E9 honeycomb#162 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

class="wikitable" align="right" style="margin-left:10px" width="250" !bgcolor=#e7dcc3 colspan=2\|6₂₁ honeycomb
bgcolor=#e7dcc3\|Family	k₂₁ polytope
bgcolor=#e7dcc3\|Schläfli symbol	{3,3,3,3,3,3,3^2,1}
bgcolor=#e7dcc3\|Coxeter symbol	6₂₁
bgcolor=#e7dcc3\|Coxeter-Dynkin diagram	{{CDD\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea_1}}
bgcolor=#e7dcc3\|9-faces	6₁₁ 25px {3⁸} 25px
bgcolor=#e7dcc3\|8-faces	{3⁷} 25px
bgcolor=#e7dcc3\|7-faces	{3⁶} 25px
bgcolor=#e7dcc3\|6-faces	{3⁵} 25px
bgcolor=#e7dcc3\|5-faces	{3⁴} 25px
bgcolor=#e7dcc3\|4-faces	{3³} 25px
bgcolor=#e7dcc3\|Cells	{3²} 25px
bgcolor=#e7dcc3\|Faces	{3} 25px
bgcolor=#e7dcc3\|Vertex figure	5₂₁
bgcolor=#e7dcc3\|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.Conway, 2008, The Gosset series, p 413

=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.

: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}

Removing the node on the end of the 2-length branch leaves the 9-orthoplex, 7₁₁.

: {{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}

Removing the node on the end of the 1-length branch leaves the 9-simplex.

: {{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5₂₁ honeycomb.

: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}

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

: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}

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

: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}

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

: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}

= 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.

261 honeycomb

class="wikitable" align="right" style="margin-left:10px" width="280" !bgcolor=#e7dcc3 colspan=2\|2₆₁ honeycomb
bgcolor=#e7dcc3\|Family	2_k1 polytope
bgcolor=#e7dcc3\|Schläfli symbol	{3,3,3^6,1}
bgcolor=#e7dcc3\|Coxeter symbol	2₆₁
bgcolor=#e7dcc3\|Coxeter-Dynkin diagram	{{CDD\|nodea_1\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}}
bgcolor=#e7dcc3\|9-face types	2₅₁ {3⁷}25px
bgcolor=#e7dcc3\|8-face types	2₄₁25px, {3⁷}25px
bgcolor=#e7dcc3\|7-face types	2₃₁25px, {3⁶}25px
bgcolor=#e7dcc3\|6-face types	2₂₁25px, {3⁵}25px
bgcolor=#e7dcc3\|5-face types	2₁₁25px, {3⁴}25px
bgcolor=#e7dcc3\|4-face type	{3³}25px
bgcolor=#e7dcc3\|Cells	{3²}25px
bgcolor=#e7dcc3\|Faces	{3}25px
bgcolor=#e7dcc3\|Vertex figure	1₆₁ 30px
bgcolor=#e7dcc3\|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.

: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

Removing the node on the short branch leaves the 9-simplex.

: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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.

: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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

: {{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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

: {{CDD|branch_10|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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.

: {{CDD|node_1|2|node_1|3|node|3|node|3|node|3|node}}

= Related polytopes and honeycombs =

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

162 honeycomb

class="wikitable" align="right" style="margin-left:10px" width="280" !bgcolor=#e7dcc3 colspan=2\|1₆₂ honeycomb
bgcolor=#e7dcc3\|Family	1_k2 polytope
bgcolor=#e7dcc3\|Schläfli symbol	{3,3^6,2}
bgcolor=#e7dcc3\|Coxeter symbol	1₆₂
bgcolor=#e7dcc3\|Coxeter-Dynkin diagram	{{CDD\|nodea\|3a\|nodea\|3a\|branch_01lr\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}}
bgcolor=#e7dcc3\|9-face types	1₅₂, 1₆₁25px
bgcolor=#e7dcc3\|8-face types	1₄₂25px, 1₅₁25px
bgcolor=#e7dcc3\|7-face types	1₃₂25px, 1₄₁25px
bgcolor=#e7dcc3\|6-face types	1₂₂25px, {3^1,3,1}25px {3⁵}25px
bgcolor=#e7dcc3\|5-face types	1₂₁25px, {3⁴}25px
bgcolor=#e7dcc3\|4-face type	1₁₁25px, {3³}25px
bgcolor=#e7dcc3\|Cells	{3²}25px
bgcolor=#e7dcc3\|Faces	{3}25px
bgcolor=#e7dcc3\|Vertex figure	t₂{3⁸} 25px
bgcolor=#e7dcc3\|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.

: {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

Removing the node on the end of the 2-length branch leaves the 9-demicube, 1₆₁.

: {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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

: {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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

: {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

= Related polytopes and honeycombs =

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

Notes

References

The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{ISBN|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]
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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Category:10-polytopes

class="wikitable" align="right" style="margin-left:10px" width="250" !bgcolor=#e7dcc3 colspan=2\|6₂₁ honeycomb
bgcolor=#e7dcc3\|Family	k₂₁ polytope
bgcolor=#e7dcc3\|Schläfli symbol	{3,3,3,3,3,3,3^2,1}
bgcolor=#e7dcc3\|Coxeter symbol	6₂₁
bgcolor=#e7dcc3\|Coxeter-Dynkin diagram	{{CDD\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea_1}}
bgcolor=#e7dcc3\|9-faces	6₁₁ 25px {3⁸} 25px
bgcolor=#e7dcc3\|8-faces	{3⁷} 25px
bgcolor=#e7dcc3\|7-faces	{3⁶} 25px
bgcolor=#e7dcc3\|6-faces	{3⁵} 25px
bgcolor=#e7dcc3\|5-faces	{3⁴} 25px
bgcolor=#e7dcc3\|4-faces	{3³} 25px
bgcolor=#e7dcc3\|Cells	{3²} 25px
bgcolor=#e7dcc3\|Faces	{3} 25px
bgcolor=#e7dcc3\|Vertex figure	5₂₁
bgcolor=#e7dcc3\|Symmetry group	${\bar{T}}_9$ , [3^6,2,1]

class="wikitable" align="right" style="margin-left:10px" width="280" !bgcolor=#e7dcc3 colspan=2\|2₆₁ honeycomb
bgcolor=#e7dcc3\|Family	2_k1 polytope
bgcolor=#e7dcc3\|Schläfli symbol	{3,3,3^6,1}
bgcolor=#e7dcc3\|Coxeter symbol	2₆₁
bgcolor=#e7dcc3\|Coxeter-Dynkin diagram	{{CDD\|nodea_1\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}}
bgcolor=#e7dcc3\|9-face types	2₅₁ {3⁷}25px
bgcolor=#e7dcc3\|8-face types	2₄₁25px, {3⁷}25px
bgcolor=#e7dcc3\|7-face types	2₃₁25px, {3⁶}25px
bgcolor=#e7dcc3\|6-face types	2₂₁25px, {3⁵}25px
bgcolor=#e7dcc3\|5-face types	2₁₁25px, {3⁴}25px
bgcolor=#e7dcc3\|4-face type	{3³}25px
bgcolor=#e7dcc3\|Cells	{3²}25px
bgcolor=#e7dcc3\|Faces	{3}25px
bgcolor=#e7dcc3\|Vertex figure	1₆₁ 30px
bgcolor=#e7dcc3\|Coxeter group	${\bar{T}}_9$ , [3^6,2,1]

class="wikitable" align="right" style="margin-left:10px" width="280" !bgcolor=#e7dcc3 colspan=2\|1₆₂ honeycomb
bgcolor=#e7dcc3\|Family	1_k2 polytope
bgcolor=#e7dcc3\|Schläfli symbol	{3,3^6,2}
bgcolor=#e7dcc3\|Coxeter symbol	1₆₂
bgcolor=#e7dcc3\|Coxeter-Dynkin diagram	{{CDD\|nodea\|3a\|nodea\|3a\|branch_01lr\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}}
bgcolor=#e7dcc3\|9-face types	1₅₂, 1₆₁25px
bgcolor=#e7dcc3\|8-face types	1₄₂25px, 1₅₁25px
bgcolor=#e7dcc3\|7-face types	1₃₂25px, 1₄₁25px
bgcolor=#e7dcc3\|6-face types	1₂₂25px, {3^1,3,1}25px {3⁵}25px
bgcolor=#e7dcc3\|5-face types	1₂₁25px, {3⁴}25px
bgcolor=#e7dcc3\|4-face type	1₁₁25px, {3³}25px
bgcolor=#e7dcc3\|Cells	{3²}25px
bgcolor=#e7dcc3\|Faces	{3}25px
bgcolor=#e7dcc3\|Vertex figure	t₂{3⁸} 25px
bgcolor=#e7dcc3\|Coxeter group	${\bar{T}}_9$ , [3^6,2,1]

E9 honeycomb#162 honeycomb

6<sub>21</sub> honeycomb

=Construction=

= Related polytopes and honeycombs =

2<sub>61</sub> honeycomb

=Construction=

= Related polytopes and honeycombs =

1<sub>62</sub> honeycomb

=Construction=

= Related polytopes and honeycombs =

Notes

References