# 5-demicubic honeycomb

The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.

Demipenteractic honeycomb
(No image)
TypeUniform 5-honeycomb
FamilyAlternated hypercubic honeycomb
Schläfli symbolsh{4,3,3,3,4}
h{4,3,3,31,1}
ht0,5{4,3,3,3,4}
h{4,3,3,4}h{}
h{4,3,31,1}h{}
ht0,4{4,3,3,4}h{}
h{4,3,4}h{}h{}
h{4,31,1}h{}h{}
Coxeter diagrams

=
=

Facets{3,3,3,4}
h{4,3,3,3}
Vertex figuret1{3,3,3,4}
Coxeter group${\displaystyle {\tilde {B}}_{5}}$ [4,3,3,31,1]
${\displaystyle {\tilde {D}}_{5}}$ [31,1,3,31,1]

It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h{4,3,3,3} and the alternated vertices create 5-orthoplex {3,3,3,4} facets.

## D5 lattice

The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions.[1] The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 of this lattice.[2]

The D+
5
packing (also called D2
5
) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]

The D*
5
[4] lattice (also called D4
5
and C2
5
) can be constructed by the union of all four 5-demicubic lattices:[5] It is also the 5-dimensional body centered cubic, the union of two 5-cube honeycombs in dual positions.

= .

The kissing number of the D*
5
lattice is 10 (2n for n≥5) and it Voronoi tessellation is a tritruncated 5-cubic honeycomb, , containing all with bitruncated 5-orthoplex, Voronoi cells.[6]

## Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 32 5-demicube facets around each vertex.

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
${\displaystyle {\tilde {B}}_{5}}$ = [31,1,3,3,4]
= [1+,4,3,3,4]
h{4,3,3,3,4} =
[3,3,3,4]
32: 5-demicube
10: 5-orthoplex
${\displaystyle {\tilde {D}}_{5}}$ = [31,1,3,31,1]
= [1+,4,3,31,1]
h{4,3,3,31,1} =
[32,1,1]
16+16: 5-demicube
10: 5-orthoplex
2×½${\displaystyle {\tilde {C}}_{5}}$ = [[(4,3,3,3,4,2<sup>+</sup>)]]ht0,5{4,3,3,3,4} 16+8+8: 5-demicube
10: 5-orthoplex

This honeycomb is one of 20 uniform honeycombs constructed by the ${\displaystyle {\tilde {D}}_{5}}$ Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:

Regular and uniform honeycombs in 5-space:

## References

1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.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/Ds5.html
5. Conway (1998), p. 120
6. Conway (1998), p. 466
• 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}={31,1,4}, 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.