7-cubic honeycomb
The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space.
7-cubic honeycomb | |
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Type | Regular 7-honeycomb Uniform 7-honeycomb |
Family | Hypercube honeycomb |
Schläfli symbol | {4,3^{5},4} {4,3^{4},3^{1,1}} {∞}^{7} |
Coxeter-Dynkin diagrams | |
7-face type | {4,3,3,3,3,3} |
6-face type | {4,3,3,3,3} |
5-face type | {4,3,3,3} |
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 | 128 {4,3^{5}} (7-orthoplex) |
Coxeter group | [4,3^{5},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.
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3^{5},4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,3^{4},3^{1,1}}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}^{7}.
Related honeycombs
The [4,3^{5},4],
The 7-cubic honeycomb can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets.
Quadritruncated 7-cubic honeycomb
A quadritruncated 7-cubic honeycomb,
See also
- List of regular polytopes
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
- (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 | / / | ||||
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |