2 31 polytope

321		231		132
Rectified 321			birectified 321
File:Up2 2 31 t1 E6.svg Rectified 231			Rectified 132
Orthogonal projections in E6 Coxeter plane

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

Its Coxeter symbol is 2₃₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.

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

These polytopes are part of a family of 127 (or 2⁷−1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

2_31 polytope

Gosset 231 polytope
Type	Uniform 7-polytope
Family	2k1 polytope
Schläfli symbol	{3,3,33,1}
Coxeter symbol	231
Coxeter diagram
6-faces	632: 56 221 576 {35}
5-faces	4788: 756 211 4032 {34}
4-faces	16128: 4032 201 12096 {33}
Cells	20160 {32}
Faces	10080 {3}
Edges	2016
Vertices	126
Vertex figure	131
Petrie polygon	Octadecagon
Coxeter group	E7, [33,2,1]
Properties	convex

The 2₃₁ is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 2₂₁). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E₇.

This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 3₃₁.

Alternate names

• E. L. Elte 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, 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 upon a set of 7 hyperplane mirrors in 7-dimensional space.

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

Removing the node on the short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 3₂₁ polytope, .

Removing the node on the end of the 3-length branch leaves the 2₂₁. There are 56 of these facets. These facets are centered on the locations of the vertices of the 1₃₂ polytope, .

The vertex figure 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
200px [6]	200px [12/2]	200px [10]
D5 / B4 / A4	D4 / B3 / A2 / G2	D3 / B2 / A3
200px [8]	200px [6]	200px [4]

Related polytopes and honeycombs

2k1 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde{E}}_{8}$ = E8+	E10 = ${\bar{T}}_8$ = E8++
Coxeter diagram
Symmetry	[3−1,2,1]	[30,2,1]	[[31,2,1]]	[32,2,1]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
Order	12	120	384	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	2-1,1	201	211	221	231	241	251	261

Rectified 2_31 polytope

Rectified 231 polytope
Type	Uniform 7-polytope
Family	2k1 polytope
Schläfli symbol	{3,3,33,1}
Coxeter symbol	t1(231)
Coxeter diagram
6-faces	758
5-faces	10332
4-faces	47880
Cells	100800
Faces	90720
Edges	30240
Vertices	2016
Vertex figure	6-demicube
Petrie polygon	Octadecagon
Coxeter group	E7, [33,2,1]
Properties	convex

The rectified 2₃₁ is a rectification 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 upon a set of 7 hyperplane mirrors in 7-dimensional space.

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

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

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 is determined by removing the ringed node and ringing the neighboring node.

Images

Coxeter plane projections

E7	E6 / F4	B6 / A6
File:Up2 2 31 t1 E7.svg [18]	File:Up2 2 31 t1 E6.svg [12]	200px [7x2]
A5	D7 / B6	D6 / B5
200px [6]	200px [12/2]	File:Up2 2 31 t1 D6.svg [10]
D5 / B4 / A4	D4 / B3 / A2 / G2	D3 / B2 / A3
File:Up2 2 31 t1 D5.svg [8]	200px [6]	200px [4]

See also

• List of E7 polytopes

References

• H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Richard Klitzing, 7D, uniform polytopes (polyexa) x3o3o3o *c3o3o3o - laq, o3x3o3o *c3o3o3o - rolaq