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7-simplex
In 7-dimensional geometry , a 7-simplex
7-simplex
is a self-dual regular 7-polytope . It has 8 vertices , 28 edges , 56 triangle faces , 70 tetrahedral cells , 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. 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 OCTA-7-TOPE, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym OCA
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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
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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 (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
Petrie polygon
becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane
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 higher-dimensional regular polytopes
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7-polytope
In seven-dimensional geometry , a 7-POLYTOPE is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets . A UNIFORM 7-POLYTOPE is one which is vertex-transitive , and constructed from uniform 6-polytope facets. CONTENTS * 1 Regular 7-polytopes * 2 Characteristics * 3 Uniform 7-polytopes 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 7-polytopes * 10 References * 11 External links REGULAR 7-POLYTOPESRegular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with U {p,q,r,s,t} 6-polytopes facets around each 4-face
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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 K-SIMPLEX is a k-dimensional 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 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell
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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 19th-century Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas
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Coxeter-Dynkin 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 order-3. 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
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6-simplex
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
Vertex figure
5-simplex
5-simplex
Circumradius 0.645497 Properties convex , isogonal self-dual In geometry , a 6-simplex
6-simplex
is a self-dual regular 6-polytope . It has 7 vertices , 21 edges , 35 triangle faces , 35 tetrahedral cells , 21 5-cell
5-cell
4-faces, and 7 5-simplex
5-simplex
5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. CONTENTS * 1 Alternate names * 2 Coordinates * 3 Images * 4 Related uniform 6-polytopes * 5 Notes * 6 References * 7 External links ALTERNATE NAMESIt can also be called a HEPTAPETON, or HEPTA-6-TOPE, as a 7-facetted polytope in 6-dimensions
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5-simplex
In five-dimensional geometry , a 5-simplex
5-simplex
is a self-dual regular 5-polytope . It has six vertices , 15 edges , 20 triangle faces , 15 tetrahedral cells , and 6 5-cell 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 5-polytopes * 5 Other forms * 6 Notes * 7 References * 8 External links ALTERNATE NAMESIt can also be called a HEXATERON, or HEXA-5-TOPE, as a 6-facetted polytope in 5-dimensions. The name hexateron is derived from hexa- for having six facets and teron (with ter- being a corruption of tetra- ) for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym HIX
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5-cell
In geometry , the 5-CELL is a four-dimensional object bounded by 5 tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE, PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4-SIMPLEX , the simplest possible convex regular 4-polytope (four-dimensional 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 5-CELL is bounded by regular tetrahedra , and is one of the six regular convex 4-polytopes , represented by Schläfli symbol {3,3,3}
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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 three-dimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3-SIMPLEX. 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
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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 non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space ). This article is about triangles in Euclidean geometry except where otherwise noted
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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
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Octagon
In geometry , an OCTAGON (from the Greek ὀκτάγωνον _oktágōnon_, "eight angles") is an eight-sided polygon or 8-gon. 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}
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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 )
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Self-dual 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) Kepler-Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual . The dual of an isogonal polyhedron, having equivalent