# 5-cubic honeycomb

(Redirected from 5-cube honeycomb)
5-cubic honeycomb
(no image)
Type Regular 5-space honeycomb
Uniform 5-honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,33,4}
t0,5{4,33,4}
{4,3,3,31,1}
{4,3,4}x{∞}
{4,3,4}x{4,4}
{4,3,4}x{∞}2
{4,4}2x{∞}
{∞}5
Coxeter-Dynkin diagrams

5-face type {4,33}
4-face type {4,3,3}
Cell type {4,3}
Face type {4}
Face figure {4,3}
(octahedron)
Edge figure 8 {4,3,3}
(16-cell)
Vertex figure 32 {4,33}
(5-orthoplex)
Coxeter group ${\displaystyle {\tilde {C}}_{5}}$, [4,33,4]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.

## Constructions

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,33,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,31,1}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}5.

## Related polytopes and honeycombs

The [4,33,4],            , Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.

The 5-cubic honeycomb can be alternated into the 5-demicubic honeycomb, replacing the 5-cubes with 5-demicubes, and the alternated gaps are filled by 5-orthoplex facets.

It is also related to the regular 6-cube which exists in 6-space with 3 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,34}.

### Tritruncated 5-cubic honeycomb

A tritruncated 5-cubic honeycomb,      , contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled ${\displaystyle {\tilde {C}}_{5}}$ ×2, [[4,33,4]] symmetry, alternately colored from ${\displaystyle {\tilde {C}}_{5}}$ , [4,33,4] symmetry, three colors from ${\displaystyle {\tilde {B}}_{5}}$ , [4,3,3,31,1] symmetry, and 4 colors from ${\displaystyle {\tilde {D}}_{5}}$ , [31,1,3,31,1] symmetry.

Regular and uniform honeycombs in 5-space:

## References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• 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]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21