7-demicubic honeycomb

7-demicubic honeycomb
(No image)
Type	Uniform honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h{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 heptacross
Coxeter group	[4,3,3,3,3,31,1] , [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.

Its vertex arrangement is called the D7 lattice.^[1]

Kissing number

This tessellation represents a dense sphere packing (With a Kissing number of 84, compared to the best known of 126), with each vertex of this polytope represents the center point for one of the 84 6-spheres, and the central radius, equal to the edge length exactly fits one more 6-sphere.

See also

• Cubic honeycomb
• Alternated cubic honeycomb
• Demitesseractic honeycomb
• Demipenteractic honeycomb
• Demihexeractic honeycomb
• Demihepteractic honeycomb
• Demiocteractic honeycomb
• Uniform polytope

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 [1]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Notes

External links

• Olshevsky, George, Half measure polytope at Glossary for Hyperspace.