E9 honeycomb

In geometry, an E₉ honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. ${\bar {T}}_{9}$, also (E₁₀) is a paracompact 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 is 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	{3,3,3,3,3,3,3^2,1}
Coxeter symbol	621
Coxeter-Dynkin diagram
9-faces	611 {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	521
Symmetry group	${\bar {T}}_{9}$, [3^6,2,1]

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, listing polytopes and honeycombs constructed entirely of regular facets, although his list ended with the 8-dimensional the Euclidean honeycomb, 5₂₁.^[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, 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.

Related polytopes and honeycombs

The 6₂₁ 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.

k21 figures in n dimensional
Space	Finite						Euclidean	Hyperbolic
En	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde {E}}_{8}$ = E8^+	E10 = ${\bar {T}}_{8}$ = E8^++
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	−121	021	121	221	321	421	521	621

2₆₁ honeycomb

261 honeycomb
Family	2k1 polytope
Schläfli symbol	{3,3,3^6,1}
Coxeter symbol	261
Coxeter-Dynkin diagram
9-face types	251 {3^7}
8-face types	241, {3^7}
7-face types	231, {3^6}
6-face types	221, {3^5}
5-face types	211, {3^4}
4-face type	{3^3}
Cells	{3^2}
Faces	{3}
Vertex figure	161
Coxeter group	${\bar {T}}_{9}$, [3^6,2,1]

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 paracompact 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.

Related polytopes and honeycombs

The 2₆₁ is last in a dimensional series of uniform polytopes and honeycombs.

2k1 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde {E}}_{8}$ = E8^+	E10 = ${\bar {T}}_{8}$ = E8^++
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	201	211	221	231	241	251	261

1₆₂ honeycomb

162 honeycomb
Family	1k2 polytope
Schläfli symbol	{3,3^6,2}
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, {3^1,3,1} {3^5}
5-face types	121, {3^4}
4-face type	111, {3^3}
Cells	{3^2}
Faces	{3}
Vertex figure	t2{3^8}
Coxeter group	${\bar {T}}_{9}$, [3^6,2,1]

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.

Related polytopes and honeycombs

The 1₆₂ is last in a dimensional series of uniform polytopes and honeycombs.

1k2 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde {E}}_{8}$ = E8^+	E10 = ${\bar {T}}_{8}$ = E8^++
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	102	112	122	132	142	152	162

Notes

1. 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]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / E9 / E10 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds