# 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$_3$A$_2$A$_1$ | { } | f$_1$ | 2 | 10080 | 12 | 12 | 18 | 4 | 12 | 12 | 6 | 12 | 3 | 4 | 3 | {3,3}x{3} | E$_7$/A$_3$A$_2$A$_1$ = 72*8!/4!/3!/2 = 10080 |
| A$_2$A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | 40320 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | { }∨{3} | E$_7$/A$_2$A$_2$A$_1$ = 72*8!/3!/3!/2 = 40320 |
| A$_3$A$_2$ | {3,3} | f$_3$ | 4 | 6 | 4 | 20160 | * | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}∨( ) | E$_7$/A$_3$A$_2$ = 72*8!/4!/3! = 20160 |
| A$_3$A$_1$A$_1$ | {3,3} | f$_3$ | 4 | 6 | 4 | * | 30240 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | Phyllic disphenoid | E$_7$/A$_3$A$_1$A$_1$ = 72*8!/4!/2/2 = 30240 |
| A$_4$A$_2$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 5 | 0 | 4032 | * | * | 3 | 0 | 0 | 3 | 0 | {3} | E$_7$/A$_4$A$_2$ = 72*8!/5!/3! = 4032 |
| D$_4$A$_1$ | {3,3,4} | f$_4$ | 8 | 24 | 32 | 8 | 8 | * | 7560 | * | 1 | 2 | 0 | 2 | 1 | { }∨( ) | E$_7$/D$_4$A$_1$ = 72*8!/8/4!/2 = 7560 |
| A$_4$A$_1$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 0 | 5 | * | * | 12096 | 0 | 2 | 1 | 1 | 2 | { }∨( ) | E$_7$/A$_4$A$_1$ = 72*8!/5!/2 = 12096 |
| D$_5$A$_1$ | h{4,3,3,3} | f$_5$ | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | 756 | * | * | 2 | 0 | { } | E$_7$/D$_5$A$_1$ = 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$_5$A$_1$ | {3,3,3,3,3} | f$_5$ | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | * | * | 2016 | 0 | 2 | { } | E$_7$/A$_5$A$_1$ = 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$_3$A$_1$ | 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$_2$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 (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$_3$A$_2$A$_1$ | ( ) | 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$_3$A$_2$A$_1$ = 72*8!/4!/3!/2 = 10080 |
| A$_2$A$_1$A$_1$ | { } | 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$_2$A$_1$A$_1$ = 72*8!/3!/2/2 = 120960 |
| A$_2$A$_2$ | 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$_2$A$_2$ = 72*8!/3!/3! = 80640 |
| A$_2$A$_2$A$_1$ | 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$_2$A$_2$A$_1$ = 72*8!/3!/3!/2 = 40320 |
| A$_2$A$_1$A$_1$ | 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$_2$A$_1$A$_1$ = 72*8!/3!/2/2 = 120960 |
| A$_3$A$_2$ | 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$_3$A$_2$ = 72*8!/4!/3! = 20160 |
| A$_3$A$_2$ | 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$_3$A$_2$ = 72*8!/4!/3! = 20160 |
| A$_3$A$_1$ | 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$_3$A$_1$ = 72*8!/4!/2 = 60480 |
| A$_3$A$_1$A$_1$ | 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$_3$A$_1$A$_1$ = 72*8!/4!/2/2 = 30240 |
| A$_3$A$_1$ | 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$_3$A$_1$ = 72*8!/4!/2 = 60480 |
| A$_4$A$_2$ | 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$_4$A$_2$ = 72*8!/5!/3! = 4032 |
| A$_4$A$_1$ | 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$_4$A$_1$ = 72*8!/5!/2 = 12096 |
| D$_4$A$_1$ | 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$_4$A$_1$ = 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$_4$A$_1$ | 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$_4$A$_1$ = 72*8!/5!/2 = 12096 |
| A$_4$A$_1$ | 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$_4$A$_1$ = 72*8!/5!/2 = 12096 |
| D$_5$A$_1$ | 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$_5$A$_1$ = 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$_5$A$_1$ | 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$_5$A$_1$ = 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] |