8-demicubic_honeycomb Knowpia

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 edit

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.^[4] 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).^[5] 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:^[6] 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).^[7] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, , containing all trirectified 8-orthoplex Voronoi cell, .^[8]

Symmetry constructions edit

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⁺)]]	ht_0,8{4,3,3,3,3,3,3,4}			128+64+64: 8-demicube 16: 8-orthoplex

Notes edit

^ "The Lattice D8".
^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
^ Johnson (2015) p.177
^ Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
^ Conway (1998), p. 119
^ "The Lattice D8".
^ Conway (1998), p. 120
^ Conway (1998), p. 466

References edit

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 [2]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: Geometries and Transformations, (2018)
Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.

External links edit

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

[1] "The Lattice D8".

[2] Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]

[3] Johnson (2015) p.177

[4] Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)

[5] Conway (1998), p. 119

[6] "The Lattice D8".

[7] Conway (1998), p. 120

[8] Conway (1998), p. 466

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Summary

D8 lattice edit

Symmetry constructions edit

See also edit

Notes edit

References edit

External links edit