Cyclotruncated 6-simplex honeycomb

Cyclotruncated 6-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Cyclotruncated simplectic honeycomb
Schläfli symbol	t0,1{3[7]}
Coxeter diagram
6-face types	{35} t{35} 2t{35} 3t{35}
Vertex figure	Elongated 5-simplex antiprism
Symmetry	${\displaystyle {\tilde {A}}_{6}}$×2, [[3[7]]]
Properties	vertex-transitive

In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

Structure[]

It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane.

Related polytopes and honeycombs[]

This honeycomb is one of 17 unique uniform honeycombs^[1] constructed by the ${\displaystyle {\tilde {A}}_{6}}$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

A6 honeycombs
Heptagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycombs
a1	[3[7]]		${\displaystyle {\tilde {A}}_{6}}$
i2	[[3[7]]]		${\displaystyle {\tilde {A}}_{6}}$×2	1 2
r14	[7[3[7]]]		${\displaystyle {\tilde {A}}_{6}}$×14	3

See also[]

Regular and uniform honeycombs in 6-space:

• 6-cubic honeycomb
• 6-demicubic honeycomb
• 6-simplex honeycomb
• Omnitruncated 6-simplex honeycomb
• 2₂₂ honeycomb

Notes[]

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]

v t Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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
E10	Uniform 10-honeycomb	{3[11]}	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21