#
E_{9} honeycomb

In geometry, an **E _{9} honeycomb** is a tessellation of uniform polytopes in hyperbolic 9-dimensional space.
, 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

_{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 |
, [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}.[1]

### 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 dimensional | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | 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} =
= E_{8}^{+} |
E_{10} =
= 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 | 192 | 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

_{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}^{7}} |

7-face types | 2_{31}^{6}} |

6-face types | 2_{21}^{5}} |

5-face types | 2_{11}^{4}} |

4-face type | {3^{3}} |

Cells | {3^{2}} |

Faces | {3} |

Vertex figure | 1_{61} |

Coxeter group |
, [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 | 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} =
= E_{8}^{+} |
E_{10} =
= E_{8}^{++} | |||

Coxeter diagram |
|||||||||||

Symmetry | [3^{−1,2,1}] |
[3^{0,2,1}] |
[[3<sup>1,2,1</sup>]] | [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

_{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}^{1,3,1}}{3 ^{5}} |

5-face types | 1_{21}^{4}} |

4-face type | 1_{11}^{3}} |

Cells | {3^{2}} |

Faces | {3} |

Vertex figure | t_{2}{3^{8}} |

Coxeter group |
, [3^{6,2,1}] |

The **1 _{62} honeycomb** contains

**1**(9-honeycomb) and

_{52}**1**9-demicube facets. It is the final figure in the 1

_{61}_{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 | 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} =
= E_{8}^{+} |
E_{10} =
= E_{8}^{++} | |||

Coxeter diagram |
|||||||||||

Symmetry (order) |
[3^{−1,2,1}] |
[3^{0,2,1}] |
[3^{1,2,1}] |
[[3<sup>2,2,1</sup>]] | [3^{3,2,1}] |
[3^{4,2,1}] |
[3^{5,2,1}] |
[3^{6,2,1}] | |||

Order | 12 | 120 | 192 | 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} |

## Notes

- Conway, 2008, The Gosset series, p 413

## References

*The Symmetries of Things*2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5- Coxeter
*The Beauty of Geometry: Twelve Essays*, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes) - Coxeter
*Regular Polytopes*(1963), Macmillan Company*Regular Polytopes*, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope)

**Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6- (Paper 24) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3-45]

- (Paper 24) H.S.M. Coxeter,