6-cubic honeycomb
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!bgcolor=#e7dcc3 colspan=2|6-cubic honeycomb | |
| bgcolor=#ffffff align=center colspan=2|(no image) | |
| bgcolor=#e7dcc3|Type | Regular 6-honeycomb Uniform 6-honeycomb |
| bgcolor=#e7dcc3|Family | Hypercube honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | {4,34,4} {4,33,31,1} |
| bgcolor=#e7dcc3|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|4|node_1}} {{CDD|node_1|4|node|3|node|3|node|3|node|split1|nodes}} {{CDD|node_1|4|node|3|node|3|node|3|node|4|node|2|node_1|infin|node}} {{CDD|node_1|4|node|3|node|3|node|3|node|4|node_1|2|node_1|infin|node}}
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| bgcolor=#e7dcc3|6-face type | {4,34} |
| bgcolor=#e7dcc3|5-face type | {4,33} |
| bgcolor=#e7dcc3|4-face type | {4,3,3} |
| bgcolor=#e7dcc3|Cell type | {4,3} |
| bgcolor=#e7dcc3|Face type | {4} |
| bgcolor=#e7dcc3|Face figure | {4,3} (octahedron) |
| bgcolor=#e7dcc3|Edge figure | 8 {4,3,3} (16-cell) |
| bgcolor=#e7dcc3|Vertex figure | 64 {4,34} (6-orthoplex) |
| bgcolor=#e7dcc3|Coxeter group | , [4,34,4] , [4,33,31,1] |
| bgcolor=#e7dcc3|Dual | self-dual |
| bgcolor=#e7dcc3|Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive |
The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space.
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,34,4}. Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol {4,33,31,1}. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(6).
Related honeycombs
The [4,34,4], {{CDD|node|4|node|3|node|3|node|3|node|3|node|4|node}}, Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb.
The 6-cubic honeycomb can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex facets.
= Trirectified 6-cubic honeycomb =
A trirectified 6-cubic honeycomb, {{CDD|node_1|split1|nodes|3ab|nodes|4a4b|nodes}}, contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the D6* lattice. Facets can be identically colored from a doubled ×2,
See also
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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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