Cyclotruncated 5-simplex honeycomb

Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells.

It can be constructed as six sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane.

This honeycomb is one of 12 unique uniform honeycombs[1] constructed by the ${\displaystyle {\tilde {A}}_{5}}$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\displaystyle {\tilde {A}}_{5}}$ Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:

A5 honeycombs
Hexagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
a1	[3[6]]		${\displaystyle {\tilde {A}}_{5}}$
d2	<[3[6]]>		${\displaystyle {\tilde {A}}_{5}}$×21	1, , , ,
p2	[[3[6]]]		${\displaystyle {\tilde {A}}_{5}}$×22	2,
i4	[<[3[6]]>]		${\displaystyle {\tilde {A}}_{5}}$×21×22	,
d6	<3[3[6]]>		${\displaystyle {\tilde {A}}_{5}}$×61
r12	[6[3[6]]]		${\displaystyle {\tilde {A}}_{5}}$×12	3

Regular and uniform honeycombs in 5-space:

• 5-cubic honeycomb
• 5-demicubic honeycomb
• 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb

• 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
  • (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	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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