6-cubic honeycomb

6-cubic honeycomb
(no image)
Type	Regular 6-honeycomb Uniform 6-honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,34,4} {4,33,31,1}
Coxeter-Dynkin diagrams
6-face type	{4,34}
5-face type	{4,33}
4-face type	{4,3,3}
Cell type	{4,3}
Face type	{4}
Face figure	{4,3} (octahedron)
Edge figure	8 {4,3,3} (16-cell)
Vertex figure	64 {4,34} (6-orthoplex)
Coxeter group	${\displaystyle {\tilde {C}}_{6}}$, [4,34,4] ${\displaystyle {\tilde {B}}_{6}}$, [4,33,31,1]
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3⁴,4}. Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol {4,3³,3^1,1}. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}⁽⁶⁾.

The [4,3⁴,4], , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb.

The 6-cubic honeycomb can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex facets.

A trirectified 6-cubic honeycomb, , contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the D₆^* lattice. Facets can be identically colored from a doubled ${\displaystyle {\tilde {C}}_{6}}$×2, [[4,3⁴,4]] symmetry, alternately colored from ${\displaystyle {\tilde {C}}_{6}}$, [4,3⁴,4] symmetry, three colors from ${\displaystyle {\tilde {B}}_{6}}$, [4,3³,3^1,1] symmetry, and 4 colors from ${\displaystyle {\tilde {D}}_{6}}$, [3^1,1,3,3,3^1,1] symmetry.

• List of regular polytopes

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• 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 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

v t e 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	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21