title: 1 32 polytope
text: # 1 32 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, $1_{32}$ is a uniform polytope, constructed from the E7 group.
  
  Its Coxeter symbol is $1_{32},$ describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.
  
  The rectified $1_{32}$ is constructed by points at the mid-edges of the $1_{32}.$
  
  These polytopes are part of a family of 127 $(2^7-1)$ 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: .
  
  ## 1_32 polytope
  
  $1_{32}$ :
  
  | Type | Uniform 7-polytope |
  | - | - |
  | Family | $1_{k2}$ polytope |
  | Schläfli symbol | ${3,3^{3,2}}$ |
  | Coxeter symbol | $1_{32}$ |
  | Coxeter diagram | |
  | 6-faces | 182: 56 $1_{22}$ 126 $1_{31}$ |
  | 5-faces | 4284: 756 $1_{21}$ 1512 $1_{21}$ 2016 ${3^4}$ |
  | 4-faces | 23688: 4032 ${3^3}$ 7560 $1_{11}$ 12096 ${3^3}$ |
  | Cells | 50400: 20160 ${3^2}$ 30240 ${3^2}$ |
  | Faces | 40320 {3} |
  | Edges | 10080 |
  | Vertices | 576 |
  | Vertex figure | $t_2{3^5}$ |
  | Petrie polygon | Octadecagon |
  | Coxeter group | $E_7,$ $[3^{3,2,1}],$ order 2903040 |
  | Properties | convex |
  
  This polytope can tessellate 7-dimensional space, with symbol $1_{33},$ and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual $E_7^*$ lattice.
  
  ### Alternate names
  
  * Emanuel Lodewijk Elte named it $V_{576}$ (for its 576 vertices) in his 1912 listing of semiregular polytopes.
  * Coxeter called it $1_{32}$ for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
  * Pentacontihexa-hecatonicosihexa-exon (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)
  
  ### Images
  
  Coxeter plane projections :
  
  | E7 | E6 / F4 | B7 / 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] |
  
  ### 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 end of the 2-length branch leaves the 6-demicube, $1_{31},$
  
  Removing the node on the end of the 3-length branch leaves the $1_{22},$
  
  The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, $0_{32},$
  
  Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
  
  | $E_7$ | k-face | $f_k$ | $f_0$ | $f_1$ | $f_2$ | $f_3$ | $f_3$ | $f_4$ | $f_4$ | $f_4$ | $f_5$ | $f_5$ | $f_5$ | $f_6$ | $f_6$ | k-figures | notes |
  | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
  | $A_6$ | ( ) | $f_0$ | 576 | 35 | 210 | 140 | 210 | 35 | 105 | 105 | 21 | 42 | 21 | 7 | 7 | 2r{3,3,3,3,3} | $E_7/A_6$ = 72*8!/7! = 576 |
  | $A_3A_2A_1$ | { } | $f_1$ | 2 | 10080 | 12 | 12 | 18 | 4 | 12 | 12 | 6 | 12 | 3 | 4 | 3 | {3,3}x{3} | $E_7/A_3A_2A_1$ = 72*8!/4!/3!/2 = 10080 |
  | $A_2A_2A_1$ | {3} | $f_2$ | 3 | 3 | 40320 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | { }∨{3} | $E_7/A_2A_2A_1$ = 72*8!/3!/3!/2 = 40320 |
  | $A_3A_2$ | {3,3} | $f_3$ | 4 | 6 | 4 | 20160 | * | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}∨( ) | $E_7/A_3A_2$ = 72*8!/4!/3! = 20160 |
  | $A_3A_1A_1$ | {3,3} | $f_3$ | 4 | 6 | 4 | * | 30240 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | Phyllic disphenoid | $E_7/A_3A_1A_1$ = 72*8!/4!/2/2 = 30240 |
  | $A_4A_2$ | {3,3,3} | $f_4$ | 5 | 10 | 10 | 5 | 0 | 4032 | * | * | 3 | 0 | 0 | 3 | 0 | {3} | $E_7/A_4A_2$ = 72*8!/5!/3! = 4032 |
  | $D_4A_1$ | {3,3,4} | $f_4$ | 8 | 24 | 32 | 8 | 8 | * | 7560 | * | 1 | 2 | 0 | 2 | 1 | { }∨( ) | $E_7/D_4A_1$ = 72*8!/8/4!/2 = 7560 |
  | $A_4A_1$ | {3,3,3} | $f_4$ | 5 | 10 | 10 | 0 | 5 | * | * | 12096 | 0 | 2 | 1 | 1 | 2 | { }∨( ) | $E_7/A_4A_1$ = 72*8!/5!/2 = 12096 |
  | $D_5A_1$ | h{4,3,3,3} | $f_5$ | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | 756 | * | * | 2 | 0 | { } | $E_7/D_5A_1$ = 72*8!/16/5!/2 = 756 |
  | $D_5$ | h{4,3,3,3} | $f_5$ | 16 | 80 | 160 | 40 | 80 | 0 | 10 | 16 | * | 1512 | * | 1 | 1 | { } | $E_7/D_5$ = 72*8!/16/5! = 1512 |
  | $A_5A_1$ | {3,3,3,3,3} | $f_5$ | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | * | * | 2016 | 0 | 2 | { } | $E_7/A_5A_1$ = 72*8!/6!/2 = 2016 |
  | $E_6$ | ${3,3^{2,2}}$ | $f_6$ | 72 | 720 | 2160 | 1080 | 1080 | 216 | 270 | 216 | 27 | 27 | 0 | 56 | * | ( ) | $E_7/E_6$ = 72*8!/72/6! = 56 |
  | $D_6$ | h{4,3,3,3,3} | $f_6$ | 32 | 240 | 640 | 160 | 480 | 0 | 60 | 192 | 0 | 12 | 32 | * | 126 | ( ) | $E_7/D_6$ = 72*8!/32/6! = 126 |
  
  ### Related polytopes and honeycombs
  
  The $1_{32}$ is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as $1_{3k}$ series. The next figure is the Euclidean honeycomb $1_{33}$ and the final is a noncompact hyperbolic honeycomb, $1_{34}.$
  
  $1_{3k}$ dimensional figures :
  
  | Space | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
  | - | - | - | - | - | - | - |
  | n | 4 | 5 | 6 | 7 | 8 | 9 |
  | Coxeter group | $A_3A_1$ | $A_5$ | $D_6$ | $E_7$ | ${\tilde {E}}_{7}$$=E_7^+$ | ${\bar {T}}_{8}$$=E_7^{++}$ |
  | Coxeter diagram | | | | | | |
  | Symmetry | $[3^{−1,3,1}]$ | $[3^{0,3,1}]$ | $[3^{1,3,1}]$ | $[3^{2,3,1}]$ | $[[3^{3,3,1}]]$ | $[3^{4,3,1}]$ |
  | Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | ∞ |
  | Graph | | | | | - | - |
  | Name | $1_{3,-1}$ | $1_{30}$ | $1_{31}$ | $1_{32}$ | $1_{33}$ | $1_{34}$ |
  
  $1_{k2}$ 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 (order) | $[3^{−1,2,1}]$ | $[3^{0,2,1}]$ | $[3^{1,2,1}]$ | $[[3^{2,2,1}]]$ | $[3^{3,2,1}]$ | $[3^{4,2,1}]$ | $[3^{5,2,1}]$ | $[3^{6,2,1}]$ |
  | Order | 12 | 120 | 1,920 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ∞ |
  | Graph | | | | | | | - | - |
  | Name | $1_{−1,2}$ | $1_{02}$ | $1_{12}$ | $1_{22}$ | $1_{32}$ | $1_{42}$ | $1_{52}$ | $1_{62}$ |
  
  ## Rectified 1_32 polytope
  
  Rectified $1_{32}$ :
  
  | Type | Uniform 7-polytope |
  | - | - |
  | Schläfli symbol | $t_1{3,3^{3,2}}$ |
  | Coxeter symbol | $0_{321}$ |
  | Coxeter-Dynkin diagram | |
  | 6-faces | 758 |
  | 5-faces | 12348 |
  | 4-faces | 72072 |
  | Cells | 191520 |
  | Faces | 241920 |
  | Edges | 120960 |
  | Vertices | 10080 |
  | Vertex figure | {3,3}×{3}× |
  | Coxeter group | $E_7,$ $[3^{3,2,1}],$ order 2903040 |
  | Properties | convex |
  
  The rectified $1_{32}$ (also called $0_{321})$ is a rectification of the $1_{32}$ polytope, creating new vertices on the center of edge of the $1_{32}.$ Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×.
  
  ### Alternate names
  
  * Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (Jonathan Bowers)
  
  ### Construction
  
  It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).
  
  Removing the node on the end of the 3-length branch leaves the rectified $1_{22}$ polytope,
  
  Removing the node on the end of the 2-length branch leaves the demihexeract, $1_{31},$
  
  Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,
  
  The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×,
  
  Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
  
  | $E_7$ | k-face | $f_k$ | $f_0$ | $f_1$ | $f_2$ | $f_2$ | $f_2$ | $f_3$ | $f_3$ | $f_3$ | $f_3$ | $f_3$ | $f_4$ | $f_4$ | $f_4$ | $f_4$ | $f_4$ | $f_4$ | $f_5$ | $f_5$ | $f_5$ | $f_5$ | $f_5$ | $f_6$ | $f_6$ | $f_6$ | k-figures | notes |
  | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
  | $A_3A_2A_1$ | ( ) | $f_0$ | 10080 | 24 | 24 | 12 | 36 | 8 | 12 | 36 | 18 | 24 | 4 | 12 | 18 | 24 | 12 | 6 | 6 | 8 | 12 | 6 | 3 | 4 | 2 | 3 | {3,3}x{3}x{ } | $E_7/A_3A_2A_1$ = 72*8!/4!/3!/2 = 10080 |
  | $A_2A_1A_1$ | { } | $f_1$ | 2 | 120960 | 2 | 1 | 3 | 1 | 2 | 6 | 3 | 3 | 1 | 3 | 6 | 6 | 3 | 1 | 3 | 3 | 6 | 2 | 1 | 3 | 1 | 2 | ( )v{3}v{ } | $E_7/A_2A_1A_1$ = 72*8!/3!/2/2 = 120960 |
  | $A_2A_2$ | $0_1$ | $f_2$ | 3 | 3 | 80640 | * | * | 1 | 1 | 3 | 0 | 0 | 1 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 1 | 0 | 3 | 1 | 1 | {3}v( )v( ) | $E_7/A_2A_2$ = 72*8!/3!/3! = 80640 |
  | $A_2A_2A_1$ | $0_1$ | $f_2$ | 3 | 3 | * | 40320 | * | 0 | 2 | 0 | 3 | 0 | 1 | 0 | 6 | 0 | 3 | 0 | 3 | 0 | 6 | 0 | 1 | 3 | 0 | 2 | {3}v{ } | $E_7/A_2A_2A_1$ = 72*8!/3!/3!/2 = 40320 |
  | $A_2A_1A_1$ | $0_1$ | $f_2$ | 3 | 3 | * | * | 120960 | 0 | 0 | 2 | 1 | 2 | 0 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | { }v{ }v( ) | $E_7/A_2A_1A_1$ = 72*8!/3!/2/2 = 120960 |
  | $A_3A_2$ | $0_2$ | $f_3$ | 4 | 6 | 4 | 0 | 0 | 20160 | * | * | * | * | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 3 | 1 | 0 | {3}v( ) | $E_7/A_3A_2$ = 72*8!/4!/3! = 20160 |
  | $A_3A_2$ | $0_{11}$ | $f_3$ | 6 | 12 | 4 | 4 | 0 | * | 20160 | * | * | * | 1 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 0 | 3 | 0 | 1 | {3}v( ) | $E_7/A_3A_2$ = 72*8!/4!/3! = 20160 |
  | $A_3A_1$ | $0_{11}$ | $f_3$ | 6 | 12 | 4 | 0 | 4 | * | * | 60480 | * | * | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 2 | 2 | 1 | 0 | 2 | 1 | 1 | Sphenoid | $E_7/A_3A_1$ = 72*8!/4!/2 = 60480 |
  | $A_3A_1A_1$ | $0_{11}$ | $f_3$ | 6 | 12 | 0 | 4 | 4 | * | * | * | 30240 | * | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 0 | 1 | 2 | 0 | 2 | { }v{ } | $E_7/A_3A_1A_1$ = 72*8!/4!/2/2 = 30240 |
  | $A_3A_1$ | $0_2$ | $f_3$ | 4 | 6 | 0 | 0 | 4 | * | * | * | * | 60480 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | Sphenoid | $E_7/A_3A_1$ = 72*8!/4!/2 = 60480 |
  | $A_4A_2$ | $0_{21}$ | $f_4$ | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 4032 | * | * | * | * | * | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | {3} | $E_7/A_4A_2$ = 72*8!/5!/3! = 4032 |
  | $A_4A_1$ | $0_{21}$ | $f_4$ | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 12096 | * | * | * | * | 1 | 2 | 0 | 0 | 0 | 2 | 1 | 0 | { }v | $E_7/A_4A_1$ = 72*8!/5!/2 = 12096 |
  | $D_4A_1$ | $0_{111}$ | $f_4$ | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 8 | 0 | * | * | 7560 | * | * | * | 1 | 0 | 2 | 0 | 0 | 2 | 0 | 1 | { }v | $E_7/D_4A_1$ = 72*8!/8/4!/2 = 7560 |
  | $A_4$ | $0_{21}$ | $f_4$ | 10 | 30 | 10 | 0 | 20 | 0 | 0 | 5 | 0 | 5 | * | * | * | 24192 | * | * | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | ( )v( )v( ) | $E_7/A_4$ = 72*8!/5! = 34192 |
  | $A_4A_1$ | $0_{21}$ | $f_4$ | 10 | 30 | 0 | 10 | 20 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 12096 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | { }v | $E_7/A_4A_1$ = 72*8!/5!/2 = 12096 |
  | $A_4A_1$ | $0_3$ | $f_4$ | 5 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 5 | * | * | * | * | * | 12096 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | { }v | $E_7/A_4A_1$ = 72*8!/5!/2 = 12096 |
  | $D_5A_1$ | $0_{211}$ | $f_5$ | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 40 | 0 | 16 | 16 | 10 | 0 | 0 | 0 | 756 | * | * | * | * | 2 | 0 | 0 | { } | $E_7/D_5A_1$ = 72*8!/16/5!/2 = 756 |
  | $A_5$ | $0_{22}$ | $f_5$ | 20 | 90 | 60 | 0 | 60 | 15 | 0 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | * | 4032 | * | * | * | 1 | 1 | 0 | { } | $E_7/A_5$ = 72*8!/6! = 4032 |
  | $D_5$ | $0_{211}$ | $f_5$ | 80 | 480 | 160 | 160 | 320 | 0 | 40 | 80 | 80 | 80 | 0 | 0 | 10 | 16 | 16 | 0 | * | * | 1512 | * | * | 1 | 0 | 1 | { } | $E_7/D_5$ = 72*8!/16/5! = 1512 |
  | $A_5$ | $0_{31}$ | $f_5$ | 15 | 60 | 20 | 0 | 60 | 0 | 0 | 15 | 0 | 30 | 0 | 0 | 0 | 6 | 0 | 6 | * | * | * | 4032 | * | 0 | 1 | 1 | { } | $E_7/A_5$ = 72*8!/6! = 4032 |
  | $A_5A_1$ | $0_{31}$ | $f_5$ | 15 | 60 | 0 | 20 | 60 | 0 | 0 | 0 | 15 | 30 | 0 | 0 | 0 | 0 | 6 | 6 | * | * | * | * | 2016 | 0 | 0 | 2 | { } | $E_7/A_5A_1$ = 72*8!/6!/2 = 2016 |
  | $E_6$ | $0_{221}$ | $f_6$ | 720 | 6480 | 4320 | 2160 | 4320 | 1080 | 1080 | 2160 | 1080 | 1080 | 216 | 432 | 270 | 432 | 216 | 0 | 27 | 72 | 27 | 0 | 0 | 56 | * | * | ( ) | $E_7/E_6$ = 72*8!/72/6! = 56 |
  | $A_6$ | $0_{32}$ | $f_6$ | 35 | 210 | 140 | 0 | 210 | 35 | 0 | 105 | 0 | 105 | 0 | 21 | 0 | 42 | 0 | 21 | 0 | 7 | 0 | 7 | 0 | * | 576 | * | ( ) | $E_7/A_6$ = 72*8!/7! = 576 |
  | $D_6$ | $0_{311}$ | $f_6$ | 240 | 1920 | 640 | 640 | 1920 | 0 | 160 | 480 | 480 | 960 | 0 | 0 | 60 | 192 | 192 | 192 | 0 | 0 | 12 | 32 | 32 | * | * | 126 | ( ) | $E_7/D_6$ = 72*8!/32/6! = 126 |
  
  ### Images
  
  Coxeter plane projections :
  
  | E7 | E6 / F4 | B7 / A6 |
  | - | - | - |
  | [18] | [12] | [14] |
  | A5 | D7 / B6 | D6 / B5 |
  | [6] | [12/2] | [10] |
  | D5 / B4 / A4 | D4 / B3 / A2 / G2 | D3 / B2 / A3 |
  | [8] | [6] | [4] |
  