43 relations: Configuration (polytope), Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, E7 (mathematics), E7 polytope, Edge (geometry), Emanuel Lodewijk Elte, Face (geometry), Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, Isosceles triangle, Octadecagon, Petrie polygon, Projection (linear algebra), Rectification (geometry), Rectified 5-simplexes, Rectified 6-simplexes, Schläfli symbol, Simple Lie group, Tetrahedral prism, Tetrahedron, Triangle, Uniform 2 k1 polytope, Uniform 7-polytope, Uniform honeycomb, Uniform polytope, Vertex (geometry), Vertex figure, Wythoff construction, 1 32 polytope, 2 21 polytope, 2 31 polytope, 3 21 polytope, 3 31 honeycomb, 5-cell, 5-orthoplex, 5-simplex, 6-demicube, 6-simplex.

## Configuration (polytope)

In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.

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## Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

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## Coxeter element

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.

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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).

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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).

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## E7 (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.

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## E7 polytope

In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry.

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## Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.

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## Emanuel Lodewijk Elte

Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) at joodsmonument.nl was a Dutch mathematician.

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## Face (geometry)

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.

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## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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## Gosset–Elte figures

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles.

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## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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## Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

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## Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

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## Octadecagon

An octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon.

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## Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every (n – 1) consecutive sides (but no n) belongs to one of the facets.

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## Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

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## Rectification (geometry)

In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.

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## Rectified 5-simplexes

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

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## Rectified 6-simplexes

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

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## Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

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## Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

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## Tetrahedral prism

In geometry, a tetrahedral prism is a convex uniform 4-polytope.

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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.

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## Triangle

A triangle is a polygon with three edges and three vertices.

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## Uniform 2 k1 polytope

In geometry, 2k1 polytope is a uniform polytope in n dimensions (n.

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## Uniform 7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets.

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## Uniform honeycomb

In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets.

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## Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.

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## Vertex (geometry)

In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.

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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.

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## Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.

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## 1 32 polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

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## 2 21 polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group.

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## 2 31 polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

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## 3 21 polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group.

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## 3 31 honeycomb

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

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## 5-cell

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.

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## 5-orthoplex

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

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## 5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

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## 6-demicube

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed.

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## 6-simplex

In geometry, a 6-simplex is a self-dual regular 6-polytope.

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## Redirects here:

Gosset 2 31 polytope, Rectified 2 31 polytope.

## References

[1] https://en.wikipedia.org/wiki/2_31_polytope