title: 2 31 polytope
text: # 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 |
  | $A_5A_1$ | { } | $f_1$ | 2 | 2016 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | rectified 5-simplex | $E_7/A_5A_1$ = 72x8!/6!/2 = 2016 |
  | $A_3A_2A_1$ | {3} | $f_2$ | 3 | 3 | 10080 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | tetrahedral prism | $E_7/A_3A_2A_1$ = 72x8!/4!/3!/2 = 10080 |
  | $A_3A_2$ | {3,3} | $f_3$ | 4 | 6 | 4 | 20160 | 1 | 3 | 3 | 3 | 3 | 1 | tetrahedron | $E_7/A_3A_2$ = 72x8!/4!/3! = 20160 |
  | $A_4A_2$ | {3,3,3} | $f_4$ | 5 | 10 | 10 | 5 | 4032 | * | 3 | 0 | 3 | 0 | {3} | $E_7/A_4A_2$ = 72x8!/5!/3! = 4032 |
  | $A_4A_1$ | {3,3,3} | $f_4$ | 5 | 10 | 10 | 5 | * | 12096 | 1 | 2 | 2 | 1 | Isosceles triangle | $E_7/A_4A_1$ = 72x8!/5!/2 = 12096 |
  | $D_5A_1$ | {3,3,3,4} | $f_5$ | 10 | 40 | 80 | 80 | 16 | 16 | 756 | * | 2 | 0 | { } | $E_7/D_5A_1$ = 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 | $E_3=A_2A_1$ | $E_4=A_4$ | $E_5=D_5$ | $E_6$ | $E_7$ | $E_8$ | $E_9$ = ${\tilde {E}}_{8}$ = $E_8^+$ | $E_{10}$ = ${\bar {T}}_{8}$ = $E_8^{++}$ |
  | 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] |
  