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	${\displaystyle {\tilde {B}}_{8}}$ [4,3,3,3,3,3,3^1,1] ${\displaystyle {\tilde {D}}_{8}}$ [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.

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

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
${\displaystyle {\tilde {B}}_{8}}$  = [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
${\displaystyle {\tilde {D}}_{8}}$  = [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${\displaystyle {\tilde {C}}_{8}}$  = [[(4,3,3,3,3,3,4,2^+)]]	ht0,8{4,3,3,3,3,3,3,4}			128+64+64: 8-demicube 16: 8-orthoplex

See also

• 8-cubic honeycomb
• Uniform polytope

Notes

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

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 [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

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
E^2	Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
E^3	Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
E^4	Uniform 4-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E^5	Uniform 5-honeycomb	{3^[6]}	δ6	hδ6	qδ6
E^6	Uniform 6-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
E^7	Uniform 7-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
E^8	Uniform 8-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E^9	Uniform 9-honeycomb	{3^[10]}	δ10	hδ10	qδ10
E^10	Uniform 10-honeycomb	{3^[11]}	δ11	hδ11	qδ11
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21