Gosset 2 31 polytope - AbsoluteAstronomy.com

Orthogonal projections in E₆ Coxeter plane
3₂₁	Rectified 3₂₁	Birectified 3₂₁	Rectified 1₃₂
1₃₂	2₃₁	Rectified 2₃₁	Rectified 1₃₂

In 7-dimensional geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, 2₃₁ is a uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

, constructed from the E7 group.

Coxeter named it 2₃₁ by its bifurcating Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, with a single ring on the end of one of the 2-node sequences.

The rectified 2₃₁ is constructed by points at the mid-edges of the 2₃₁.

These polytopes are part of a family of 127 (2⁷-1) convex uniform polytopes in 7-dimensions, made of uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

facets and vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

s, defined by all permutations of rings in this Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

: .

2_31 polytope

Gosset 2₃₁ polytope
Type	Uniform 7-polytope
Family	2_k1 polytope Uniform 2 k1 polytope In geometry, 2k1 polytope is a uniform polytope in n dimensions constructed from the En Coxeter group. The family was named by Coxeter as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence...
Schläfli symbol	{3,3,3^3,1}
Coxeter symbol	2₃₁
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
6-faces	632: 56 2₂₁ 576 {3⁵}
5-faces	4788: 756 2₁₁ Pentacross 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 hypercells.... 4032 {3⁴}
4-faces	16128: 4032 2₀₁ Pentachoron In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid... 12096 {3³} Pentachoron In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid...
Cells	20160 {3²} Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
Faces	10080 {3} Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
Edges	2016
Vertices	126
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	1₃₁
Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...	Octadecagon Octadecagon An octadecagon is a polygon with 18 sides and 18 vertices. Another name for an octadecagon is octakaidecagon.- Construction :A regular octadecagon cannot be constructed using compass and straightedge.- Petrie polygons :...
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...	E₇, [3^3,2,1]
Properties	convex 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...

The 2₃₁ is composed of 126 vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

, 2016 edges
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....

, 10080 faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

(Triangles), 20160 cells
Cell (geometry)
In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...

(tetrahedra
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacross
Pentacross
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 hypercells....

es, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 2₂₁). Its vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

is a 6-demicube.
Its 126 vertices represent the root vectors of the simple Lie group

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.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...

E₇.

This polytope is the vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

for a uniform tessellation

Uniform tessellation

In geometry, a uniform tessellation is a vertex-transitive tessellations made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex....

of 7-dimensional space, 3₃₁.

Alternate names

E. L. Elte
E. L. Elte
Emanuel Lodewijk Elte was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher....

named it V₁₂₆ (for its 126 vertices) in his 1912 listing of semiregular polytopes.
It was called 2₃₁ by Coxeter for its bifurcating Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, with a single ring on the end of the 2-node sequence.
Pentacontihexa-pentacosiheptacontihexa-exon (Acronym laq) - 56-576 facetted polyexon (Jonathan Bowers)

Construction

It is created by a Wythoff construction

Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.- Construction process :...

upon a set of 7 hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, .

Removing the node on the short branch leaves the 6-simplex. There are 56 of these facets. These facets are centered on the locations of the vertices of the 3₂₁

Gosset 3 21 polytope

In 7-dimensional geometry, the 321 polytope is a uniform 6-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper...

polytope, .

Removing the node on the end of the 3-length branch leaves the 2₂₁

Gosset 2 21 polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure....

. There are 576 of these facets. These facets are centered on the locations of the vertices of the 1₃₂

Gosset 1 32 polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.Coxeter named it 132 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences....

polytope, .

The vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 1₃₁, .

Images

Coxeter plane projections
E7	E6 / F4	B6 / A6
[18]	[12]	[7x2]
A5	D7 / B6	D6 / B5
[6]	[12/2]	[10]
D5 / B4 / A4	D4 / B3 / A2 / G2	D3 / B2 / A3
[8]	[6]	[4]

Rectified 2_31 polytope

Rectified 2₃₁ polytope
Type	Uniform 7-polytope
Family	2_k1 polytope Uniform 2 k1 polytope In geometry, 2k1 polytope is a uniform polytope in n dimensions constructed from the En Coxeter group. The family was named by Coxeter as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence...
Schläfli symbol	{3,3,3^3,1}
Coxeter symbol	2₃₁
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
6-faces	758
5-faces	10332
4-faces	47880
Cells	100800
Faces	90720
Edges	30240
Vertices	2016
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	6-demicube
Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...	Octadecagon Octadecagon An octadecagon is a polygon with 18 sides and 18 vertices. Another name for an octadecagon is octakaidecagon.- Construction :A regular octadecagon cannot be constructed using compass and straightedge.- Petrie polygons :...
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...	E₇, [3^3,2,1]
Properties	convex 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...

The rectified 2₃₁ is a rectification

Rectification (geometry)

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

of the 2₃₁ polytope, creating new vertices on the center of edge of the 2₃₁.

Alternate names

Rectified pentacontihexa-pentacosiheptacontihexa-exon - as a rectified 56-576 facetted polyexon (acronym rolaq) (Jonathan Bowers)

Construction

It is created by a Wythoff construction

Wythoff construction

upon a set of 7 hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, .

Removing the node on the short branch leaves the rectified 6-simplex

Rectified 6-simplex

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the...

, .

Removing the node on the end of the 2-length branch leaves the, 6-demicube,
.

Removing the node on the end of the 3-length branch leaves the rectified 2₂₁, .

The vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 1₃₁, .

Images

Coxeter plane projections
E7	E6 / F4	B6 / A6
[18]	[12]	[7x2]
A5	D7 / B6	D6 / B5
[6]	[12/2]	[10]
D5 / B4 / A4	D4 / B3 / A2 / G2	D3 / B2 / A3
[8]	[6]	[4]

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