Omnitruncated simplectic honeycomb
In geometry an omnitruncated simplectic honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the affine Coxeter group. Each is composed of omnitruncated simplex facets. The vertex figure for each is an irregular n-simplex.
The facets of an omnitruncated simplectic honeycomb are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
n | Image | Tessellation | Facets | Vertex figure | Facets per vertex figure | Vertices per vertex figure | |
---|---|---|---|---|---|---|---|
1 | Apeirogon |
Line segment | Line segment | 1 | 2 | ||
2 | Hexagonal tiling |
hexagon |
Equilateral triangle |
3 hexagons | 3 | ||
3 | Bitruncated cubic honeycomb |
Truncated octahedron |
irr. tetrahedron |
4 truncated octahedron | 4 | ||
4 | Omnitruncated 4-simplex honeycomb |
Omnitruncated 4-simplex |
irr. 5-cell |
5 omnitruncated 4-simplex | 5 | ||
5 | Omnitruncated 5-simplex honeycomb |
Omnitruncated 5-simplex |
irr. 5-simplex |
6 omnitruncated 5-simplex | 6 | ||
6 | Omnitruncated 6-simplex honeycomb |
Omnitruncated 6-simplex |
irr. 6-simplex |
7 omnitruncated 6-simplex | 7 | ||
7 | Omnitruncated 7-simplex honeycomb |
Omnitruncated 7-simplex |
irr. 7-simplex |
8 omnitruncated 7-simplex | 8 | ||
8 | Omnitruncated 8-simplex honeycomb |
Omnitruncated 8-simplex |
irr. 8-simplex |
9 omnitruncated 8-simplex | 9 |
Projection by folding
The (2n-1)-simplex honeycombs can be projected into the n-dimensional omnitruncated hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
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See also
References
- George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
- Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
- 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
- (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 | ||||||
---|---|---|---|---|---|---|
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} |
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