7-cubic honeycomb

7-cubic honeycomb
(no image)
Type	Regular 7-honeycomb Uniform 7-honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,3⁵,4} {4,3⁴,3^1,1} {∞}⁽⁷⁾
Coxeter-Dynkin diagrams
7-face type	{4,3,3,3,3,3}
6-face type	{4,3,3,3,3}
5-face type	{4,3,3,3}
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	128 {4,3⁵} (7-orthoplex)
Coxeter group	[4,3⁵,4]
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-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 {4,3⁵,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,3⁴,3^1,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}⁽⁷⁾.

Related honeycombs

The [4,3⁵,4], , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb.

The 7-cubic honeycomb can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets.

Quadritruncated 7-cubic honeycomb

A quadritruncated 7-cubic honeycomb, , contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D₇^* lattice. Facets can be identically colored from a doubled ${\tilde {C}}_{7}$ ×2, [[4,3⁵,4]] symmetry, alternately colored from ${\tilde {C}}_{7}$ , [4,3⁵,4] symmetry, three colors from ${\tilde {B}}_{7}$ , [4,3⁴,3^1,1] symmetry, and 4 colors from ${\tilde {D}}_{7}$ , [3^1,1,3³,3^1,1] symmetry.

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 [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	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
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
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

Related honeycombs

Quadritruncated 7-cubic honeycomb

See also

References