# E9 honeycomb

In geometry, an E$_9$ honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. ${\bar {T}}_{9}$, also (E$_{10}$) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.
E$_{10}$ is last of the series of Coxeter groups with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E$_{10}$ honeycombs by all combinations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram is a nonlinear graph, but there are three simplest ones, with a single ring at the end of its 3 branches: 6$_{21}$, 2$_{61}$, 1$_{62}$.

## 6$_{21}$ honeycomb

6$_{21}$ honeycomb :
| Family | k$_{21}$ polytope |
| Schläfli symbol | {3,3,3,3,3,3,3$^{2,1}$} |
| Coxeter symbol | 6$_{21}$ |
| Coxeter-Dynkin diagram | |
| 9-faces | 6$_{11}$ {3$^8$} |
| 8-faces | {3$^7$} |
| 7-faces | {3$^6$} |
| 6-faces | {3$^5$} |
| 5-faces | {3$^4$} |
| 4-faces | {3$^3$} |
| Cells | {3$^2$} |
| Faces | {3} |
| Vertex figure | 5$_{21}$ |
| Symmetry group | ${\bar {T}}_{9}$, [3$^{6,2,1}$] |
The 6$_{21}$ honeycomb is constructed from alternating 9-simplex and 9-orthoplex facets within the symmetry of the E$_{10}$ Coxeter group.
This honeycomb is highly regular in the sense that its symmetry group (the affine E$_9$ Weyl group) acts transitively on the k-faces for k ≤ 7. All of the k-faces for k ≤ 8 are simplices.
This honeycomb is last in the series of k$_{21}$ polytopes, enumerated by Thorold Gosset in 1900, listing polytopes and honeycombs constructed entirely of regular facets, although his list ended with the 8-dimensional the Euclidean honeycomb, 5$_{21}$.

### Construction

It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic 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 9-orthoplex, 7$_{11}$.
Removing the node on the end of the 1-length branch leaves the 9-simplex.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5$_{21}$ honeycomb.
The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 4$_{21}$ polytope.
The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 3$_{21}$ polytope.
The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 2$_{21}$ polytope.

### Related polytopes and honeycombs

The 6$_{21}$ is last in a dimensional series of semiregular polytopes and honeycombs, identified in 1900 by Thorold Gosset. Each member of the sequence has the previous member as its vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes.
k$_{21}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| E$_n$ | 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 | [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 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ∞ |
| Graph | | | | | | | - | - |
| Name | −1$_{21}$ | 0$_{21}$ | 1$_{21}$ | 2$_{21}$ | 3$_{21}$ | 4$_{21}$ | 5$_{21}$ | 6$_{21}$ |

## 2$_{61}$ honeycomb

2$_{61}$ honeycomb :
| Family | 2$_{k1}$ polytope |
| Schläfli symbol | {3,3,3$^{6,1}$} |
| Coxeter symbol | 2$_{61}$ |
| Coxeter-Dynkin diagram | |
| 9-face types | 2$_{51}$ {3$^7$} |
| 8-face types | 2$_{41}$, {3$^7$} |
| 7-face types | 2$_{31}$, {3$^6$} |
| 6-face types | 2$_{21}$, {3$^5$} |
| 5-face types | 2$_{11}$, {3$^4$} |
| 4-face type | {3$^3$} |
| Cells | {3$^2$} |
| Faces | {3} |
| Vertex figure | 1$_{61}$ |
| Coxeter group | ${\bar {T}}_{9}$, [3$^{6,2,1}$] |
The 2$_{61}$ honeycomb is composed of 2$_{51}$ 9-honeycomb and 9-simplex facets. It is the final figure in the 2$_{k1}$ family.

### Construction

It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the 9-simplex.
Removing the node on the end of the 6-length branch leaves the 2$_{51}$ honeycomb. This is an infinite facet because E10 is a paracompact hyperbolic group.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 9-demicube, 1$_{61}$.
The edge figure is the vertex figure of the edge figure. This makes the rectified 8-simplex, 0$_{51}$.
The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 5-simplex prism.

### Related polytopes and honeycombs

The 2$_{61}$ is last in a dimensional series of uniform 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$_2$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 | [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$_{61}$ |

## 1$_{62}$ honeycomb

1$_{62}$ honeycomb :
| Family | 1$_{k2}$ polytope |
| Schläfli symbol | {3,3$^{6,2}$} |
| Coxeter symbol | 1$_{62}$ |
| Coxeter-Dynkin diagram | |
| 9-face types | 1$_{52}$, 1$_{61}$ |
| 8-face types | 1$_{42}$, 1$_{51}$ |
| 7-face types | 1$_{32}$, 1$_{41}$ |
| 6-face types | 1$_{22}$, {3$^{1,3,1}$} {3$^5$} |
| 5-face types | 1$_{21}$, {3$^4$} |
| 4-face type | 1$_{11}$, {3$^3$} |
| Cells | {3$^2$} |
| Faces | {3} |
| Vertex figure | t$_2${3$^8$} |
| Coxeter group | ${\bar {T}}_{9}$, [3$^{6,2,1}$] |
The 1$_{62}$ honeycomb contains 1$_{52}$ (9-honeycomb) and 1$_{61}$ 9-demicube facets. It is the final figure in the 1$_{k2}$ polytope family.

### Construction

It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-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 9-demicube, 1$_{61}$.
Removing the node on the end of the 6-length branch leaves the 1$_{52}$ honeycomb.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 9-simplex, 0$_{62}$.

### Related polytopes and honeycombs

The 1$_{62}$ is last in a dimensional series of uniform polytopes and honeycombs.
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$_{62}$ |