even numerator. The Euclidean norm from above falls into class and is the 2-norm. The L∞-norm or maximum norm is the limit of the Lp-norms for p → ∞ and it turns out that this limit is equivalent to the following definition, ∥ x ∥ ∞ = max See L-infinity. Abstractly speaking, this means that Rn together with the p-norm is a Banach space and this Banach space is the Lp-space over Rn. The grid distance or rectilinear distance between two points is never shorter than the length of the segment between them. Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm, ∥ x ∥2 ≤ ∥ x ∥1. This fact generalizes to p-norms in that the p-norm ||x||p of any vector x does not grow with p, ||x||p+a ≤ ||x||p for any vector x and real numbers p ≥1
Lp space
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Unit circle (
superellipse) in p = 3 / 2 norm
102.
Parallelogram law
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In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals, using the notation in the diagram on the right, the sides are. For the general quadrilateral with four sides not necessarily equal,2 +2 +2 +2 =2 +2 +4 x 2, where x is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that, for a parallelogram, x =0, and the general formula simplifies to the parallelogram law. In a normed space, the statement of the law is an equation relating norms,2 ∥ x ∥2 +2 ∥ y ∥2 = ∥ x + y ∥2 + ∥ x − y ∥2. In an inner space, the norm is determined using the inner product. Most real and complex normed vector spaces do not have inner products, for example, a commonly used norm is the p-norm, ∥ x ∥ p =1 / p, where the x i are the components of vector x. Given a norm, one can evaluate both sides of the law above. A remarkable fact is that if the law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the if and only if p =2. For any norm satisfying the law, the inner product generating the norm is unique as a consequence of the polarization identity. Commutative property Inner product space Normed vector space Polarization identity Weisstein, Eric W. Parallelogram Law
Parallelogram law
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A parallelogram. The sides are shown in blue and the diagonals in red.
103.
Pythagorean tiling
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A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name and it is commonly used as a pattern for floor tiles. When used for this, it is known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling. This tiling has four-way rotational symmetry around each of its squares, generalizations of this tiling to three dimensions have also been studied. The Pythagorean tiling is the tiling by squares of two different sizes that is both unilateral and equitransitive. Topologically, the Pythagorean tiling has the structure as the truncated square tiling by squares. However, the two tilings have different sets of symmetries, because the square tiling is symmetric under mirror reflections whereas the Pythagorean tiling isnt. It is a pattern, meaning that it is impossible to superpose it on top of its mirror image using only translations and rotations. A uniform tiling is a tiling in which each tile is a regular polygon, usually, uniform tilings additionally are required to have tiles that meet edge-to-edge, but if this requirement is relaxed then there are eight additional uniform tilings. Four are formed from strips of squares or equilateral triangles. The remaining one is the Pythagorean tiling, similarly, overlaying two Pythagorean tilings may be used to generate a six-piece dissection of two unequal squares into a different two unequal squares. Although the Pythagorean tiling is itself periodic its cross sections can be used to generate one-dimensional aperiodic sequences. In the Klotz construction for aperiodic sequences, one forms a Pythagorean tiling with two squares whose sizes are chosen to make the ratio between the two side lengths be a number x. In this sequence, the proportion of 0s and 1s will be in the ratio x,1. This proportion cannot be achieved by a sequence of 0s and 1s, because it is irrational. According to Kellers conjecture, any tiling of the plane by congruent squares must include two squares that meet edge-to-edge. None of the squares in the Pythagorean tiling meet edge-to-edge, but this fact does not violate Kellers conjecture because the tiles have different sizes, so they are not all congruent to each other. The Pythagorean tiling may be generalized to a tiling of Euclidean space by cubes of two different sizes, which also is unilateral and equitransitive
Pythagorean tiling
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Street Musicians at the Doorway of a House,
Jacob Ochtervelt, 1665. As observed by Nelsen the floor tiles in this painting are set in the Pythagorean tiling
Pythagorean tiling
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A Pythagorean tiling
Pythagorean tiling
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Periodic
Pythagorean tiling
104.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an
EAN-13 bar code