E9 honeycomb

In geometry, an E₉ honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. ${\tilde {E}}_{9}$, also (E₁₀) is a noncompact hyperbolic group, so either facets or vertex figures will not be bounded.

E₁₀ is last of the series of Coxeter groups with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E₁₀ honeycombs by all combinations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but there are three simplest ones, with a single ring at the end of its 3 branches: 6₂₁, 2₆₁, 1₆₂.

6₂₁ honeycomb

621 honeycomb
Family	k21 polytope
Schläfli symbol	{3,3,3,3,3,3,32,1}
Coxeter symbol	621
Coxeter-Dynkin diagram
9-faces	611 {38}
8-faces	{37}
7-faces	{36}
6-faces	{35}
5-faces	{34}
4-faces	{33}
Cells	{32}
Faces	{3}
Vertex figure	521
Symmetry group	E10, [36,2,1]

The 6₂₁ honeycomb is constructed from alternating 9-simplex and 9-orthoplex facets within the symmetry of the E₁₀ Coxeter group.

This honeycomb is highly regular in the sense that its symmetry group (the affine E₉ 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 k₂₁ polytopes, enumerated by Thorold_Gosset in 1900, although his list ended with the 8-dimensional the Euclidean honeycomb, 5₂₁.

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, 7₁₁.

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 5₂₁ honeycomb.

The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 4₂₁ polytope.

The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 3₂₁ polytope.

The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 2₂₁ polytope.

2₆₁ honeycomb

261 honeycomb
Family	2k1 polytope
Schläfli symbol	{3,3,36,1}
Coxeter symbol	261
Coxeter-Dynkin diagram
9-face types	251 {37}
8-face types	241, {37}
7-face types	231, {36}
6-face types	221, {35}
5-face types	211, {34}
4-face type	{33}
Cells	{32}
Faces	{3}
Vertex figure	161
Coxeter group	$E_{10}$, [36,2,1]

The 2₆₁ honeycomb is composed of 2₅₁ 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 2₅₁ honeycomb. This is an infinite facet because E10 is a noncompact hyperbolic group.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 9-demicube, 1₆₁.

The edge figure is the vertex figure of the edge figure. This makes the rectified 8-simplex, 0₅₁.

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.

1₆₂ honeycomb

162 honeycomb
Family	1k2 polytope
Schläfli symbol	{3,36,2}
Coxeter symbol	162
Coxeter-Dynkin diagram
9-face types	152, 161
8-face types	142, 151
7-face types	132, 141
6-face types	122, {31,3,1} {35}
5-face types	121, {34}
4-face type	111, {33}
Cells	{32}
Faces	{3}
Vertex figure	t2{38}
Coxeter group	$E_{10}$, [36,2,1]

The 1₆₂ honeycomb contains 1₅₂ (9-honeycomb) and 1₆₁ 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, 1₆₁.

Removing the node on the end of the 6-length branch leaves the 1₅₂ honeycomb.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 9-simplex, 0₆₂.

References

• 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), Macmillian 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 [2]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]