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7simplex In 7dimensional geometry , a 7simplex 7simplex is a selfdual regular 7polytope . It has 8 vertices , 28 edges , 56 triangle faces , 70 tetrahedral cells , 56 5cell 5faces, 28 5simplex 5simplex 6faces, and 8 6simplex 6simplex 7faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°. CONTENTS * 1 Alternate names * 2 Coordinates * 3 Images * 4 Related polytopes * 5 Notes * 6 External links ALTERNATE NAMESIt can also be called an OCTAEXON, or OCTA7TOPE, as an 8facetted polytope in 7dimensions. The name octaexon is derived from octa for eight facets in Greek and ex for having sixdimensional facets, and on. Jonathan Bowers gives an octaexon the acronym OCA [...More...]  "7simplex" on: Wikipedia Yahoo 

Orthogonal Projection In linear algebra and functional analysis , a PROJECTION is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent ). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection . One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object [...More...]  "Orthogonal Projection" on: Wikipedia Yahoo 

Petrie Polygon In geometry , a PETRIE POLYGON for a regular polytope of _n_ dimensions is a skew polygon such that every (_n_1) 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 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 . These polygons and projected graphs are useful in visualizing symmetric structure of the higherdimensional regular polytopes [...More...]  "Petrie Polygon" on: Wikipedia Yahoo 

7polytope In sevendimensional geometry , a 7POLYTOPE is a polytope contained by 6polytope facets. Each 5polytope ridge being shared by exactly two 6polytope facets . A UNIFORM 7POLYTOPE is one which is vertextransitive , and constructed from uniform 6polytope facets. CONTENTS * 1 Regular 7polytopes * 2 Characteristics * 3 Uniform 7polytopes by fundamental Coxeter groups * 4 The A7 family * 5 The B7 family * 6 The D7 family * 7 The E7 family * 8 Regular and uniform honeycombs * 8.1 Regular and uniform hyperbolic honeycombs * 9 Notes on the Wythoff construction Wythoff construction for the uniform 7polytopes * 10 References * 11 External links REGULAR 7POLYTOPESRegular 7polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with U {p,q,r,s,t} 6polytopes facets around each 4face [...More...]  "7polytope" 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 

Schläfli Symbol In geometry , the SCHLäFLI SYMBOL is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations . The Schläfli symbol Schläfli symbol is named after the 19thcentury Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas [...More...]  "Schläfli Symbol" on: Wikipedia Yahoo 

CoxeterDynkin Diagram In geometry , a COXETER–DYNKIN DIAGRAM (or COXETER DIAGRAM, COXETER GRAPH) is a graph with numerically labeled edges (called BRANCHES) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes ). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet ) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge ). An unlabeled branch implicitly represents order3. Each diagram represents a Coxeter group , and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed , while Coxeter diagrams are undirected ; secondly, Dynkin diagrams must satisfy an additional (crystallographic ) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6 [...More...]  "CoxeterDynkin Diagram" on: Wikipedia Yahoo 

6simplex f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7 (χ=0) Coxeter group A6, , order 5040 Bowers name and (acronym) Heptapeton (hop) Vertex figure 5simplex 5simplex Circumradius 0.645497 Properties convex , isogonal selfdual In geometry , a 6simplex 6simplex is a selfdual regular 6polytope . It has 7 vertices , 21 edges , 35 triangle faces , 35 tetrahedral cells , 21 5cell 4faces, and 7 5simplex 5simplex 5faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. CONTENTS * 1 Alternate names * 2 Coordinates * 3 Images * 4 Related uniform 6polytopes * 5 Notes * 6 References * 7 External links ALTERNATE NAMESIt can also be called a HEPTAPETON, or HEPTA6TOPE, as a 7facetted polytope in 6dimensions [...More...]  "6simplex" on: Wikipedia Yahoo 

5simplex In fivedimensional geometry , a 5simplex 5simplex is a selfdual regular 5polytope . It has six vertices , 15 edges , 20 triangle faces , 15 tetrahedral cells , and 6 5cell facets . It has a dihedral angle of cos−1(1/5), or approximately 78.46°. CONTENTS * 1 Alternate names * 2 Regular hexateron cartesian coordinates * 3 Projected images * 4 Related uniform 5polytopes * 5 Other forms * 6 Notes * 7 References * 8 External links ALTERNATE NAMESIt can also be called a HEXATERON, or HEXA5TOPE, as a 6facetted polytope in 5dimensions. The name hexateron is derived from hexa for having six facets and teron (with ter being a corruption of tetra ) for having fourdimensional facets. By Jonathan Bowers, a hexateron is given the acronym HIX [...More...]  "5simplex" on: Wikipedia Yahoo 

5cell In geometry , the 5CELL is a fourdimensional object bounded by 5 tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE, PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4SIMPLEX , the simplest possible convex regular 4polytope (fourdimensional analogue of a Platonic solid Platonic solid ), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base. The REGULAR 5CELL is bounded by regular tetrahedra , and is one of the six regular convex 4polytopes , represented by Schläfli symbol {3,3,3} [...More...]  "5cell" 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 

Triangle A TRIANGLE is a polygon with three edges and three vertices . It is one of the basic shapes in geometry . A triangle with vertices _A_, _B_, and _C_ is denoted A B C {displaystyle triangle ABC} . In Euclidean geometry any three points, when noncollinear, determine a unique triangle and a unique plane (i.e. a twodimensional Euclidean space ). This article is about triangles in Euclidean geometry except where otherwise noted [...More...]  "Triangle" on: Wikipedia Yahoo 

Vertex Figure In geometry , a VERTEX FIGURE, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. CONTENTS* 1 Definitions – theme and variations * 1.1 As a flat slice * 1.2 As a spherical polygon * 1.3 As the set of connected vertices * 1.4 Abstract definition * 2 General properties * 3 Dorman Luke construction * 4 Regular polytopes * 5 An example vertex figure of a honeycomb * 6 Edge figure * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 External links DEFINITIONS – THEME AND VARIATIONSTake some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g [...More...]  "Vertex Figure" on: Wikipedia Yahoo 

Octagon In geometry , an OCTAGON (from the Greek ὀκτάγωνον _oktágōnon_, "eight angles") is an eightsided polygon or 8gon. A _regular octagon_ has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square , t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon , t{16} [...More...]  "Octagon" on: Wikipedia Yahoo 

Coxeter Group In mathematics , a COXETER GROUP, named after H. S. M. Coxeter , is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors ). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups ; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934 ) as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 (Coxeter 1935 ) [...More...]  "Coxeter Group" on: Wikipedia Yahoo 

Selfdual Polytope In geometry , any polyhedron is associated with a second DUAL figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra , but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals also belong to a symmetric class. Thus, the regular polyhedra – the (convex) Platonic solids and (star) KeplerPoinsot polyhedra KeplerPoinsot polyhedra – form dual pairs, where the regular tetrahedron is selfdual . The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces [...More...]  "Selfdual Polytope" on: Wikipedia Yahoo 