8-cubic honeycomb
8-cubic honeycomb | |
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Type | Regular 8-honeycomb Uniform 8-honeycomb |
Family | Hypercube honeycomb |
Schläfli symbol | {4,3^{6},4} {4,3^{5},3^{1,1}} t_{0,8}{4,3^{6},4} {∞}^{(8)} |
Coxeter-Dynkin diagrams | |
8-face type | {4,3^{6}} |
7-face type | {4,3^{5}} |
6-face type | {4,3^{4}} |
5-face type | {4,3^{3}} |
4-face type | {4,3^{2}} |
Cell type | {4,3} |
Face type | {4} |
Face figure | {4,3} (octahedron) |
Edge figure | 8 {4,3,3} (16-cell) |
Vertex figure | 256 {4,3^{6}} (8-orthoplex) |
Coxeter group | [4,3^{6},4] |
Dual | self-dual |
Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive |
In geometry, the 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space.
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.
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3^{6},4}. Another form has two alternating hypercube facets (like a checkerboard) with Schläfli symbol {4,3^{5},3^{1,1}}. The lowest symmetry Wythoff construction has 256 types of facets around each vertex and a prismatic product Schläfli symbol {∞}^{(8)}.
Related honeycombs
The [4,3^{6},4], , Coxeter group generates 511 permutations of uniform tessellations, 271 with unique symmetry and 270 with unique geometry. The expanded 8-cubic honeycomb is geometrically identical to the 8-cubic honeycomb.
The 8-cubic honeycomb can be alternated into the 8-demicubic honeycomb, replacing the 8-cubes with 8-demicubes, and the alternated gaps are filled by 8-orthoplex facets.
Quadrirectified 8-cubic honeycomb
A quadrirectified 8-cubic honeycomb, , contains all trirectified 8-orthoplex facets and is the Voronoi tessellation of the D_{8}^{*} lattice. Facets can be identically colored from a doubled [math]\displaystyle{ {\tilde{C}}_8 }[/math]×2, 4,3^{6},4 symmetry, alternately colored from [math]\displaystyle{ {\tilde{C}}_8 }[/math], [4,3^{6},4] symmetry, three colors from [math]\displaystyle{ {\tilde{B}}_8 }[/math], [4,3^{5},3^{1,1}] symmetry, and 4 colors from [math]\displaystyle{ {\tilde{D}}_8 }[/math], [3^{1,1},3^{4},3^{1,1}] symmetry.
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 [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] |
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} |
Original source: https://en.wikipedia.org/wiki/8-cubic honeycomb.
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