Английская Википедия:5-cubic honeycomb
| 5-cubic honeycomb | |
|---|---|
| (no image) | |
| Type | Regular 5-space honeycomb Uniform 5-honeycomb |
| Family | Hypercube honeycomb |
| Schläfli symbol | Шаблон:Math |
| Coxeter-Dynkin diagrams |
Шаблон:CDD |
| 5-face type | Шаблон:Math (5-cube) |
| 4-face type | Шаблон:Math (tesseract) |
| Cell type | Шаблон:Math (cube) |
| Face type | Шаблон:Math (square) |
| Face figure | Шаблон:Math (octahedron) |
| Edge figure | Шаблон:Math (16-cell) |
| Vertex figure | Шаблон:Math (5-orthoplex) |
| Coxeter group | <math>{\tilde{C}}_5</math> Шаблон:Math |
| 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 {4,33,4}. 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], Шаблон:CDD, 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, Шаблон:CDD, contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled <math>{\tilde{C}}_5</math>×2, [[4,33,4]] symmetry, alternately colored from <math>{\tilde{C}}_5</math>, [4,33,4] symmetry, three colors from <math>{\tilde{B}}_5</math>, [4,3,3,31,1] symmetry, and 4 colors from <math>{\tilde{D}}_5</math>, [31,1,3,31,1] symmetry.
See also
Regular and uniform honeycombs in 5-space:
- 5-demicubic honeycomb
- 5-simplex honeycomb
- Truncated 5-simplex honeycomb
- Omnitruncated 5-simplex honeycomb
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Шаблон:ISBN 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, Шаблон:ISBN [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
