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	Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD
Facets	{3,3,3,3,3,3,4} h{4,3,3,3,3,3,3}
Vertex figure	Rectified 8-orthoplex
Coxeter group	${\tilde{B}}_8$ [4,3,3,3,3,3,31,1] ${\tilde{D}}_8$ [31,1,3,3,3,3,31,1]

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} File:Demiocteract ortho petrie.svg and the alternated vertices create 8-orthoplex {3,3,3,3,3,3,4} facets File:Cross graph 8 Nodes highlighted.svg.

D8 lattice

The vertex arrangement of the 8-demicubic honeycomb is the D₈ lattice. The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice. 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. Both ${\tilde{E}}_8$ and ${\tilde{D}}_8$ can be seen as affine extensions of $D_8$ from different nodes: File:Affine D8 E8 relations.png

The D lattice (also called D) 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). 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)).

Template:CDD ∪ Template:CDD = Template:CDD.

The D lattice (also called D and C) can be constructed by the union of all four D8 lattices: It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD = Template:CDD ∪ Template:CDD.

The kissing number of the D lattice is 16 (2n for n≥5). and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, Template:CDD, containing all trirectified 8-orthoplex Voronoi cell, Template:CDD.

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
${\tilde{B}}_8$ = [31,1,3,3,3,3,3,4] = [1+,4,3,3,3,3,3,3,4]	h{4,3,3,3,3,3,3,4}	Template:CDD = Template:CDD	Template:CDD [3,3,3,3,3,3,4]	256: 8-demicube 16: 8-orthoplex
${\tilde{D}}_8$ = [31,1,3,3,3,31,1] = [1+,4,3,3,3,3,31,1]	h{4,3,3,3,3,3,31,1}	Template:CDD = Template:CDD	Template:CDD [36,1,1]	128+128: 8-demicube 16: 8-orthoplex
2×½${\tilde{C}}_8$ = [[(4,3,3,3,3,3,4,2+)]]	ht0,8{4,3,3,3,3,3,3,4}	Template:CDD		128+64+64: 8-demicube 16: 8-orthoplex

See also

• 8-cubic honeycomb
• Uniform polytope

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Template:ISBN
  • 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, Template:ISBN [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)