1.
Regular polygon
–
In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
Regular polygon
–
The zig-zagging side edges of a n -
antiprism represent a regular skew 2 n -gon, as shown in this 17-gonal antiprism.
Regular polygon
–
Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols
2.
Edge (geometry)
–
For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
Edge (geometry)
–
Three edges AB, BC, and CA, each between two
vertices of a
triangle.
3.
Vertex (geometry)
–
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
Vertex (geometry)
–
A vertex of an angle is the endpoint where two line segments or rays come together.
4.
Coxeter diagram
–
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction, each node represents a mirror. An unlabeled branch implicitly represents order-3, each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams correspond to and are used to root systems. Branches of a Coxeter–Dynkin diagram are labeled with a number p. When p =2 the angle is 90° and the mirrors have no interaction, if a branch is unlabeled, it is assumed to have p =3, representing an angle of 60°. Two parallel mirrors have a branch marked with ∞, in principle, n mirrors can be represented by a complete graph in which all n /2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes, plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs. Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements ai, j = aj, as a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the cases of p =2,3,4, and 6. The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite, affine and this rule is called Schläflis Criterion. The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of type, affine type. The indefinite type is further subdivided, e. g. into hyperbolic. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups and we use the following definition, A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite, Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact groups in 1950
Coxeter diagram
–
Coxeter–Dynkin diagrams for the fundamental finite Coxeter groups
5.
List of planar symmetry groups
–
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes, International notation, orbifold notation, and Coxeter notation. There are three kinds of groups of the plane,2 rosette groups – 2D point groups 7 frieze groups – 2D line groups 17 wallpaper groups – 2D space groups. There are two families of discrete point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group. The 7 frieze groups, the line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups, the yellow regions represent the infinite fundamental domain in each. The p1 and p2 groups, with no symmetry, are repeated in all classes. The related pure reflectional Coxeter group are given with all classes except oblique, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Coxeter, H. S. M. & Moser, W. O. J. Generators and Relations for Discrete Groups. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Conways manuscript on Orbifold notation The 17 Wallpaper Groups
List of planar symmetry groups
–
C 1, [] + (•)
6.
Dihedral symmetry
–
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
Dihedral symmetry
–
The
symmetry group of a
snowflake is Dih 6, a dihedral symmetry, the same as for a regular
hexagon.
7.
Internal angle
–
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple polygon, regardless of whether it is convex or non-convex, a polygon has exactly one internal angle per vertex. If every internal angle of a polygon is less than 180°. In contrast, an angle is an angle formed by one side of a simple polygon. The sum of the angle and the external angle on the same vertex is 180°. The sum of all the angles of a simple polygon is 180° where n is the number of sides. The formula can be proved using induction and starting with a triangle for which the angle sum is 180°. The sum of the angles of any simple convex or non-convex polygon is 360°. The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles, in other words, 360k° represents the sum of all the exterior angles. For example, for convex and concave polygons k =1, since the exterior angle sum is 360°
Internal angle
–
Internal and External angles
8.
Degree (angle)
–
A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of measure is the radian. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians, the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the year. Some ancient calendars, such as the Persian calendar, used 360 days for a year, the use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today
Degree (angle)
–
One degree (shown in red) and eighty nine (shown in blue)
9.
Dual polygon
–
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. The dual of a polygon is an isotoxal polygon. For example, the rectangle and rhombus are duals, in a cyclic polygon, longer sides correspond to larger exterior angles in the dual, and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, for example, the dual of a highly acute isosceles triangle is an obtuse isosceles triangle. In the Dorman Luke construction, each face of a polyhedron is the dual polygon of the corresponding vertex figure. As an example of the duality of polygons we compare properties of the cyclic. This duality is perhaps more clear when comparing an isosceles trapezoid to a kite. The simplest qualitative construction of a polygon is a rectification operation. New edges are formed between these new vertices and that is, the polygon generated by applying it twice is in general not similar to the original polygon. As with dual polyhedra, one can take a circle and perform polar reciprocation in it. Combinatorially, one can define a polygon as a set of vertices, a set of edges, then the dual polygon is obtained by simply switching the vertices and edges. Thus for the triangle with vertices and edges, the triangle has vertices, and edges, where B connects AB & BC. This is not a particularly fruitful avenue, as combinatorially, there is a family of polygons, geometric duality of polygons is more varied. Dual curve Dual polyhedron Self-dual polygon Dual Polygon Applet by Don Hatch
Dual polygon
–
Dorman Luke construction, showing a
rhombus face being dual to a
rectangle vertex figure.
10.
Convex polygon
–
A convex polygon is a simple polygon in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a polygon whose interior is a convex set. In a convex polygon, all angles are less than or equal to 180 degrees. A simple polygon which is not convex is called concave, the following properties of a simple polygon are all equivalent to convexity, Every internal angle is less than or equal to 180 degrees. Every point on line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. The polygon is contained in a closed half-plane defined by each of its edges. For each edge, the points are all on the same side of the line that the edge defines. The angle at each vertex contains all vertices in its edges. The polygon is the hull of its edges. Additional properties of convex polygons include, The intersection of two convex polygons is a convex polygon, a convex polygon may br triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Hellys theorem, For every collection of at least three convex polygons, if the intersection of three of them is nonempty, then the whole collection has a nonempty intersection. Krein–Milman theorem, A convex polygon is the hull of its vertices. Thus it is defined by the set of its vertices. Hyperplane separation theorem, Any two convex polygons with no points in common have a separator line, if the polygons are closed and at least one of them is compact, then there are even two parallel separator lines. Inscribed triangle property, Of all triangles contained in a convex polygon, inscribing triangle property, every convex polygon with area A can be inscribed in a triangle of area at most equal to 2A. Equality holds for a parallelogram.5 × Area ≤ Area ≤2 × Area, the mean width of a convex polygon is equal to its perimeter divided by pi. So its width is the diameter of a circle with the perimeter as the polygon. Every polygon inscribed in a circle, if not self-intersecting, is convex, however, not every convex polygon can be inscribed in a circle
Convex polygon
–
An example of a convex polygon: a
regular pentagon
11.
Cyclic polygon
–
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Cyclic polygon
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
12.
Equilateral polygon
–
In geometry, three or more than three straight lines make a polygon and an equilateral polygon is a polygon which has all sides of the same length. Except in the case, it need not be equiangular. If the number of sides is at least five, an equilateral polygon need not be a convex polygon, all regular polygons and isotoxal polygons are equilateral. An equilateral triangle is a triangle with 60° internal angles. An equilateral quadrilateral is called a rhombus, an isotoxal polygon described by an angle α and it includes the square as a special case. A convex equilateral pentagon can be described by two angles α and β, which determine the other angles. Concave equilateral pentagons exist, as do concave equilateral polygons with any number of sides. An equilateral polygon which is cyclic is a regular polygon, a tangential polygon is equilateral if and only if the alternate angles are equal. Thus if the number of n is odd, a tangential polygon is equilateral if. The principal diagonals of a hexagon each divide the hexagon into quadrilaterals, in any convex equilateral hexagon with common side a, there exists a principal diagonal d1 such that d 1 a ≤2 and a principal diagonal d2 such that d 2 a >3. Triambi are equilateral hexagons with trigonal symmetry, Equilateral triangle With interactive animation A Property of Equiangular Polygons, a discussion of Vivianis theorem at Cut-the-knot
Equilateral polygon
13.
Isogonal figure
–
In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
Isogonal figure
14.
Isotoxal figure
–
In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek τοξον meaning arc, an isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons, in general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
Isotoxal figure
–
A
rhombic dodecahedron is an isohedral and isotoxal polyhedron
Isotoxal figure
15.
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.
16.
Ancient Greek
–
Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
Ancient Greek
–
Inscription about the construction of the statue of
Athena Parthenos in the
Parthenon, 440/439 BC
Ancient Greek
–
Ostracon bearing the name of
Cimon,
Stoa of Attalos
Ancient Greek
–
The words ΜΟΛΩΝ ΛΑΒΕ as they are inscribed on the marble of the 1955
Leonidas Monument at
Thermopylae
17.
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
18.
Truncation (geometry)
–
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Keplers names for the Archimedean solids, in general any polyhedron can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, there are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation, for example, the icosidodecahedron, represented as Schläfli symbols r or, and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedron, represented as tr or t. In the Coxeter-Dynkin diagram, the effect of a truncation is to ring all the adjacent to the ringed node. A truncated n-sided polygon will have 2n sides, a regular polygon uniformly truncated will become another regular polygon, t is. A complete truncation, r, is another regular polygon in its dual position, a regular polygon can also be represented by its Coxeter-Dynkin diagram, and its uniform truncation, and its complete truncation. Star polygons can also be truncated, a truncated pentagram will look like a pentagon, but is actually a double-covered decagon with two sets of overlapping vertices and edges. A truncated great heptagram gives a tetradecagram and this sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron, the middle image is the uniform truncated cube. It is represented by a Schläfli symbol t, a bitruncation is a deeper truncation, removing all the original edges, but leaving an interior part of the original faces. The truncated octahedron is a cube, 2t is an example. A complete bitruncation is called a birectification that reduces original faces to points, for polyhedra, this becomes the dual polyhedron. An octahedron is a birectification of the cube, = 2r is an example, another type of truncation is called cantellation, cuts edge and vertices, removing original edges and replacing them with rectangles. Higher dimensional polytopes have higher truncations, runcination cuts faces, edges, in 5-dimensions sterication cuts cells, faces, and edges. Edge-truncation is a beveling or chamfer for polyhedra, similar to cantellation but retains original vertices, in 4-polytopes edge-truncation replaces edges with elongated bipyramid cells. Alternation or partial truncation only removes some of the original vertices, a partial truncation or alternation - Half of the vertices and connecting edges are completely removed. The operation only applies to polytopes with even-sided faces, faces are reduced to half as many sides, and square faces degenerate into edges
Truncation (geometry)
–
Truncated cubic honeycomb t{4,3,4} or
Truncation (geometry)
–
Truncated square is a regular octagon: t{4} = {8} =
19.
Square
–
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
Square
–
A regular
quadrilateral (tetragon)
20.
Hexadecagon
–
In mathematics, a hexadecagon or 16-gon is a sixteen-sided polygon. A regular hexadecagon is a hexadecagon in which all angles are equal and its Schläfli symbol is and can be constructed as a truncated octagon, t, and a twice-truncated square tt. A truncated hexadecagon, t, is a triacontadigon, as 16 =24, a regular hexadecagon is constructible using compass and straightedge, this was already known to ancient Greek mathematicians. Each angle of a regular hexadecagon is 157.5 degrees, the area of a regular hexadecagon with edge length t is A =4 t 2 cot π16 =4 t 2. Since the area of the circumcircle is π R2, the regular hexadecagon fills approximately 97. 45% of its circumcircle, the regular hexadecagon has Dih16 symmetry, order 32. There are 4 dihedral subgroups, Dih8, Dih4, Dih2, and Dih1, and 5 cyclic subgroups, Z16, Z8, Z4, Z2, and Z1, on the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1, the dihedral symmetries are divided depending on whether they pass through vertices or edges Cyclic symmetries in the middle column are labeled as g for their central gyration orders. These two forms are duals of each other and have half the order of the regular hexadecagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g16 subgroup has no degrees of freedom but can seen as directed edges. A skew hexadecagon is a polygon with 24 vertices and edges. The interior of such an hexadecagon is not generally defined, a skew zig-zag hexadecagon has vertices alternating between two parallel planes. A regular skew hexadecagon is vertex-transitive with equal edge lengths, in 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D8d, symmetry, order 32. The octagrammic antiprism, s and octagrammic crossed-antiprism, s also have regular skew octagons, there are three regular star polygons, using the same vertices, but connecting every third, fifth or seventh points. There are also three compounds, is reduced to 2 as two octagons, is reduced to 4 as four squares and reduces to 2 as two octagrams, and finally is reduced to 8 as eight digons. Deeper truncations of the octagon and octagram can produce isogonal intermediate hexadecagram forms with equally spaced vertices. A truncated octagon is a hexadecagon, t=, a quasitruncated octagon, inverted as, is a hexadecagram, t=. A truncated octagram is a hexadecagram, t= and a quasitruncated octagram, inverted as, is a hexadecagram, hexadecagrams are included in the Girih patterns in the Alhambra. An octagonal star can be seen as a concave hexadecagon, Weisstein, Eric W. Hexadecagon
Hexadecagon
–
The hexadecagonal tower from Raphael's
The Marriage of the Virgin
Hexadecagon
–
A regular hexadecagon
Hexadecagon
–
A hexadecagrammic pattern from the
Alhambra
21.
Equidiagonal quadrilateral
–
In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal, examples of equidiagonal quadrilaterals include the isosceles trapezoids, rectangles and squares. Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is a kite with angles π/3, 5π/12, 5π/6. A convex quadrilateral is equidiagonal if and only if its Varignon parallelogram, an equivalent condition is that the bimedians of the quadrilateral are perpendicular. A convex quadrilateral with diagonal lengths p and q and bimedian lengths m and n is equidiagonal if, the area K of an equidiagonal quadrilateral can easily be calculated if the length of the bimedians m and n are known. A quadrilateral is equidiagonal if and only if K = m n, other area formulas may be obtained from setting p = q in the formulas for the area of a convex quadrilateral. A parallelogram is equidiagonal if and only if it is a rectangle, the cyclic equidiagonal quadrilaterals are exactly the isosceles trapezoids. Equivalently, a quadrilateral has equal diagonals if and only if it has perpendicular bimedians, silvester gives further connections between equidiagonal and orthodiagonal quadrilaterals, via a generalization of van Aubels theorem. The square is one such quadrilateral, but there are many others. Equidiagonal, orthodiagonal quadrilaterals have been referred to as midsquare quadrilaterals because they are the ones for which the Varignon parallelogram is a square. Such a quadrilateral, with sides a, b, c, d, has area K = a 2 + c 2 +4 −24
Equidiagonal quadrilateral
–
An equidiagonal quadrilateral, showing its equal diagonals, Varignon rhombus, and perpendicular bimedians
22.
Orthodiagonal quadrilateral
–
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a figure in which the line segments between non-adjacent vertices are orthogonal to each other. A kite is a quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides, that is, a rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides. A square is a case of both a kite and a rhombus. The square is one such quadrilateral, but there are many others. Conversely, any quadrilateral in which a2 + c2 = b2 + d2 must be orthodiagonal and this can be proved in a number of ways, including using the law of cosines, vectors, an indirect proof, and complex numbers. The diagonals of a quadrilateral are perpendicular if and only if the two bimedians have equal length. A convex quadrilateral is orthodiagonal if and only if its Varignon parallelogram is a rectangle, the center of this circle is the centroid of the quadrilateral. The quadrilateral formed by the feet of the maltitudes is called the principal orthic quadrilateral, a related characterization states that a convex quadrilateral is orthodiagonal if and only if RSTU is a rectangle whose sides are parallel to the diagonals of ABCD. There are several metric characterizations regarding the four triangles formed by the diagonal intersection P, denote by m1, m2, m3, m4 the medians in triangles ABP, BCP, CDP, DAP from P to the sides AB, BC, CD, DA respectively. A few metric characterizations of tangential quadrilaterals and orthodiagonal quadrilaterals are very similar in appearance, as can be seen in this table. The notations on the sides a, b, c, d, the circumradii R1, R2, R3, R4, and the altitudes h1, h2, h3, h4 are the same as above in both types of quadrilaterals. The area K of an orthodiagonal quadrilateral equals one half the product of the lengths of the p and q, K = p ⋅ q 2. Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal, the orthodiagonal quadrilateral has the biggest area of all convex quadrilaterals with given diagonals. Orthodiagonal quadrilaterals are the only quadrilaterals for which the sides and the angle formed by the diagonals do not uniquely determine the area, for example, two rhombi both having common side a, but one having a smaller acute angle than the other, have different areas. If squares are erected outward on the sides of any quadrilateral and this is called Van Aubels theorem. Then D2 = p 12 + p 22 + q 12 + q 22 = a 2 + c 2 = b 2 + d 2 where D is the diameter of the circumcircle and this holds because the diagonals are perpendicular chords of a circle
Orthodiagonal quadrilateral
–
An orthodiagonal quadrilateral. According to the characterization of these quadrilaterals, the two red squares on two opposite sides of the quadrilateral have the same total area as the two blue squares on the other pair of opposite sides.
23.
Midpoint polygon
–
In geometry, the midpoint polygon of a polygon P is the polygon whose vertices are the midpoints of the edges of P. It is sometimes called the Kasner polygon after Edward Kasner, who termed it the inscribed polygon for brevity, the midpoint polygon of a triangle is called the medial triangle. It shares the same centroid and medians with the original triangle, the perimeter of the medial triangle equals the semiperimeter of the original triangle, and the area is one quarter of the area of the original triangle. This can be proven by the midpoint theorem of triangles and Herons formula, the orthocenter of the medial triangle coincides with the circumcenter of the original triangle. The midpoint polygon of a quadrilateral is a parallelogram called its Varignon parallelogram, if the quadrilateral is simple, the area of the parallelogram is one half the area of the original quadrilateral. The perimeter of the parallelogram equals the sum of the diagonals of the original quadrilateral, a Polygon Problem, American Mathematical Monthly,72, 233–241, doi,10. 2307/2313689, JSTOR2313689 Cadwell, J. H. The Sidesplitting Story of the Midpoint Polygon, Mathematics Teacher,87, 249–256 Weisstein, Eric W. Midpoint Polygon
Midpoint polygon
–
The
medial triangle
24.
Reflective symmetry
–
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points, this set is called the axis or plane of reflection. The image of a figure by a reflection is its image in the axis or plane of reflection. For example the image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b, a reflection is an involution, when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term reflection is used for a larger class of mappings from a Euclidean space to itself. Such isometries have a set of fixed points that is an affine subspace, for instance a reflection through a point is an involutive isometry with just one fixed point, the image of the letter p under it would look like a d. This operation is known as a central inversion, and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation, other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term reflection means reflection in a hyperplane, a figure that does not change upon undergoing a reflection is said to have reflectional symmetry. Some mathematicians use flip as a synonym for reflection, in a plane geometry, to find the reflection of a point drop a perpendicular from the point to the line used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure, step 2, construct circles centered at A′ and B′ having radius r. P and Q will be the points of intersection of two circles. Point Q is then the reflection of point P through line AB, the matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1,1,1. The product of two matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an number of reflections in hyperplanes through the origin. Thus reflections generate the group, and this result is known as the Cartan–Dieudonné theorem. Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes, in general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups, note that the second term in the above equation is just twice the vector projection of v onto a
Reflective symmetry
–
A reflection through an axis followed by a reflection across a second axis parallel to the first one results in a total motion that is a
translation.
25.
Rotational symmetry
–
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
Rotational symmetry
–
The starting position in
shogi
Rotational symmetry
–
The
triskelion appearing on the
Isle of Man flag.
26.
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.
27.
Radian
–
The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
Radian
–
A chart to convert between degrees and radians
Radian
–
An arc of a
circle with the same length as the
radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to an angle of 2
π radians.
28.
Central angle
–
Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one. The central angle is known as the arcs angular distance. The size of a central angle Θ is 0° < Θ < 360° оr 0 < Θ < 2π. When defining or drawing a central angle, in addition to specifying the points A and B, equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise. If the intersection points A and B of the legs of the angle with the form a diameter. Let L be the arc of the circle between points A and B, and let R be the radius of the circle. If the central angle Θ is subtended by L, then 0 ∘ < Θ <180 ∘, Θ = ∘ = L R. If the central angle Θ is not subtended by the minor arc L, if a tangent at A and a tangent at B intersect at the exterior point P, then denoting the center as O, the angles ∠BOA and ∠BPA are supplementary. A regular polygon with n sides has a circle upon which all its vertices lie. The central angle of the polygon is formed at the center by the radii to two adjacent vertices. The measure of this angle is 2 π / n, Inscribed angle Great-circle navigation Central angle. Interactive Inscribed and Central Angles in a Circle
Central angle
–
Angle AOB forms a central angle
29.
Circumscribed circle
–
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Circumscribed circle
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
30.
Apothem
–
The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides, the word apothem can also refer to the length of that line segment. Regular polygons are the polygons that have apothems. Because of this, all the apothems in a polygon will be congruent. For a regular pyramid, which is a pyramid base is a regular polygon, the apothem is the slant height of a lateral face, that is. For a truncated pyramid, the apothem is the height of a trapezoidal lateral face. An apothem of a polygon will always be a radius of the inscribed circle. It is also the distance between any side of the polygon and its center. A = p a 2 = r 2 = π r 2 The apothem of a regular polygon can be multiple ways. The apothem a of a regular n-sided polygon with side length s, or circumradius R, can be using the following formula. The apothem can also be found by a = s 2 tan and these formulae can still be used even if only the perimeter p and the number of sides n are known because s = p n. Circumradius of a regular polygon Sagitta Chord Apothem of a regular polygon With interactive animation Apothem of pyramid or truncated pyramid Pegg, Jr. Ed
Apothem
–
Apothem of a
hexagon
31.
Inscribed figure
–
In geometry, an inscribed planar shape or solid is one that is enclosed by and fits snugly inside another geometric shape or solid. To say that figure F is inscribed in figure G means precisely the same thing as figure G is circumscribed about figure F, a circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, a circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a polygon. The inradius or filling radius of a given outer figure is the radius of the circle or sphere. The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces. For an alternative usage of the term inscribed, see the inscribed square problem, every circle has an inscribed triangle with any three given angle measures, and every triangle can be inscribed in some circle. Every triangle has a circle, called the incircle. Every circle has a regular polygon of n sides, for any n≥3. Every regular polygon has a circle, and every circle can be inscribed in some regular polygon of n sides. Not every polygon with more than three sides has a circle, those polygons that do are called tangential polygons. Not every polygon with more than three sides is a polygon of a circle, those polygons that are so inscribed are called cyclic polygons. Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangles centroid, every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides, every acute triangle has three inscribed squares. In a right triangle two of them are merged and coincide with other, so there are only two distinct inscribed squares. An obtuse triangle has an inscribed square, with one side coinciding with part of the triangles longest side. A Reuleaux triangle, or more generally any curve of constant width, circumconic and inconic Cyclic quadrilateral Inscribed and circumscribed figures
Inscribed figure
–
Inscribed circles of various polygons
32.
Pi
–
The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
Pi
–
The constant π is represented in this
mosaic outside the Mathematics Building at the
Technical University of Berlin.
Pi
–
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
Pi
–
Archimedes developed the polygonal approach to approximating π.
Pi
–
Isaac Newton used
infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
33.
Unit circle
–
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π
Unit circle
–
Illustration of a unit circle. The variable t is an
angle measure.
34.
Area
–
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
Area
–
A square metre
quadrat made of PVC pipe.
Area
–
The combined area of these three
shapes is
approximately 15.57
squares.
35.
Square (geometry)
–
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
Square (geometry)
–
A regular
quadrilateral (tetragon)
36.
Special right triangles
–
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a triangle may have angles that form simple relationships. This is called a right triangle. A side-based right triangle is one in which the lengths of the sides form ratios of numbers, such as 3,4,5. Angle-based special right triangles are specified by the relationships of the angles of which the triangle is composed, the angles of these triangles are such that the larger angle, which is 90 degrees or π/2 radians, is equal to the sum of the other two angles. The side lengths are generally deduced from the basis of the circle or other geometric methods. This approach may be used to reproduce the values of trigonometric functions for the angles 30°, 45°. In plane geometry, constructing the diagonal of a results in a triangle whose three angles are in the ratio 1,1,2, adding up to 180° or π radians. Hence, the angles respectively measure 45°, 45°, and 90°, the sides in this triangle are in the ratio 1,1, √2, which follows immediately from the Pythagorean theorem. Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, of all right triangles, the 45°–45°–90° degree triangle has the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4. Triangles with these angles are the only right triangles that are also isosceles triangles in Euclidean geometry. However, in geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles. This is a triangle whose three angles are in the ratio 1,2,3 and respectively measure 30°, 60°, the sides are in the ratio 1, √3,2. The proof of this fact is clear using trigonometry, the geometric proof is, Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D, then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1. The fact that the remaining leg AD has length √3 follows immediately from the Pythagorean theorem, the 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α + δ, α + 2δ are the angles in the then the sum of the angles 3α + 3δ = 180°. After dividing by 3, the angle α + δ must be 60°, the right angle is 90°, leaving the remaining angle to be 30°
Special right triangles
–
Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering
trigonometric functions of multiples of 30 and 45 degrees.
37.
Circumradius
–
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Circumradius
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
38.
Inradius
–
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle, it touches the three sides. The center of the incircle is a center called the triangles incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides, every triangle has three distinct excircles, each tangent to one of the triangles sides. The center of the incircle, called the incenter, can be found as the intersection of the three angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle, the center of this excircle is called the excenter relative to the vertex A, or the excenter of A. Because the internal bisector of an angle is perpendicular to its external bisector, polygons with more than three sides do not all have an incircle tangent to all sides, those that do are called tangential polygons. See also Tangent lines to circles, suppose △ A B C has an incircle with radius r and center I. The distance from vertex A to the incenter I is, d = c sin cos = b sin cos The trilinear coordinates for a point in the triangle is the ratio of distances to the triangle sides. Because the Incenter is the distance of all sides the trilinear coordinates for the incenter are 1,1,1. The barycentric coordinates for a point in a triangle give weights such that the point is the average of the triangle vertex positions. The Cartesian coordinates of the incenter are an average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i. e. Using the barycentric coordinates given above, normalized to sum to unity—as weights. If the three vertices are located at, and, and the sides opposite these vertices have corresponding lengths a, b, additionally, I A ⋅ I B ⋅ I C =4 R r 2, where R and r are the triangles circumradius and inradius respectively. The collection of triangle centers may be given the structure of a group under multiplication of trilinear coordinates, in this group. Then the incircle has the radius r = x y z x + y + z, the product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is r R = a b c 2. Some relations among the sides, incircle radius, and circumcircle radius are, a b + b c + c a = s 2 + r, any line through a triangle that splits both the triangles area and its perimeter in half goes through the triangles incenter. There are either one, two, or three of these for any given triangle, the distance from any vertex to the incircle tangency on either adjacent side is half the sum of the vertexs adjacent sides minus half the opposite side. Thus for example for vertex B and adjacent tangencies TA and TC, the incircle radius is no greater than one-ninth the sum of the altitudes
Inradius
–
A triangle (black) with incircle (blue),
incenter (I), excircles (orange), excenters (J A,J B,J C), internal
angle bisectors (red) and external angle bisectors (green)
39.
Straightedge
–
A straightedge is a tool with an edge free from curves, or straight, used for transcribing straight lines, or checking the straightness of lines. If it has equally spaced markings along its length, it is called a ruler. Straightedges are used in the service and machining industry to check the flatness of machined mating surfaces. A pair of straightedges called winding sticks are used in woodworking to amplify twist in pieces of wood, an idealized straightedge is used in compass-and-straightedge constructions in plane geometry. It may be used, Given two points, to draw the line connecting them, Given a point and a circle, to draw either tangent. Given two circles, to any of their common tangents. It may not be marked or used together with the compass so as to transfer the length of one segment to another and it is possible to do all compass and straightedge constructions without the straightedge. That is, it is possible, using only a compass and it is not, however, possible to do all constructions using only a straightedge. It is possible to do them with straightedge alone given one circle, chalk line Geometrography Wayne R. Moore, Foundations of Mechanical Accuracy, Moore Special Tool Company, Bridgeport, CT Making Accurate Straight-Edges from Scratch
Straightedge
–
A triangular
architect's scale
40.
Compass (drawing tool)
–
A pair of compasses, also known simply as a compass, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as tools to measure distances, Compasses can be used for mathematics, drafting, navigation and other purposes. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. Prior to computerization, compasses and other tools for manual drafting were often packaged as a bow set with interchangeable parts, today these facilities are more often provided by computer-aided design programs, so the physical tools serve mainly a didactic purpose in teaching geometry, technical drawing, etc. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. The handle is usually half a inch long. Users can grip it between their pointer finger and thumb, there are two types of legs in a pair of compasses, the straight or the steady leg and the adjustable one. Each has a purpose, the steady leg serves as the basis or support for the needle point. The screw on your hinge holds the two legs in its position, the hinge can be adjusted depending on desired stiffness, the tighter the screw the better the compass’ performance. The needle point is located on the leg, and serves as the center point of circles that are drawn. The pencil lead draws the circle on a paper or material. This holds the lead or pen in place. Circles can be made by fastening one leg of the compasses into the paper with the spike, putting the pencil on the paper, the radius of the circle can be adjusted by changing the angle of the hinge. Distances can be measured on a map using compasses with two spikes, also called a dividing compass, to use a pair of compasses, place the points on a ruler and open it to the measurement of ½ of the measurement of the circle that is desired. For instance, if one desires to draw a 3 inch circle, next, place the point on the spot that you wish the center of your circle to be, and then rotate the section that has the pencil lead around the point, using the handle. Compasses-and-straightedge constructions are used to illustrate principles of plane geometry, although a real pair of compasses is used to draft visible illustrations, the ideal compass used in proofs is an abstract creator of perfect circles
Compass (drawing tool)
–
A beam compass and a regular compass
Compass (drawing tool)
–
A thumbscrew compass for setting and maintaining a precise radius
Compass (drawing tool)
–
Compass for tracing a line.
Compass (drawing tool)
–
Flat branch, pivot wing nut, pencil sleeve branch of the scribe-compass.
41.
Power of two
–
In mathematics, a power of two means a number of the form 2n where n is an integer, i. e. the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to values, so we have 1,2. Because two is the base of the numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100…000 or 0. 00…001, just like a power of ten in the decimal system, a word, interpreted as an unsigned integer, can represent values from 0 to 2n −1 inclusively. Corresponding signed integer values can be positive, negative and zero, either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this show up frequently in computer software. For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees at any time. Powers of two are used to measure computer memory. A byte is now considered eight bits (an octet, resulting in the possibility of 256 values, the prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024. However, in general, the term kilo has been used in the International System of Units to mean 1,000, binary prefixes have been standardized, such as kibi meaning 1,024. Nearly all processor registers have sizes that are powers of two,32 or 64 being most common, powers of two occur in a range of other places as well. For many disk drives, at least one of the size, number of sectors per track. The logical block size is almost always a power of two. Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of two or three powers of two, or powers of two minus one. For example,640 =512 +128 =128 ×5, put another way, they have fairly regular bit patterns. A prime number that is one less than a power of two is called a Mersenne prime, for example, the prime number 31 is a Mersenne prime because it is 1 less than 32. Similarly, a number that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational, the numbers that can be represented as sums of consecutive positive integers are called polite numbers, they are exactly the numbers that are not powers of two
Power of two
–
Visualization of powers of two from 1 to 1024 (2 0 to 2 10).
42.
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
Coxeter
–
Harold Scott MacDonald Coxeter
43.
Petrie polygon
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In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets. The Petrie polygon of a polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the interior to it. The plane in question is the Coxeter plane of the group of the polygon. These polygons and projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability, in periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the skew polygons which appear on the surface of regular polyhedra. When my incredulity had begun to subside, he described them to me, one consisting of squares, six at each vertex, in 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication, realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes, the Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon, three of the Kepler–Poinsot polyhedra have hexagonal, and decagrammic, petrie polygons. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and this table represents Petrie polygon projections of 3 regular families, and the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed, Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. The Beauty of Geometry, Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, ISBN 0-521-81496-0 Steinberg, Robert, ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. Petrie polygon. Weisstein, Eric W. Cross polytope graphs, Weisstein, Eric W. Gosset graph 3_21
Petrie polygon
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Net
44.
Tesseract
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In geometry, the tesseract is the four-dimensional analog of the cube, the tesseract is to the cube as the cube is to the square. Just as the surface of the consists of six square faces. The tesseract is one of the six convex regular 4-polytopes, the tesseract is also called an 8-cell, C8, octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the family of hypercubes or measure polytopes. In this publication, as well as some of Hintons later work, the tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384, constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64, as an orthotope it can be represented by composite Schläfli symbol × × × or 4, with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the hull of the points. That is, it consists of the points, A tesseract is bounded by eight hyperplanes, each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge, there are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes,24 squares,32 edges, the construction of a hypercube can be imagined the following way, 1-dimensional, Two points A and B can be connected to a line, giving a new line segment AB. 2-dimensional, Two parallel line segments AB and CD can be connected to become a square, 3-dimensional, Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional, Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube and it is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices, the scheme is similar to the construction of a cube from two squares, juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length, the regular complex polytope 42, in C2 has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 42 has 16 vertices, and 8 4-edges and its symmetry is 42, order 32. It also has a lower construction, or 4×4, with symmetry 44
Tesseract
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Schlegel diagram
45.
Square antiprism
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In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube, if all its faces are regular, it is a semiregular polyhedron or uniform polyhedron. One molecule with this geometry is the ion in the salt nitrosonium octafluoroxenate, however. Very few ions are cubical because such a shape would cause large repulsion between ligands, PaF3−8 is one of the few examples, in addition, the element sulfur forms octatomic S8 molecules as its most stable allotrope. The main building block of the One World Trade Center has the shape of a tall tapering square antiprism. It is not a true antiprism because of its taper, the top square has half the area of the bottom one, a twisted prism can be made with the same vertex arrangement. It can be seen as the form with 4 tetrahedrons excavated around the sides. However, after this it can no longer be triangulated into tetrahedra without adding new vertices and it has half of the symmetry of the uniform solution, Dn, +, order 8. The gyroelongated square pyramid is a Johnson solid constructed by augmenting one a square pyramid, similarly, the gyroelongated square bipyramid is a deltahedron constructed by replacing both squares of a square antiprism with a square pyramid. The snub disphenoid is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles, the snub square antiprism can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle. The sphenocorona and the sphenomegacorona are other Johnson solids that, like the square antiprism, the square antiprism is first in a series of snub polyhedra and tilings with vertex figure 3.3.4.3. n. Compound of three square antiprisms Weisstein, Eric W. Antiprism, square Antiprism interactive model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, A4
Square antiprism
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One World Trade Center
Square antiprism
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Uniform Square antiprism
46.
Skew polygon
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In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices, the interior surface of such a polygon is not uniquely defined. Skew infinite polygons have vertices which are not all collinear, a zig-zag skew polygon or antiprismatic polygon has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygon in 3 dimensions are always zig-zag, a regular skew polygon is isogonal with equal edge lengths. In 3 dimensions a regular polygon is a zig-zag skew. The sides of an n-antiprism can define a regular skew 2n-gons, a regular skew n-gonal can be given a symbol # as a blend of a regular polygon, and an orthogonal line segment. The symmetry operation between sequential vertices is glide reflection, examples are shown on the uniform square and pentagon antiprisms. The star antiprisms also generate regular skew polygons with different connection order of the top, the filled top and bottom polygons are drawn for structural clarity, and are not part of the skew polygons. A regular compound skew 2n-gon can be constructed by adding a second skew polygon by a rotation. These shares the same vertices as the compound of antiprisms. Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes, for example, the 5 Platonic solids have 4,6, and 10-sided regular skew polygons, as seen in these orthogonal projections with red edges around the projective envelope. The tetrahedron and octahedron include all the vertices in the zig-zag skew polygon and can be seen as a digonal, the regular skew polyhedron have regular faces, and regular skew polygon vertex figures. Three are infinite space-filling in 3-space and others exist in 4-space, an isogonal skew polygon is a skew polygon with one type of vertex, connected by two types of edges. Isogonal skew polygons with equal edge lengths can also be considered quasiregular and it is similar to a zig-zag skew polygon, existing on two planes, except allowing one edge to cross to the opposite plane, and the other edge to stay on the same plane. Isogonal skew polygons can be defined on even-sided n-gonal prisms, alternatingly following an edge of one side polygon, for example, on the vertices of a cube. Vertices alternate between top and bottom squares with red edges between sides, and blue edges along each side, in 4 dimensions a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike zig-zag skew polygons, skew polygons on double rotations can include an odd-number of sides, the petrie polygons of the regular 4-polytope define regular skew polygons. The Coxeter number for each coxeter group symmetry expresses how many sides a petrie polygon has and this is 5 sides for a 5-cell,8 sides for a tesseract and 16-cell,12 sides for a 24-cell, and 30 sides for a 120-cell and 600-cell
Skew polygon
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The (red) side edges of
tetragonal disphenoid represent a regular zig-zag skew quadrilateral.
47.
Vertex-transitive
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
Vertex-transitive
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Isogonal
apeirogons
48.
Uniform polytope
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A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons and this is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures are allowed, which expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs of Euclidean, nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the antiprism in four dimensions. Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension and this approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation. Regular n-polytopes have n orders of rectification, the zeroth rectification is the original form. The th rectification is the dual, an extended Schläfli symbol can be used for representing rectified forms, with a single subscript, k-th rectification = tk = kr. Truncation operations that can be applied to regular n-polytopes in any combination, the resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them. Truncation cuts vertices, cantellation cuts edges, runcination cuts faces, each higher operation also cuts lower ones too, so a cantellation also truncates vertices. T0,1 or t, Truncation - applied to polygons, a truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges and it can be seen as rectifying its rectification. A cantellation truncates both vertices and edges and replaces them with new facets, cells are replaced by topologically expanded copies of themselves. There are higher cantellations also, bicantellation t1,3 or r2r, tricantellation t2,4 or r3r, quadricantellation t3,5 or r4r, etc. t0,1,2 or tr, Cantitruncation - applied to polyhedra and higher. It can be seen as a truncation of its rectification, a cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves, runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves, There are higher runcinations also, biruncination t1,4, triruncination t2,5, etc. t0,4 or 2r2r, Sterication - applied to Uniform 5-polytopes and higher. It can be seen as birectifying its birectification, Sterication truncates vertices, edges, faces, and cells, replacing each with new facets
Uniform polytope
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Truncated triangle is a uniform hexagon, with
Coxeter diagram.
49.
Orthogonal projection
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In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. That is, whenever P is applied twice to any value, though abstract, this definition of projection formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on an object by examining the effect of the projection on points in the object. For example, the function maps the point in three-dimensional space R3 to the point is an orthogonal projection onto the x–y plane. This function is represented by the matrix P =, the action of this matrix on an arbitrary vector is P =. To see that P is indeed a projection, i. e. P = P2, a simple example of a non-orthogonal projection is P =. Via matrix multiplication, one sees that P2 = = = P. proving that P is indeed a projection, the projection P is orthogonal if and only if α =0. Let W be a finite dimensional space and P be a projection on W. Suppose the subspaces U and V are the range and kernel of P respectively, then P has the following properties, By definition, P is idempotent. P is the identity operator I on U ∀ x ∈ U, P x = x and we have a direct sum W = U ⊕ V. Every vector x ∈ W may be decomposed uniquely as x = u + v with u = P x and v = x − P x = x, the range and kernel of a projection are complementary, as are P and Q = I − P. The operator Q is also a projection and the range and kernel of P become the kernel and range of Q and we say P is a projection along V onto U and Q is a projection along U onto V. In infinite dimensional spaces, the spectrum of a projection is contained in as −1 =1 λ I +1 λ P. Only 0 or 1 can be an eigenvalue of a projection, the corresponding eigenspaces are the kernel and range of the projection. Decomposition of a space into direct sums is not unique in general. Therefore, given a subspace V, there may be many projections whose range is V, if a projection is nontrivial it has minimal polynomial x 2 − x = x, which factors into distinct roots, and thus P is diagonalizable. The product of projections is not, in general, a projection, if projections commute, then their product is a projection. When the vector space W has a product and is complete the concept of orthogonality can be used
Orthogonal projection
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The transformation P is the orthogonal projection onto the line m.
50.
Coxeter plane
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. Note that this assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple classes of Coxeter elements. There are many different ways to define the Coxeter number h of a root system. A Coxeter element is a product of all simple reflections, the product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the number of roots divided by the rank, the number of reflections in the Coxeter group is half the number of roots. The Coxeter number is the order of any Coxeter element, if the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi The dimension of the corresponding Lie algebra is n, where n is the rank and h is the Coxeter number. The Coxeter number is the highest degree of an invariant of the Coxeter group acting on polynomials. Notice that if m is a degree of a fundamental invariant then so is h +2 − m, the eigenvalues of a Coxeter element are the numbers e2πi/h as m runs through the degrees of the fundamental invariants. Since this starts with m =2, these include the primitive hth root of unity, ζh = e2πi/h, an example, has h=30, so 64*30/g =12 -3 -6 -5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 =14400. Coxeter elements of A n −1 ≅ S n, considered as the group on n elements, are n-cycles, for simple reflections the adjacent transpositions, …. The dihedral group Dihm is generated by two reflections that form an angle of 2 π /2 m, and thus their product is a rotation by 2 π / m. For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h and this is called the Coxeter plane and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi/h. This plane was first systematically studied in, and subsequently used in to provide uniform proofs about properties of Coxeter elements, for polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids, in three dimensions, the symmetry of a regular polyhedron, with one directed petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, order h. Adding a mirror, the symmetry can be doubled to symmetry, Dhd. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, in four dimension, the symmetry of a regular polychoron, with one directed petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h, order h. In five dimension, the symmetry of a regular polyteron, with one directed petrie polygon marked, is represented by the composite of 5 reflections
Coxeter plane
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