7-demicubic honeycomb

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.

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} ht0,7{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	${\displaystyle {\tilde {B}}_{7}}$ [4,3,3,3,3,3^1,1] ${\displaystyle {\tilde {D}}_{7}}$, [3^1,1,3,3,3,3^1,1]

It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h{4,3,3,3,3,3} and the alternated vertices create 7-orthoplex {3,3,3,3,3,4} facets.

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
${\displaystyle {\tilde {B}}_{7}}$ = [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
${\displaystyle {\tilde {D}}_{7}}$ = [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${\displaystyle {\tilde {C}}_{7}}$ = [[(4,3,3,3,3,4,2<sup>+</sup>)]]	ht0,7{4,3,3,3,3,3,4}			64+32+32: 7-demicube 14: 7-orthoplex

See also

• 7-cubic honeycomb

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

1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D7.html
2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
3. Conway (1998), p. 119
4. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds7.html
5. Conway (1998), p. 466

External links

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^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21