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2 31 polytope

Index 2 31 polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. [1]

Table of Contents

  1. 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.

  2. 7-polytopes

Configuration (polytope)

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

See 2 31 polytope and Configuration (polytope)

Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n.

See 2 31 polytope and Convex polytope

Coxeter element

In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections.

See 2 31 polytope and Coxeter element

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).

See 2 31 polytope and Coxeter group

Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

See 2 31 polytope and Coxeter–Dynkin diagram

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.

See 2 31 polytope and E7 (mathematics)

E7 polytope

In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry. 2 31 polytope and E7 polytope are 7-polytopes.

See 2 31 polytope and E7 polytope

Edge (geometry)

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

See 2 31 polytope and Edge (geometry)

Emanuel Lodewijk Elte

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

See 2 31 polytope and Emanuel Lodewijk Elte

Face (geometry)

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

See 2 31 polytope and Face (geometry)

Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

See 2 31 polytope and Geometry

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.

See 2 31 polytope and Gosset–Elte figures

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.

See 2 31 polytope and Harold Scott MacDonald Coxeter

Hyperplane

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension.

See 2 31 polytope and Hyperplane

Isosceles triangle

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

See 2 31 polytope and Isosceles triangle

Octadecagon

In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon.

See 2 31 polytope and Octadecagon

Petrie polygon

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

See 2 31 polytope and Petrie polygon

Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P.

See 2 31 polytope and Projection (linear algebra)

Rectification (geometry)

In Euclidean geometry, rectification, also known as critical truncation 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.

See 2 31 polytope and Rectification (geometry)

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.

See 2 31 polytope and Rectified 5-simplexes

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.

See 2 31 polytope and Rectified 6-simplexes

Schläfli symbol

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

See 2 31 polytope and Schläfli symbol

Simple Lie group

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

See 2 31 polytope and Simple Lie group

Tetrahedral prism

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

See 2 31 polytope and Tetrahedral prism

Tetrahedron

In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.

See 2 31 polytope and Tetrahedron

Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.

See 2 31 polytope and Triangle

Uniform 2 k1 polytope

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

See 2 31 polytope and Uniform 2 k1 polytope

Uniform 7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. 2 31 polytope and Uniform 7-polytope are 7-polytopes.

See 2 31 polytope and Uniform 7-polytope

Uniform honeycomb

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

See 2 31 polytope and Uniform honeycomb

Uniform polytope

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

See 2 31 polytope and Uniform polytope

Vertex (geometry)

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

See 2 31 polytope and Vertex (geometry)

Vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

See 2 31 polytope and Vertex figure

Wythoff construction

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

See 2 31 polytope and Wythoff construction

1 32 polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. 2 31 polytope and 1 32 polytope are 7-polytopes.

See 2 31 polytope and 1 32 polytope

2 21 polytope

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

See 2 31 polytope and 2 21 polytope

2 31 polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. 2 31 polytope and 2 31 polytope are 7-polytopes.

See 2 31 polytope and 2 31 polytope

3 21 polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. 2 31 polytope and 3 21 polytope are 7-polytopes.

See 2 31 polytope and 3 21 polytope

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.

See 2 31 polytope and 3 31 honeycomb

5-cell

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol.

See 2 31 polytope and 5-cell

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.

See 2 31 polytope and 5-orthoplex

5-simplex

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

See 2 31 polytope and 5-simplex

6-demicube

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

See 2 31 polytope and 6-demicube

6-simplex

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

See 2 31 polytope and 6-simplex

See also

7-polytopes

References

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

Also known as Gosset 2 31 polytope, Rectified 2 31 polytope.