8-demicubic honeycomb

8-demicubic honeycomb
(No image)
Type	Uniform 8-honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h{4,3,3,3,3,3,3,4}
Coxeter diagrams	= =
Facets	{3,3,3,3,3,3,4} h{4,3,3,3,3,3,3}
Vertex figure	Rectified 8-orthoplex
Coxeter group	${{\tilde {B}}}_{8}$ [4,3,3,3,3,3,3^1,1] ${{\tilde {D}}}_{8}$ [3^1,1,3,3,3,3,3^1,1]

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb.

It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h{4,3,3,3,3,3,3} and the alternated vertices create 8-orthoplex {3,3,3,3,3,3,4} facets .

D8 lattice

The vertex arrangement of the 8-demicubic honeycomb is the D₈ lattice.^[1] The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice.^[2] The best known is 240, from the E₈ lattice and the 5₂₁ honeycomb.

${\tilde {E}}_{8}$ contains ${{\tilde {D}}}_{8}$ as a subgroup of index 270.^[3] Both ${\tilde {E}}_{8}$ and ${{\tilde {D}}}_{8}$ can be seen as affine extensions of $D_{8}$ from different nodes:

The D+
8 lattice (also called D2
8) can be constructed by the union of two D8 lattices. This packing is only a lattice for even dimensions. The kissing number is 240. (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).^[4] It is identical to the E8 lattice. At 8-dimensions, the 240 contacts contain both the 2⁷=128 from lower dimension contact progression (2^n-1), and 16*7=112 from higher dimensions (2n(n-1)).

∪

The D*
8 lattice (also called D4
8 and C2
8) can be constructed by the union of all four D8 lattices:^[5] 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*
8 lattice is 16 (2n for n≥5).^[6] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, , containing all trirectified 8-orthoplex Voronoi cell, .^[7]

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 256 8-demicube facets around each vertex.

Coxeter group	Schläfli symbol	Coxeter-Dynkin diagram	Vertex figure Symmetry	Facets/verf
${{\tilde {B}}}_{8}$ = [3^1,1,3,3,3,3,3,4] = [1⁺,4,3,3,3,3,3,3,4]	h{4,3,3,3,3,3,3,4}	=	[3,3,3,3,3,3,4]	256: 8-demicube 16: 8-orthoplex
${{\tilde {D}}}_{8}$ = [3^1,1,3,3,3,3^1,1] = [1⁺,4,3,3,3,3,3^1,1]	h{4,3,3,3,3,3,3^1,1}	=	[3^6,1,1]	128+128: 8-demicube 16: 8-orthoplex
2×½ ${{\tilde {C}}}_{8}$ = [[(4,3,3,3,3,3,4,2<sup>+</sup>)]]	ht_0,8{4,3,3,3,3,3,3,4}			128+64+64: 8-demicube 16: 8-orthoplex

Notes

↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D8.html
↑ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
↑ Johnson (2015) p.177
↑ Conway (1998), p. 119
↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds8.html
↑ Conway (1998), p. 120
↑ Conway (1998), p. 466

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]
N.W. Johnson: Geometries and Transformations, (2015)
Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.

External links

Olshevsky, George. "Half measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.

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

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