# 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$_2$A$_2$A$_1$ | { } | f$_1$ | 2 | 720 | 9 | 9 | 9 | 3 | 3 | 9 | 3 | 3 | {3}×{3} | E$_6$/A$_2$A$_2$A$_1$ = 72*6!/3!/3!/2 = 720 |
| A$_2$A$_1$A$_1$ | {3} | f$_2$ | 3 | 3 | 2160 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | s{2,4} | E$_6$/A$_2$A$_1$A$_1$ = 72*6!/3!/2/2 = 2160 |
| A$_3$A$_1$ | {3,3} | f$_3$ | 4 | 6 | 4 | 1080 | * | 1 | 0 | 2 | 2 | 1 | { }∨( ) | E$_6$/A$_3$A$_1$ = 72*6!/4!/2 = 1080 |
| A$_3$A$_1$ | {3,3} | f$_3$ | 4 | 6 | 4 | * | 1080 | 0 | 1 | 2 | 1 | 2 | { }∨( ) | E$_6$/A$_3$A$_1$ = 72*6!/4!/2 = 1080 |
| A$_4$A$_1$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 5 | 0 | 216 | * | * | 2 | 0 | { } | E$_6$/A$_4$A$_1$ = 72*6!/5!/2 = 216 |
| A$_4$A$_1$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 0 | 5 | * | 216 | * | 0 | 2 | { } | E$_6$/A$_4$A$_1$ = 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$_2$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$_{62}$ |

#### 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$_2$A$_2$A$_1$ | ( ) | 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$_2$A$_2$A$_1$ = 72*6!/3!/3!/2 = 720 |
| A$_1$A$_1$A$_1$ | { } | f$_1$ | 2 | 6480 | 2 | 2 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | { }∨{ }∨( ) | E$_6$/A$_1$A$_1$A$_1$ = 72*6!/2/2/2 = 6480 |
| A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | 4320 | * | * | 1 | 2 | 1 | 0 | 0 | 2 | 1 | 1 | 2 | 0 | 1 | 2 | 1 | Sphenoid | E$_6$/A$_2$A$_1$ = 72*6!/3!/2 = 4320 |
| A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | * | 4320 | * | 0 | 2 | 0 | 1 | 1 | 1 | 0 | 2 | 2 | 1 | 1 | 1 | 2 | Sphenoid | E$_6$/A$_2$A$_1$ = 72*6!/3!/2 = 4320 |
| A$_2$A$_1$A$_1$ | {3} | f$_2$ | 3 | 3 | * | * | 2160 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 1 | 0 | 2 | 2 | { }∨{ } | E$_6$/A$_2$A$_1$A$_1$ = 72*6!/3!/2/2 = 2160 |
| A$_2$A$_1$ | {3,3} | f$_3$ | 4 | 6 | 4 | 0 | 0 | 1080 | * | * | * | * | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 0 | { }∨( ) | E$_6$/A$_2$A$_1$ = 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$_3$A$_1$ | r{3,3} | f$_3$ | 6 | 12 | 4 | 0 | 4 | * | * | 1080 | * | * | 0 | 1 | 0 | 2 | 0 | 0 | 2 | 1 | { }∨( ) | E$_6$/A$_3$A$_1$ = 72*6!/4!/2 = 1080 |
| A$_3$A$_1$ | {3,3} | f$_3$ | 4 | 6 | 0 | 4 | 0 | * | * | * | 1080 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | { }∨( ) | E$_6$/A$_3$A$_1$ = 72*6!/4!/2 = 1080 |
| A$_3$A$_1$ | r{3,3} | f$_3$ | 6 | 12 | 0 | 4 | 4 | * | * | * | * | 1080 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | { }∨( ) | E$_6$/A$_3$A$_1$ = 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$_4$A$_1$ | r{3,3,3} | f$_4$ | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 216 | * | * | * | 0 | 2 | 0 | { } | E$_6$/A$_4$A$_1$ = 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$_4$A$_1$ | r{3,3,3} | f$_4$ | 10 | 30 | 0 | 20 | 10 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 216 | 0 | 0 | 2 | { } | E$_6$/A$_4$A$_1$ = 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)