Cyclotruncated 8-simplex honeycomb

Cyclotruncated 8-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Cyclotruncated simplectic honeycomb
Schläfli symbol	t0,1{3[9]}
Coxeter diagram
8-face types	{37} , t0,1{37} 30px t1,2{37} 30px, t2,3{37} 30px t3,4{37}
Vertex figure	Elongated 7-simplex antiprism
Symmetry	[math]\displaystyle{ {\tilde{A}}_8 }[/math]×2, 3[9]
Properties	vertex-transitive

In eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets. These facet types occur in proportions of 2:2:2:2:1 respectively in the whole honeycomb.

Structure

It can be constructed by nine sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 7-simplex honeycomb divisions on each hyperplane.

Related polytopes and honeycombs

This honeycomb is one of 45 unique uniform honeycombs constructed by the [math]\displaystyle{ {\tilde{A}}_8 }[/math] Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

A8 honeycombs
Enneagon symmetry	Symmetry	Extended diagram	Extended group	Honeycombs
a1	[3[9]]		[math]\displaystyle{ {\tilde{A}}_8 }[/math]
i2	[[3[9]]]		[math]\displaystyle{ {\tilde{A}}_8 }[/math]×2	1 2
i6	[3[3[9]]]		[math]\displaystyle{ {\tilde{A}}_8 }[/math]×6
r18	[9[3[9]]]		[math]\displaystyle{ {\tilde{A}}_8 }[/math]×18	3

See also

Regular and uniform honeycombs in 8-space:

• 8-cubic honeycomb
• 8-demicubic honeycomb
• 8-simplex honeycomb
• Omnitruncated 8-simplex honeycomb
• 5₂₁ honeycomb
• 2₅₁ honeycomb
• 1₅₂ honeycomb

References

• Norman Johnson Uniform Polytopes, Manuscript (1991)
• 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 [1]
  • (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]

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	[math]\displaystyle{ {\tilde{A}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{C}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{B}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{D}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
E2	Uniform tiling	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21