5-cubic honeycomb

In geometry, 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.

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}×{∞} {4,3,4}×{4,4} {4,3,4}×{∞}(2) {4,4}(2)×{∞} {∞}(5)
Coxeter-Dynkin diagrams
5-face type	{4,33} (5-cube)
4-face type	{4,3,3} (tesseract)
Cell type	{4,3} (cube)
Face type	{4} (square)
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

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 {4,3,3,3^1,1}. 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 three 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,3⁴}.

The Penrose tilings are 2-dimensional aperiodic tilings that can be obtained as a projection of the 5-cubic honeycomb along a 5-fold rotational axis of symmetry. The vertices correspond to points in the 5-dimensional cubic lattice, and the tiles are formed by connecting points in a predefined manner.^[1]

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 ${\displaystyle {\tilde {B}}_{5}}$, [4,3,3,3^1,1] symmetry, and 4 colors from ${\displaystyle {\tilde {D}}_{5}}$, [3^1,1,3,3^1,1] symmetry.

See also

• List of regular polytopes

Regular and uniform honeycombs in 5-space:

• 5-demicubic honeycomb
• 5-simplex honeycomb
• Truncated 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb