5-cubic honeycomb

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5-cubic honeycomb
(no image)
Type Regular 5-space 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

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png

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 {\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[edit]

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[edit]

The [4,33,4], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, 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[edit]

A tritruncated 5-cubic honeycomb, CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png, containins all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled {\tilde{C}}_5×2, [[4,33,4]] symmetry, alternately colored from {\tilde{C}}_5, [4,33,4] symmetry, three colors from {\tilde{B}}_5, [4,3,3,31,1] symmetry, and 4 colors from {\tilde{D}}_5, [31,1,3,31,1] symmetry.

See also[edit]

Regular and uniform honeycombs in 5-space:

References[edit]