Misplaced Pages

5-cubic honeycomb

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Tiling of five-dimensional space
5-cubic honeycomb
(no image)
Type Regular 5-space honeycomb
Uniform 5-honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,3,4}
t0,5{4,3,4}
{4,3,3,3}
{4,3,4}×{∞}
{4,3,4}×{4,4}
{4,3,4}×{∞}
{4,4}×{∞}
{∞}
Coxeter-Dynkin diagrams















5-face type {4,3} (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 {4,3} (5-orthoplex)
Coxeter group C ~ 5 {\displaystyle {\tilde {C}}_{5}}
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

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.

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 {4,3,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,3}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}.

Related polytopes and honeycombs

The , , 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, {4,3}.

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.

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 C ~ 5 {\displaystyle {\tilde {C}}_{5}} ×2, ] symmetry, alternately colored from C ~ 5 {\displaystyle {\tilde {C}}_{5}} , symmetry, three colors from B ~ 5 {\displaystyle {\tilde {B}}_{5}} , symmetry, and 4 colors from D ~ 5 {\displaystyle {\tilde {D}}_{5}} , symmetry.

See also

Regular and uniform honeycombs in 5-space:

References

  1. de Bruijn, N. G. (1981). "Algebraic theory of Penrose's non-periodic tilings of the plane, I, II" (PDF). Indagationes Mathematicae. 43 (1): 39–66. doi:10.1016/1385-7258(81)90017-2.
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}}
E Uniform tiling 0 δ3 hδ3 qδ3 Hexagonal
E Uniform convex honeycomb 0 δ4 hδ4 qδ4
E Uniform 4-honeycomb 0 δ5 hδ5 qδ5 24-cell honeycomb
E Uniform 5-honeycomb 0 δ6 hδ6 qδ6
E Uniform 6-honeycomb 0 δ7 hδ7 qδ7 222
E Uniform 7-honeycomb 0 δ8 hδ8 qδ8 133 • 331
E Uniform 8-honeycomb 0 δ9 hδ9 qδ9 152 • 251 • 521
E Uniform 9-honeycomb 0 δ10 hδ10 qδ10
E Uniform 10-honeycomb 0 δ11 hδ11 qδ11
E Uniform (n-1)-honeycomb 0 δn hδn qδn 1k2 • 2k1 • k21
Categories: