5-cubic 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 | [math]\displaystyle{ {\tilde{C}}_5 }[/math] [4,33,4] |
| 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], 
, 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 [math]\displaystyle{ {\tilde{C}}_5 }[/math]×2, 4,33,4 symmetry, alternately colored from [math]\displaystyle{ {\tilde{C}}_5 }[/math], [4,33,4] symmetry, three colors from [math]\displaystyle{ {\tilde{B}}_5 }[/math], [4,3,3,31,1] symmetry, and 4 colors from [math]\displaystyle{ {\tilde{D}}_5 }[/math], [31,1,3,31,1] 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
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
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|---|---|---|---|---|---|---|
| Space | Family | [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math] |
| E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
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