1 22 polytope
# 1 22 polytope

| $1_{22}$ | Rectified $1_{22}$ | Birectified $1_{22}$ |
| - | - | - |
| $2_{21}$ | Rectified $2_{21}$ | Birectified $1_{22}$ |
| orthogonal projections in $E_6$ Coxeter plane |||

In 6-dimensional geometry, the $1_{22}$ polytope is a uniform polytope, constructed from the $E_6$ group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as $V_{72}$ (for its 72 vertices).

Its Coxeter symbol is $1_{22},$ describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the $1_{22},$ constructed by positions points on the elements of $1_{22}.$ The rectified $1_{22}$ is constructed by points at the mid-edges of the $1_{22}.$ The birectified $1_{22}$ is constructed by points at the triangle face centers of the $1_{22}.$

These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

## $1_{22}$ polytope

$1_{22}$ polytope :

| Type | Uniform 6-polytope |
| - | - |
| Family | $1_{k2}$ polytope |
| Schläfli symbol | ${3,3^{2,2}}$ |
| Coxeter symbol | $1_{22}$ |
| Coxeter-Dynkin diagram | or |
| 5-faces | 54: 27 $1_{21}$ 27 $1_{21}$ |
| 4-faces | 702: 270 $1_{11}$ 432 $1_{20}$ |
| Cells | 2160: 1080 $1_{10}$ 1080 {3,3} |
| Faces | 2160 {3} |
| Edges | 720 |
| Vertices | 72 |
| Vertex figure | Birectified 5-simplex: $0_{22}$ |
| Petrie polygon | Dodecagon |
| Coxeter group | $E_6,$ $[[3,3^{2,2}]],$ order 103680 |
| Properties | convex, isotopic |

The $1_{22}$ polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group $E_6.$

### Alternate names

* Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)

### Images

Coxeter plane orthographic projections :

| E6 [12] | D5 [8] | D4 / A2 [6] |
| - | - | - |
| (1,2) | (1,3) | (1,9,12) |
| B6 [12/2] | A5 [6] | A4 [[5]] = [10] | A3 / D3 [4] |
| (1,2) | (2,3,6) | (1,2) | (1,6,8,12) |

### Construction

It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.

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

Removing the node on either of 2-length branches leaves the 5-demicube, $1_{31},$ .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, $0_{22},$ .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.

| $E_6$ | k-face | $f_k$ | $f_0$ | $f_1$ | $f_2$ | $f_3$ | $f_3$ | $f_4$ | $f_4$ | $f_4$ | $f_5$ | $f_5$ | k-figure | notes |
| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
| $A_5$ | ( ) | $f_0$ | 72 | 20 | 90 | 60 | 60 | 15 | 15 | 30 | 6 | 6 | r{3,3,3} | $E_6/A_5$ = 72*6!/6! = 72 |
| $A_2A_2A_1$ | { } | $f_1$ | 2 | 720 | 9 | 9 | 9 | 3 | 3 | 9 | 3 | 3 | {3}×{3} | $E_6/A_2A_2A_1$ = 72*6!/3!/3!/2 = 720 |
| $A_2A_1A_1$ | {3} | $f_2$ | 3 | 3 | 2160 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | s{2,4} | $E_6/A_2A_1A_1$ = 72*6!/3!/2/2 = 2160 |
| $A_3A_1$ | {3,3} | $f_3$ | 4 | 6 | 4 | 1080 | * | 1 | 0 | 2 | 2 | 1 | { }∨( ) | $E_6/A_3A_1$ = 72*6!/4!/2 = 1080 |
| $A_3A_1$ | {3,3} | $f_3$ | 4 | 6 | 4 | * | 1080 | 0 | 1 | 2 | 1 | 2 | { }∨( ) | $E_6/A_3A_1$ = 72*6!/4!/2 = 1080 |
| $A_4A_1$ | {3,3,3} | $f_4$ | 5 | 10 | 10 | 5 | 0 | 216 | * | * | 2 | 0 | { } | $E_6/A_4A_1$ = 72*6!/5!/2 = 216 |
| $A_4A_1$ | {3,3,3} | $f_4$ | 5 | 10 | 10 | 0 | 5 | * | 216 | * | 0 | 2 | { } | $E_6/A_4A_1$ = 72*6!/5!/2 = 216 |
| $D_4$ | h{4,3,3} | $f_4$ | 8 | 24 | 32 | 8 | 8 | * | * | 270 | 1 | 1 | { } | $E_6/D_4$ = 72*6!/8/4! = 270 |
| $D_5$ | h{4,3,3,3} | $f_5$ | 16 | 80 | 160 | 80 | 40 | 16 | 0 | 10 | 27 | * | ( ) | $E_6/D_5$ = 72*6!/16/5! = 27 |
| $D_5$ | h{4,3,3,3} | $f_5$ | 16 | 80 | 160 | 40 | 80 | 0 | 16 | 10 | * | 27 | ( ) | $E_6/D_5$ = 72*6!/16/5! = 27 |

### Related complex polyhedron

The regular complex polyhedron $_3{3}_3{4}_2,$ , in $\mathbb {C} ^{2}$ has a real representation as the $1_{22}$ polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is $_3[3]_3_2,$ order 1296. It has a half-symmetry quasiregular construction as , as a rectification of the Hessian polyhedron, .

### Related polytopes and honeycomb

Along with the semiregular polytope, $2_{21},$ it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

$1_{k2}$ 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 | $1_{−1,2}$ | $1_{02}$ | $1_{12}$ | $1_{22}$ | $1_{32}$ | $1_{42}$ | $1_{52}$ | $1_{62}$ |

**Geometric folding**

The $1_{22}$ is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to $1_{22}$ in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the $1_{22}.$

E6/F4 Coxeter planes :

| $1_{22}$ | 24-cell |
| - | - |
| D4/B4 Coxeter planes ||
| $1_{22}$ | 24-cell |

**Tessellations**

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, $2_{22},$ .

## Rectified $1_{22}$ polytope

Rectified $1_{22}$ :

| Type | Uniform 6-polytope |
| - | - |
| Schläfli symbol | $2r{3,3,3^{2,1}}$ $r{3,3^{2,2}}$ |
| Coxeter symbol | $0_{221}$ |
| Coxeter-Dynkin diagram | or |
| 5-faces | 126 |
| 4-faces | 1566 |
| Cells | 6480 |
| Faces | 6480 |
| Edges | 6480 |
| Vertices | 720 |
| Vertex figure | 3-3 duoprism prism |
| Petrie polygon | Dodecagon |
| Coxeter group | $E_6,$ $[[3,3^{2,2}]],$ order 103680 |
| Properties | convex |

The rectified $1_{22}$ polytope (also called $0_{221})$ can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).

### Alternate names

* Birectified $2_{21}$ polytope
* Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)

### Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Coxeter plane orthographic projections :

| E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
| - | - | - | - |
| A5 [6] | A4 [5] | A3 / D3 [4] |

### Construction

Its construction is based on the $E_6$ group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Removing the ring on the short branch leaves the birectified 5-simplex, .

Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: $t_2(2_{11}),$ .

The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.

| $E_6$ | k-face | $f_k$ | $f_0$ | $f_1$ | $f_2$ | $f_2$ | $f_2$ | $f_3$ | $f_3$ | $f_3$ | $f_3$ | $f_3$ | $f_4$ | $f_4$ | $f_4$ | $f_4$ | $f_4$ | $f_5$ | $f_5$ | $f_5$ | k-figure | notes |
| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
| $A_2A_2A_1$ | ( ) | $f_0$ | 720 | 18 | 18 | 18 | 9 | 6 | 18 | 9 | 6 | 9 | 6 | 3 | 6 | 9 | 3 | 2 | 3 | 3 | {3}×{3}×{ } | $E_6/A_2A_2A_1$ = 72*6!/3!/3!/2 = 720 |
| $A_1A_1A_1$ | { } | $f_1$ | 2 | 6480 | 2 | 2 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | { }∨{ }∨( ) | $E_6/A_1A_1A_1$ = 72*6!/2/2/2 = 6480 |
| $A_2A_1$ | {3} | $f_2$ | 3 | 3 | 4320 | * | * | 1 | 2 | 1 | 0 | 0 | 2 | 1 | 1 | 2 | 0 | 1 | 2 | 1 | Sphenoid | $E_6/A_2A_1$ = 72*6!/3!/2 = 4320 |
| $A_2A_1$ | {3} | $f_2$ | 3 | 3 | * | 4320 | * | 0 | 2 | 0 | 1 | 1 | 1 | 0 | 2 | 2 | 1 | 1 | 1 | 2 | Sphenoid | $E_6/A_2A_1$ = 72*6!/3!/2 = 4320 |
| $A_2A_1A_1$ | {3} | $f_2$ | 3 | 3 | * | * | 2160 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 1 | 0 | 2 | 2 | { }∨{ } | $E_6/A_2A_1A_1$ = 72*6!/3!/2/2 = 2160 |
| $A_2A_1$ | {3,3} | $f_3$ | 4 | 6 | 4 | 0 | 0 | 1080 | * | * | * | * | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 0 | { }∨( ) | $E_6/A_2A_1$ = 72*6!/3!/2 = 1080 |
| $A_3$ | r{3,3} | $f_3$ | 6 | 12 | 4 | 4 | 0 | * | 2160 | * | * | * | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | {3} | $E_6/A_3$ = 72*6!/4! = 2160 |
| $A_3A_1$ | r{3,3} | $f_3$ | 6 | 12 | 4 | 0 | 4 | * | * | 1080 | * | * | 0 | 1 | 0 | 2 | 0 | 0 | 2 | 1 | { }∨( ) | $E_6/A_3A_1$ = 72*6!/4!/2 = 1080 |
| $A_3A_1$ | {3,3} | $f_3$ | 4 | 6 | 0 | 4 | 0 | * | * | * | 1080 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | { }∨( ) | $E_6/A_3A_1$ = 72*6!/4!/2 = 1080 |
| $A_3A_1$ | r{3,3} | $f_3$ | 6 | 12 | 0 | 4 | 4 | * | * | * | * | 1080 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | { }∨( ) | $E_6/A_3A_1$ = 72*6!/4!/2 = 1080 |
| $A_4$ | r{3,3,3} | $f_4$ | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 432 | * | * | * | * | 1 | 1 | 0 | { } | $E_6/A_4$ = 72*6!/5! = 432 |
| $A_4A_1$ | r{3,3,3} | $f_4$ | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 216 | * | * | * | 0 | 2 | 0 | { } | $E_6/A_4A_1$ = 72*6!/5!/2 = 216 |
| $A_4$ | r{3,3,3} | $f_4$ | 10 | 30 | 10 | 20 | 0 | 0 | 5 | 0 | 5 | 0 | * | * | 432 | * | * | 1 | 0 | 1 | { } | $E_6/A_4$ = 72*6!/5! = 432 |
| $D_4$ | {3,4,3} | $f_4$ | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 0 | 8 | * | * | * | 270 | * | 0 | 1 | 1 | { } | $E_6/D_4$ = 72*6!/8/4! = 270 |
| $A_4A_1$ | r{3,3,3} | $f_4$ | 10 | 30 | 0 | 20 | 10 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 216 | 0 | 0 | 2 | { } | $E_6/A_4A_1$ = 72*6!/5!/2 = 216 |
| $A_5$ | 2r{3,3,3,3} | $f_5$ | 20 | 90 | 60 | 60 | 0 | 15 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | 72 | * | * | ( ) | $E_6/A_5$ = 72*6!/6! = 72 |
| $D_5$ | 2r{4,3,3,3} | $f_5$ | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 0 | 40 | 16 | 16 | 0 | 10 | 0 | * | 27 | * | ( ) | $E_6/D_5$ = 72*6!/16/5! = 27 |
| $D_5$ | 2r{4,3,3,3} | $f_5$ | 80 | 480 | 160 | 320 | 160 | 0 | 80 | 40 | 80 | 80 | 0 | 0 | 16 | 10 | 16 | * | * | 27 | ( ) | $E_6/D_5$ = 72*6!/16/5! = 27 |

## Truncated $1_{22}$ polytope

Truncated $1_{22}$ :

| Type | Uniform 6-polytope |
| - | - |
| Schläfli symbol | $t{3,3^{2,2}}$ |
| Coxeter symbol | $t(1_{22})$ |
| Coxeter-Dynkin diagram | or |
| 5-faces | 72+27+27 |
| 4-faces | 32+216+432+270+216 |
| Cells | 1080+2160+1080+1080+1080 |
| Faces | 4320+4320+2160 |
| Edges | 6480+720 |
| Vertices | 1440 |
| Vertex figure | ( )v{3}x{3} |
| Petrie polygon | Dodecagon |
| Coxeter group | $E_6,$ $[[3,3^{2,2}]],$ order 103680 |
| Properties | convex |

### Alternate names

* Truncated $1_{22}$ polytope

### Construction

Its construction is based on the $E_6$ group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

### Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Coxeter plane orthographic projections :

| E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
| - | - | - | - |
| A5 [6] | A4 [5] | A3 / D3 [4] |

## Birectified $1_{22}$ polytope

Birectified $1_{22}$ polytope :

| Type | Uniform 6-polytope |
| - | - |
| Schläfli symbol | $2r{3,3^{2,2}}$ |
| Coxeter symbol | $2r(1_{22})$ |
| Coxeter-Dynkin diagram | or |
| 5-faces | 126 |
| 4-faces | 2286 |
| Cells | 10800 |
| Faces | 19440 |
| Edges | 12960 |
| Vertices | 2160 |
| Vertex figure | |
| Coxeter group | $E_6,$ $[[3,3^{2,2}]],$ order 103680 |
| Properties | convex |

### Alternate names

* Bicantellated $2_{21}$
* Birectified pentacontitetrapeton (barm) (Jonathan Bowers)

### Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Coxeter plane orthographic projections :

| E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
| - | - | - | - |
| A5 [6] | A4 [5] | A3 / D3 [4] |

## Trirectified $1_{22}$ polytope

Trirectified $1_{22}$ polytope :

| Type | Uniform 6-polytope |
| - | - |
| Schläfli symbol | $3r{3,3^{2,2}}$ |
| Coxeter symbol | $3r(1_{22})$ |
| Coxeter-Dynkin diagram | or |
| 5-faces | 558 |
| 4-faces | 4608 |
| Cells | 8640 |
| Faces | 6480 |
| Edges | 2160 |
| Vertices | 270 |
| Vertex figure | |
| Coxeter group | $E_6,$ $[[3,3^{2,2}]],$ order 103680 |
| Properties | convex |

### Alternate names

* Tricantellated $2_{21}$
* Trirectified pentacontitetrapeton (trim or cacam) (Jonathan Bowers)
