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Petrie Polygon In geometry , a PETRIE POLYGON for a regular polytope of n dimensions is a skew polygon such that every (n1) consecutive side (but no n) belongs to one of the facets . The PETRIE POLYGON of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side (but no three) belongs to one of the faces . For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, h, is Coxeter number of the Coxeter group Coxeter group . These polygons and projected graphs are useful in visualizing symmetric structure of the higherdimensional regular polytopes [...More...]  "Petrie Polygon" on: Wikipedia Yahoo 

Small Stellated Dodecahedron In geometry , the SMALL STELLATED DODECAHEDRON is a KeplerPoinsot polyhedron , named by Arthur Cayley Arthur Cayley , and with Schläfli symbol {5/2,5}. It is one of four nonconvex regular polyhedra . It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron . It also shares the same edge arrangement with the great icosahedron . It is the second of four stellations of the dodecahedron . If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron , but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar [...More...]  "Small Stellated Dodecahedron" on: Wikipedia Yahoo 

Decagram (geometry) In geometry , a DECAGRAM is a 10point star polygon . There is one regular decagram, containing the vertices of a regular decagon , but connected by every third point. Its Schläfli symbol is {10/3}. The name decagram combine a numeral prefix , deca, with the Greek suffix gram. The gram suffix derives from γραμμῆς (grammēs) meaning a line. CONTENTS * 1 Regular decagram * 2 Applications * 3 Related figures * 4 See also * 5 References REGULAR DECAGRAMFor a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below. APPLICATIONSDecagrams have been used as one of the decorative motifs in girih tiles . RELATED FIGURESA regular decagram is a 10sided polygram , represented by symbol {10/n}, containing the same vertices as regular decagon [...More...]  "Decagram (geometry)" on: Wikipedia Yahoo 

Great Dodecahedron In geometry , the GREAT DODECAHEDRON is a Kepler–Poinsot polyhedron , with Schläfli symbol Schläfli symbol {5,5/2} and Coxeter–Dynkin diagram Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra . It is composed of 12 pentagonal faces (six pairs of parallel pentagons), with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path. The discovery of the great dodecahedron is sometimes credited to Louis Poinsot Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book Perspectiva Corporum Regularium by Wenzel Jamnitzer [...More...]  "Great Dodecahedron" on: Wikipedia Yahoo 

Great Stellated Dodecahedron In geometry , the GREAT STELLATED DODECAHEDRON is a KeplerPoinsot polyhedron , with Schläfli symbol {5/2,3}. It is one of four nonconvex regular polyhedra . It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex. It shares its vertex arrangement with the regular dodecahedron , as well as being a stellation of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron , is related in a similar fashion to the icosahedron . Shaving the triangular pyramids off results in an icosahedron . If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron , with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron [...More...]  "Great Stellated Dodecahedron" on: Wikipedia Yahoo 

Great Icosahedron In geometry , the GREAT ICOSAHEDRON is one of four KeplerPoinsot polyhedra (nonconvex regular polyhedra ), with Schläfli symbol {3,5/2} and CoxeterDynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence. CONTENTS * 1 Images * 2 As a snub * 3 Related polyhedra * 4 References * 5 External links IMAGES TRANSPARENT MODEL DENSITY STELLATION DIAGRAM NET A transparent model of the great icosahedron (See also Animation ) It has a density of 7, as shown in this crosssection. It is a stellation of the icosahedron, counted by Wenninger as model and the 16th of 17 stellations of the icosahedron and 7th of 59 stellations by Coxeter . × 12 Net (surface geometry); twelve isosceles pentagrammic pyramids, arranged like the faces of a dodecahedron. Each pyramid folds up like a fan: the dotted lines fold the opposite direction from the solid lines [...More...]  "Great Icosahedron" on: Wikipedia Yahoo 

Hexagon In geometry , a HEXAGON (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a six sided polygon or 6gon. The total of the internal angles of any hexagon is 720° [...More...]  "Hexagon" on: Wikipedia Yahoo 

Kepler–Poinsot Polyhedra In geometry , a KEPLER–POINSOT POLYHEDRON is any of four regular star polyhedra . They may be obtained by stellating the regular convex dodecahedron and icosahedron , and differ from these in having regular pentagrammic faces or vertex figures . CONTENTS* 1 Characteristics * 1.1 Nonconvexity * 1.2 Euler characteristic Euler characteristic χ * 1.3 Duality * 1.4 Summary * 2 Relationships among the regular polyhedra * 3 History * 4 Regular star polyhedra in art and culture * 5 See also * 6 References * 7 External links CHARACTERISTICSNONCONVEXITYThese figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures [...More...]  "Kepler–Poinsot Polyhedra" on: Wikipedia Yahoo 

Tetrahedron In geometry , a TETRAHEDRON (plural: TETRAHEDRA or TETRAHEDRONS), also known as a TRIANGULAR PYRAMID, is a polyhedron composed of four triangular faces , six straight edges , and four vertex corners . The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the threedimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3SIMPLEX. The tetrahedron is one kind of pyramid , which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra , a tetrahedron can be folded from a single sheet of paper. It has two such nets [...More...]  "Tetrahedron" on: Wikipedia Yahoo 

Octahedron In geometry , an OCTAHEDRON (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the REGULAR octahedron, a Platonic solid composed of eight equilateral triangles , four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube . It is a rectified tetrahedron . It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the threedimensional case of the more general concept of a cross polytope . A regular octahedron is a 3ball in the Manhattan (ℓ1) metric [...More...]  "Octahedron" on: Wikipedia Yahoo 

Dodecahedron In geometry , a DODECAHEDRON (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron , which is a Platonic solid Platonic solid . There are also three regular star dodecahedra , which are constructed as stellations of the convex form. All of these have icosahedral symmetry , order 120. The pyritohedron is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry . The rhombic dodecahedron , seen as a limiting case of the pyritohedron, has octahedral symmetry . The elongated dodecahedron and trapezorhombic dodecahedron variations, along with the rhombic dodecahedra, are spacefilling . There are a large number of other dodecahedra [...More...]  "Dodecahedron" on: Wikipedia Yahoo 

Icosahedron In geometry , an ICOSAHEDRON (/ˌaɪkɒsəˈhiːdrən, kə, koʊ/ or /aɪˌkɒsəˈhiːdrən/ ) is a polyhedron with 20 faces. The name comes from Greek εἴκοσι (eíkosi), meaning 'twenty', and ἕδρα (hédra), meaning 'seat'. The plural can be either "icosahedra" (/drə/ ) or "icosahedrons". There are many kinds of icosahedra, with some being more symmetrical than others. The best known is the Platonic , convex regular icosahedron [...More...]  "Icosahedron" on: Wikipedia Yahoo 

Apeirogon In geometry , an APEIROGON (from the Greek word ἄπειρος apeiros, "infinite, boundless" and γωνία gonia, "angle") is a generalized polygon with a countably infinite number of sides. It can be considered as the limit of an nsided polygon as n approaches infinity. The interior of a linear apeirogon can be defined by a direction order of vertices, and defining half the plane as the interior. This article describes an APEIROGON in its linear form as a tessellation or partition of a line . CONTENTS * 1 Regular apeirogon * 2 Irregular apeirogon * 3 Apeirogons in hyperbolic plane * 3.1 Pseudogon * 4 See also * 5 References * 6 External links REGULAR APEIROGONA REGULAR APEIROGON has equal edge lengths, just like any regular polygon , {p}. Its Schläfli symbol is {∞}, and its CoxeterDynkin diagram is . It is the first in the dimensional family of regular hypercubic honeycombs [...More...]  "Apeirogon" on: Wikipedia Yahoo 

Order7 Triangular Tiling In geometry , the ORDER7 TRIANGULAR TILING is a regular tiling of the hyperbolic plane with a Schläfli symbol Schläfli symbol of 3,7 . The 3,3,7 honeycomb has 3,7 vertex figures. CONTENTS * 1 Hurwitz surfaces * 2 Related polyhedra and tiling * 3 See also * 4 References * 5 External links HURWITZ SURFACES Further information: Hurwitz surface Hurwitz surface The symmetry group of the tiling is the (2,3,7) triangle group (2,3,7) triangle group , and a fundamental domain for this action is the (2,3,7) Schwarz triangle Schwarz triangle . This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz\'s automorphisms theorem , the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation whose symmetry group equals their automorphism group as Riemann surfaces [...More...]  "Order7 Triangular Tiling" on: Wikipedia Yahoo 

Hypercube In geometry , a HYPERCUBE is an ndimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed , compact , convex figure whose 1skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions , perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimension is equal to n {displaystyle {sqrt {n}}} . An ndimensional hypercube is also called an NCUBE or an NDIMENSIONAL CUBE. The term "measure polytope" is also used, notably in the work of H. S. M. Coxeter Coxeter (originally from Elte, 1912), but it has now been superseded. The hypercube is the special case of a hyperrectangle (also called an northotope). A UNIT HYPERCUBE is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2n points in Rn with each coordinate equal to 0 or 1 is called "THE" UNIT HYPERCUBE [...More...]  "Hypercube" on: Wikipedia Yahoo 

Simplex In geometry , a SIMPLEX (plural: SIMPLEXES or SIMPLICES) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions . Specifically, a KSIMPLEX is a kdimensional polytope which is the convex hull of its k + 1 vertices . More formally, suppose the k + 1 points u 0 , , u k R k {displaystyle u_{0},dots ,u_{k}in mathbb {R} ^{k}} are affinely independent , which means u 1 u 0 , , u k u 0 {displaystyle u_{1}u_{0},dots ,u_{k}u_{0}} are linearly independent . Then, the simplex determined by them is the set of points C = { 0 u 0 + + k u k i = 0 k i = 1 and i 0 for all i } {displaystyle C=left{theta _{0}u_{0}+dots +theta _{k}u_{k}~{bigg }~sum _{i=0}^{k}theta _{i}=1{mbox{ and }}theta _{i}geq 0{mbox{ for all }}iright}} . For example, a 2simplex is a triangle, a 3simplex is a tetrahedron, and a 4simplex is a 5cell [...More...]  "Simplex" on: Wikipedia Yahoo 