# 2 31 polytope

| 3$_{21}$ | 3$_{21}$ | 2$_{31}$ | 2$_{31}$ | 1$_{32}$ | 1$_{32}$ |
| Rectified 3$_{21}$ | Rectified 3$_{21}$ | Rectified 3$_{21}$ | birectified 3$_{21}$ | birectified 3$_{21}$ | birectified 3$_{21}$ |
| Rectified 2$_{31}$ | Rectified 2$_{31}$ | Rectified 2$_{31}$ | Rectified 1$_{32}$ | Rectified 1$_{32}$ | Rectified 1$_{32}$ |
| Orthogonal projections in E$_7$ Coxeter plane ||||||
In 7-dimensional geometry, 2$_{31}$ is a uniform polytope, constructed from the E7 group.
Its Coxeter symbol is 2$_{31}$, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.
The rectified 2$_{31}$ is constructed by points at the mid-edges of the 2$_{31}$.
These polytopes are part of a family of 127 (or 2$^7$−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 2$_{31}$ polytope :
| Type | Uniform 7-polytope |
| Family | 2$_{k1}$ polytope |
| Schläfli symbol | {3,3,3$^{3,1}$} |
| Coxeter symbol | 2$_{31}$ |
| Coxeter diagram | |
| 6-faces | 632: 56 2$_{21}$ 576 {3$^5$} |
| 5-faces | 4788: 756 2$_{11}$ 4032 {3$^4$} |
| 4-faces | 16128: 4032 2$_{01}$ 12096 {3$^3$} |
| Cells | 20160 {3$^2$} |
| Faces | 10080 {3} |
| Edges | 2016 |
| Vertices | 126 |
| Vertex figure | 1$_{31}$ |
| Petrie polygon | Octadecagon |
| Coxeter group | E$_7$, [3$^{3,2,1}$] |
| Properties | convex |
The 2$_{31}$ 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 2$_{21}$). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E$_7$.
This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 3$_{31}$.

### Alternate names

* E. L. Elte named it V$_{126}$ (for its 126 vertices) in his 1912 listing of semiregular polytopes.
* It was called 2$_{31}$ 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 3$_{21}$ polytope, .
Removing the node on the end of the 3-length branch leaves the 2$_{21}$. There are 56 of these facets. These facets are centered on the locations of the vertices of the 1$_{32}$ polytope, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 1$_{31}$, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
| E$_7$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_3$ | f$_4$ | f$_4$ | f$_5$ | f$_5$ | f$_6$ | f$_6$ | k-figures | notes |
| D$_6$ | ( ) | f$_0$ | 126 | 32 | 240 | 640 | 160 | 480 | 60 | 192 | 12 | 32 | 6-demicube | E$_7$/D$_6$ = 72x8!/32/6! = 126 |
| A$_5$A$_1$ | { } | f$_1$ | 2 | 2016 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | rectified 5-simplex | E$_7$/A$_5$A$_1$ = 72x8!/6!/2 = 2016 |
| A$_3$A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | 10080 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | tetrahedral prism | E$_7$/A$_3$A$_2$A$_1$ = 72x8!/4!/3!/2 = 10080 |
| A$_3$A$_2$ | {3,3} | f$_3$ | 4 | 6 | 4 | 20160 | 1 | 3 | 3 | 3 | 3 | 1 | tetrahedron | E$_7$/A$_3$A$_2$ = 72x8!/4!/3! = 20160 |
| A$_4$A$_2$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 5 | 4032 | * | 3 | 0 | 3 | 0 | {3} | E$_7$/A$_4$A$_2$ = 72x8!/5!/3! = 4032 |
| A$_4$A$_1$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 5 | * | 12096 | 1 | 2 | 2 | 1 | Isosceles triangle | E$_7$/A$_4$A$_1$ = 72x8!/5!/2 = 12096 |
| D$_5$A$_1$ | {3,3,3,4} | f$_5$ | 10 | 40 | 80 | 80 | 16 | 16 | 756 | * | 2 | 0 | { } | E$_7$/D$_5$A$_1$ = 72x8!/32/5! = 756 |
| A$_5$ | {3,3,3,3} | f$_5$ | 6 | 15 | 20 | 15 | 0 | 6 | * | 4032 | 1 | 1 | { } | E$_7$/A$_5$ = 72x8!/6! = 72*8*7 = 4032 |
| E$_6$ | {3,3,3$^{2,1}$} | f$_6$ | 27 | 216 | 720 | 1080 | 216 | 432 | 27 | 72 | 56 | * | ( ) | E$_7$/E$_6$ = 72x8!/72x6! = 8*7 = 56 |
| A$_6$ | {3,3,3,3,3} | f$_6$ | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 | * | 576 | ( ) | E$_7$/A$_6$ = 72x8!/7! = 72×8 = 576 |

### 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] |

### Related polytopes and honeycombs

2$_{k1}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Coxeter group | E$_3$=A$_2$A$_1$ | E$_4$=A$_4$ | E$_5$=D$_5$ | E$_6$ | E$_7$ | E$_8$ | E$_9$ = ${\tilde {E}}_{8}$ = E$_8$$^+$ | E$_{10}$ = ${\bar {T}}_{8}$ = E$_8$$^{++}$ |
| Coxeter diagram | | | | | | | | |
| Symmetry | [3$^{−1,2,1}$] | [3$^{0,2,1}$] | [[3$^{1,2,1}$]] | [3$^{2,2,1}$] | [3$^{3,2,1}$] | [3$^{4,2,1}$] | [3$^{5,2,1}$] | [3$^{6,2,1}$] |
| Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ∞ |
| Graph | | | | | | | - | - |
| Name | 2$_{−1,1}$ | 2$_{01}$ | 2$_{11}$ | 2$_{21}$ | 2$_{31}$ | 2$_{41}$ | 2$_{51}$ | 2$_{61}$ |

## Rectified 2_31 polytope

Rectified 2$_{31}$ polytope :
| Type | Uniform 7-polytope |
| Family | 2$_{k1}$ polytope |
| Schläfli symbol | {3,3,3$^{3,1}$} |
| Coxeter symbol | t$_1$(2$_{31}$) |
| 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 | E$_7$, [3$^{3,2,1}$] |
| Properties | convex |
The rectified 2$_{31}$ is a rectification of the 2$_{31}$ polytope, creating new vertices on the center of edge of the 2$_{31}$.

### 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 2$_{21}$, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node.

### 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] |