View source for 8-simplex honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="300"
!bgcolor=#e7dcc3 colspan=2|8-simplex honeycomb
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|bgcolor=#ffffff align=center colspan=2|(No image)
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|bgcolor=#e7dcc3|Type||[[Uniform 8-honeycomb]]
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|bgcolor=#e7dcc3|Family||[[Simplectic honeycomb]]
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|bgcolor=#e7dcc3|[[Schläfli symbol]]||{3<sup>[9]</sup>} = 0<sub>[9]</sub>
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|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
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|bgcolor=#e7dcc3|6-face types||[[8-simplex|{3<sup>7</sup>}]] [[File:8-simplex t0.svg|30px]], [[rectified 8-simplex|t<sub>1</sub>{3<sup>7</sup>}]] [[File:8-simplex t1.svg|30px]]<BR>[[birectified 8-simplex|t<sub>2</sub>{3<sup>7</sup>}]] [[File:8-simplex t2.svg|30px]], [[trirectified 8-simplex|t<sub>3</sub>{3<sup>7</sup>}]] [[File:8-simplex t3.svg|30px]]
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|bgcolor=#e7dcc3|6-face types||[[7-simplex|{3<sup>6</sup>}]] [[File:7-simplex t0.svg|30px]], [[rectified 7-simplex|t<sub>1</sub>{3<sup>6</sup>}]] [[File:7-simplex t1.svg|30px]]<BR>[[birectified 7-simplex|t<sub>2</sub>{3<sup>6</sup>}]] [[File:7-simplex t2.svg|40px]], [[trirectified 7-simplex|t<sub>3</sub>{3<sup>6</sup>}]] [[File:7-simplex t2.svg|30px]]
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|bgcolor=#e7dcc3|6-face types||[[6-simplex|{3<sup>5</sup>}]] [[File:6-simplex t0.svg|30px]], [[rectified 6-simplex|t<sub>1</sub>{3<sup>5</sup>}]] [[File:6-simplex t1.svg|30px]]<BR>[[birectified 6-simplex|t<sub>2</sub>{3<sup>5</sup>}]] [[File:6-simplex t2.svg|30px]]
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|bgcolor=#e7dcc3|5-face types||[[5-simplex|{3<sup>4</sup>}]] [[File:5-simplex t0.svg|30px]], [[rectified 5-simplex|t<sub>1</sub>{3<sup>4</sup>}]] [[File:5-simplex t1.svg|30px]]<BR>[[birectified 5-simplex|t<sub>2</sub>{3<sup>4</sup>}]] [[File:5-simplex t2.svg|30px]]
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|bgcolor=#e7dcc3|4-face types||[[5-cell|{3<sup>3</sup>}]] [[File:4-simplex t0.svg|30px]], [[Rectified 5-cell|t<sub>1</sub>{3<sup>3</sup>}]] [[File:4-simplex t1.svg|30px]]
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|bgcolor=#e7dcc3|Cell types||[[tetrahedron|{3,3}]] [[File:3-simplex t0.svg|30px]], [[Octahedron|t<sub>1</sub>{3,3}]] [[File:3-simplex t1.svg|30px]]
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|bgcolor=#e7dcc3|Face types||[[triangle|{3}]] [[File:2-simplex t0.svg|30px]]
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|bgcolor=#e7dcc3|Vertex figure||[[Expanded 8-simplex|t<sub>0,7</sub>{3<sup>7</sup>}]] [[File:8-simplex t07.svg|30px]]
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|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||<math>{\tilde{A}}_8</math>×2, [[3<sup>[9]</sup>]]
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|bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
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In [[Eighth dimension|eighth-dimensional]] [[Euclidean geometry]], the '''8-simplex honeycomb''' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). The tessellation fills space by [[8-simplex]], [[rectified 8-simplex]], [[birectified 8-simplex]], and [[trirectified 8-simplex]] facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.

== A8 lattice ==
This [[vertex arrangement]] is called the [[A8 lattice]] or '''8-simplex lattice'''. The 72 vertices of the [[expanded 8-simplex]] vertex figure represent the 72 roots of the <math>{\tilde{A}}_8</math> Coxeter group.<ref>{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A8.html|title = The Lattice A8}}</ref>  It is the 8-dimensional case of a [[simplectic honeycomb]]. Around each vertex figure are 510 facets: 9+9 [[8-simplex]], 36+36 [[rectified 8-simplex]], 84+84 [[birectified 8-simplex]], 126+126 [[trirectified 8-simplex]], with the count distribution from the 10th row of [[Pascal's triangle]].

<math>{\tilde{E}}_8</math> contains <math>{\tilde{A}}_8</math> as a subgroup of index 5760.<ref>N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 12: Euclidean symmetry groups, p.294</ref> Both <math>{\tilde{E}}_8</math> and <math>{\tilde{A}}_8</math> can be seen as affine extensions of <math>A_8</math> from different nodes: [[File:Affine A8 E8 relations.png]]

The A{{sup sub|3|8}} lattice is the union of three A<sub>8</sub> lattices, and also identical to the [[E8 lattice]].<ref>Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)</ref>
: {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}} ∪ {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes_10lr|3ab|branch}} ∪ {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes_01lr|3ab|branch}} = {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.

The A{{sup sub|*|8}} lattice (also called A{{sup sub|9|8}}) is the union of nine A<sub>8</sub> lattices, and has the [[vertex arrangement]] of the dual honeycomb to the [[omnitruncated 8-simplex honeycomb]], and therefore the [[Voronoi cell]] of this lattice is an [[omnitruncated 8-simplex]]
:
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}} ∪
{{CDD|node|split1|nodes_10lr|3ab|nodes|3ab|nodes|3ab|branch}} ∪
{{CDD|node|split1|nodes_01lr|3ab|nodes|3ab|nodes|3ab|branch}} ∪
{{CDD|node|split1|nodes|3ab|nodes_10lr|3ab|nodes|3ab|branch}} ∪
{{CDD|node|split1|nodes|3ab|nodes_01lr|3ab|nodes|3ab|branch}} ∪
{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes_10lr|3ab|branch}} ∪
{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes_01lr|3ab|branch}} ∪
{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch_10l}} ∪
{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch_01l}} = dual of {{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes_11|3ab|branch_11}}.

== Related polytopes and honeycombs ==
{{8-simplex honeycomb family}}

=== Projection by folding ===

The ''8-simplex honeycomb'' can be projected into the 4-dimensional [[tesseractic honeycomb]] by a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] operation that maps two pairs of mirrors into each other, sharing the same [[vertex arrangement]]:

{|class=wikitable
|-
!<math>{\tilde{A}}_8</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
|-
!<math>{\tilde{C}}_4</math>
|{{CDD|node_1|4|node|3|node|3|node|4|node}}
|}

== See also ==
* Regular and uniform honeycombs in 8-space:
**[[8-cubic honeycomb]]
**[[8-demicubic honeycomb]]
**[[Truncated 8-simplex honeycomb]]
**[[5 21 honeycomb|5<sub>21</sub> honeycomb]]
**[[2 51 honeycomb|2<sub>51</sub> honeycomb]]
**[[1 52 honeycomb|1<sub>52</sub> honeycomb]]

== Notes ==
{{reflist}}

== References ==
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
* '''Kaleidoscopes: Selected Writings of [[Harold Scott MacDonald Coxeter|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 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3–45]

{{Honeycombs}}

[[Category:Honeycombs (geometry)]]
[[Category:9-polytopes]]