6-cubic honeycomb

6-cubic honeycomb
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Type	Regular 6-space honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,3^4,4} {4,3^3,3^1,1}
Coxeter-Dynkin diagrams
6-face type	{4,3^4}
5-face type	{4,3^3}
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 {4,3^4} (6-orthoplex)
Coxeter group	${{\tilde {C}}}_{6}$, [4,3^4,4] ${{\tilde {B}}}_{6}$, [4,3^3,3^1,1]
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 6-cube 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.

Constructions

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.

Related honeycombs

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.

Trirectified 6-cubic honeycomb

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

See also

• List of regular polytopes

References

• 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
  • (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 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
Uniform 5-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ6	hδ6	qδ6
Uniform 7-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
Uniform 8-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
Uniform 9-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
Uniform 10-honeycomb	{3^[10]}	δ10	hδ10	qδ10
Uniform n-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21