7-demicubic honeycomb

Last updated April 10, 2024

7-demicubic honeycomb
(No image)
Type	Uniform 7-honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h{4,3,3,3,3,3,4} h{4,3,3,3,3,3^1,1} ht_0,7{4,3,3,3,3,3,4}
Coxeter-Dynkin diagram	= =
Facets	{3,3,3,3,3,4} h{4,3,3,3,3,3}
Vertex figure	Rectified 7-orthoplex
Coxeter group	${\tilde {B}}_{7}$ [4,3,3,3,3,3^1,1] ${\tilde {D}}_{7}$ , [3^1,1,3,3,3,3^1,1]

The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb.

D7 lattice

The vertex arrangement of the 7-demicubic honeycomb is the D₇ lattice.^[1] The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice.^[2] The best known is 126, from the E₇ lattice and the 3₃₁ honeycomb.

The D⁺
₇ packing (also called D²
₇) can be constructed by the union of two D₇ lattices. The D⁺
_n packings form lattices only in even dimensions. The kissing number is 2⁶=64 (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).^[3]

∪

The D^*
₇ lattice (also called D⁴
₇ and C²
₇) can be constructed by the union of all four 7-demicubic lattices:^[4] It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

∪

=

∪

.

The kissing number of the D^*
₇ lattice is 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb, , containing all with tritruncated 7-orthoplex, Voronoi cells.^[5]

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 128 7-demicube facets around each vertex.

Coxeter group	Schläfli symbol	Coxeter-Dynkin diagram	Vertex figure Symmetry	Facets/verf
${\tilde {B}}_{7}$ = [3^1,1,3,3,3,3,4] = [1⁺,4,3,3,3,3,3,4]	h{4,3,3,3,3,3,4}	=	[3,3,3,3,3,4]	128: 7-demicube 14: 7-orthoplex
${\tilde {D}}_{7}$ = [3^1,1,3,3,3^1,1] = [1⁺,4,3,3,3,3^1,1]	h{4,3,3,3,3,3^1,1}	=	[3^5,1,1]	64+64: 7-demicube 14: 7-orthoplex
2×½ ${\tilde {C}}_{7}$ = [[(4,3,3,3,3,4,2⁺)]]	ht_0,7{4,3,3,3,3,3,4}			64+32+32: 7-demicube 14: 7-orthoplex

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References

Coxeter, H.S.M. Regular Polytopes , (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
- pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={3^1,1,4}, h{4,3,3,4}={3,3,4,3}, ...
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]
Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.

Notes

↑ "The Lattice D7".
↑ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
↑ Conway (1998), p. 119
↑ "The Lattice D7".
↑ Conway (1998), p. 466

External links

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	{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
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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[1] "The Lattice D7".

[2] Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai

[3] Conway (1998), p. 119

[4] "The Lattice D7".

[5] Conway (1998), p. 466

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