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5-demicubic honeycomb

## Summary

Demipenteractic honeycomb
(No image)
Type Uniform 5-honeycomb
Family Alternated hypercubic honeycomb
Schläfli symbols h{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 figure t1{3,3,3,4}
Coxeter group ${\displaystyle {\tilde {B}}_{5}}$ [4,3,3,31,1]
${\displaystyle {\tilde {D}}_{5}}$ [31,1,3,31,1]

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.

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 its Voronoi tessellation is a tritruncated 5-cubic honeycomb,      , containing all 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+)]] 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:

D5 honeycombs
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
[31,1,3,31,1]         ${\displaystyle {\tilde {D}}_{5}}$
<[31,1,3,31,1]>
↔ [31,1,3,3,4]

${\displaystyle {\tilde {D}}_{5}}$ ×21 = ${\displaystyle {\tilde {B}}_{5}}$         ,        ,        ,

,        ,        ,

[[31,1,3,31,1]]         ${\displaystyle {\tilde {D}}_{5}}$ ×22        ,
<2[31,1,3,31,1]>
↔ [4,3,3,3,4]

${\displaystyle {\tilde {D}}_{5}}$ ×41 = ${\displaystyle {\tilde {C}}_{5}}$         ,        ,        ,        ,        ,
[<2[31,1,3,31,1]>]
↔ [[4,3,3,3,4]]

${\displaystyle {\tilde {D}}_{5}}$ ×8 = ${\displaystyle {\tilde {C}}_{5}}$ ×2        ,        ,

Regular and uniform honeycombs in 5-space:

## References

1. ^ "The Lattice D5".
2. ^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
3. ^ Conway (1998), p. 119
4. ^ "The Lattice D5".
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 [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.