# 5-demicubic honeycomb

Demipenteractic honeycomb :
| (No image) ||
| Type | Uniform 5-honeycomb |
| Family | Alternated hypercubic honeycomb |
| Schläfli symbols | h{4,3,3,3,4} h{4,3,3,3$^{1,1}$} ht$_{0,5}${4,3,3,3,4} h{4,3,3,4}h{∞} h{4,3,3$^{1,1}$}h{∞} ht$_{0,4}${4,3,3,4}h{∞} h{4,3,4}h{∞}h{∞} h{4,3$^{1,1}$}h{∞}h{∞} |
| Coxeter diagrams | = = |
| Facets | {3,3,3,4} h{4,3,3,3} |
| Vertex figure | t$_1${3,3,3,4} |
| Coxeter group | ${\tilde {B}}_{5}$ [4,3,3,3$^{1,1}$] ${\tilde {D}}_{5}$ [3$^{1,1}$,3,3$^{1,1}$] |
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 D$_5$ lattice which is the densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 of this lattice.
The D$^+$ $_5$ packing (also called D$^2$ $_5$) can be constructed by the union of two D$_5$ lattices. The analogous packings form lattices only in even dimensions. The kissing number is 2$^4$=16 (2$^{n-1}$ for n<8, 240 for n=8, and 2n(n-1) for n>8).
The D$^*$ $_5$ lattice (also called D$^4$ $_5$ and C$^2$ $_5$) can be constructed by the union of all four 5-demicubic lattices: 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.

## 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 |
| ${\tilde {B}}_{5}$ = [3$^{1,1}$,3,3,4] = [1$^+$,4,3,3,4] | h{4,3,3,3,4} | = | [3,3,3,4] | 32: 5-demicube 10: 5-orthoplex |
| ${\tilde {D}}_{5}$ = [3$^{1,1}$,3,3$^{1,1}$] = [1$^+$,4,3,3$^{1,1}$] | h{4,3,3,3$^{1,1}$} | = | [3$^{2,1,1}$] | 16+16: 5-demicube 10: 5-orthoplex |
| 2×½${\tilde {C}}_{5}$ = [[(4,3,3,3,4,2$^+$)]] | ht$_{0,5}${4,3,3,3,4} | | | 16+8+8: 5-demicube 10: 5-orthoplex |

## Related honeycombs

This honeycomb is one of 20 uniform honeycombs constructed by the ${\tilde {D}}_{5}$ 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 |
| [3$^{1,1}$,3,3$^{1,1}$] | | ${\tilde {D}}_{5}$ | |
| <[3$^{1,1}$,3,3$^{1,1}$]> ↔ [3$^{1,1}$,3,3,4] | ↔ | ${\tilde {D}}_{5}$×2$_1$ = ${\tilde {B}}_{5}$ | , , , , , , |
| [[3$^{1,1}$,3,3$^{1,1}$]] | | ${\tilde {D}}_{5}$×2$_2$ | , |
| <2[3$^{1,1}$,3,3$^{1,1}$]> ↔ [4,3,3,3,4] | ↔ | ${\tilde {D}}_{5}$×4$_1$ = ${\tilde {C}}_{5}$ | , , , , , |
| [<2[3$^{1,1}$,3,3$^{1,1}$]>] ↔ [[4,3,3,3,4]] | ↔ | ${\tilde {D}}_{5}$×8 = ${\tilde {C}}_{5}$×2 | , , |