Английская Википедия:Hypercubic honeycomb
| Файл:Square tiling uniform coloring 1.png A regular square tiling. Шаблон:CDD 1 color |
Файл:Partial cubic honeycomb.png A cubic honeycomb in its regular form. Шаблон:CDD 1 color |
| Файл:Square tiling uniform coloring 7.png A checkboard square tiling Шаблон:CDD 2 colors |
Файл:Bicolor cubic honeycomb.png A cubic honeycomb checkerboard. Шаблон:CDD 2 colors |
| Файл:Square tiling uniform coloring 8.png Expanded square tiling Шаблон:CDD 3 colors |
Файл:Runcinated cubic honeycomb.png Expanded cubic honeycomb Шаблон:CDD 4 colors |
| Файл:Square tiling uniform coloring 9.png Шаблон:CDD 4 colors |
Файл:Cubic 8-color honeycomb.png Шаблон:CDD 8 colors |
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in Шаблон:Mvar-dimensional spaces with the Schläfli symbols Шаблон:Math and containing the symmetry of Coxeter group Шаблон:Math (or Шаблон:Math) for Шаблон:Math.
The tessellation is constructed from 4 Шаблон:Mvar-hypercubes per ridge. The vertex figure is a cross-polytope Шаблон:Math
The hypercubic honeycombs are self-dual.
Coxeter named this family as Шаблон:Math for an Шаблон:Mvar-dimensional honeycomb.
Wythoff construction classes by dimension
A Wythoff construction is a method for constructing a uniform polyhedron or plane tiling.
The two general forms of the hypercube honeycombs are the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard.
A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lower-dimensional elements. For example, an expanded cubic honeycomb has cubic cells centered on the original cubes, on the original faces, on the original edges, on the original vertices, creating 4 colors of cells around in vertex in 1:3:3:1 counts.
The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, in which each of the three axial directions may have different edge lengths. The facets are hyperrectangles, also called orthotopes; in 2 and 3 dimensions the orthotopes are rectangles and cuboids respectively.
See also
- Alternated hypercubic honeycomb
- Quarter hypercubic honeycomb
- Simplectic honeycomb
- Truncated simplectic honeycomb
- Omnitruncated simplectic honeycomb
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Шаблон:ISBN
- pp. 122–123. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
- pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}
- p. 296, Table II: Regular honeycombs, δn+1
