7-demicubic 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{4,3,3,3,3,31,1} ht0,7{4,3,3,3,3,3,4}
Coxeter-Dynkin diagram	File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png = File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png File:CDel label2.pngFile:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel branch.png
Facets	{3,3,3,3,3,4} h{4,3,3,3,3,3}
Vertex figure	Rectified 7-orthoplex
Coxeter group	${\tilde{B}}_7$ [4,3,3,3,3,31,1] ${\tilde{D}}_7$, [31,1,3,3,3,31,1]

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. 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. The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice. 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).

File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png ∪ File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png

The D^*
₇ lattice (also called D⁴
₇ and C²
₇) can be constructed by the union of all four 7-demicubic lattices: It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png ∪ File:CDel nodes 01rd.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png ∪ File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png ∪ File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 01ld.png = File:CDel nodes 10r.pngFile:CDel 4a4b.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel branch.png ∪ File:CDel nodes 01r.pngFile:CDel 4a4b.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel branch.png.

The kissing number of the D^*
₇ lattice is 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb, File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a4b.pngFile:CDel nodes.png, containing all with tritruncated 7-orthoplex, File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png Voronoi cells.

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
${\tilde{B}}_7$ = [31,1,3,3,3,3,4] = [1+,4,3,3,3,3,3,4]	h{4,3,3,3,3,3,4}	File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png [3,3,3,3,3,4]	128: 7-demicube 14: 7-orthoplex
${\tilde{D}}_7$ = [31,1,3,3,31,1] = [1+,4,3,3,3,31,1]	h{4,3,3,3,3,31,1}	File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png [35,1,1]	64+64: 7-demicube 14: 7-orthoplex
2×½${\tilde{C}}_7$ = [[(4,3,3,3,3,4,2+)]]	ht0,7{4,3,3,3,3,3,4}	File:CDel label2.pngFile:CDel branch hh.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a4b.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel branch.png		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 [2]
  • (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.

External links

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	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21