Cyclotruncated 5-simplex honeycomb
In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1.
Cyclotruncated 5-simplex honeycomb | |
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(No image) | |
Type | Uniform honeycomb |
Family | Cyclotruncated simplectic honeycomb |
Schläfli symbol | t_{0,1}{3^{[6]}} |
Coxeter diagram | |
5-face types | {3,3,3,3} t{3,3,3,3} 2t{3,3,3,3} |
4-face types | {3,3,3} t{3,3,3} |
Cell types | {3,3} t{3,3} |
Face types | {3} t{3} |
Vertex figure | Elongated 5-cell antiprism |
Coxeter groups | ×2_{2}, [[3^{[6]}]] |
Properties | vertex-transitive |
Structure
Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells.
It can be constructed as six sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
This honeycomb is one of 12 unique uniform honeycombs[1] constructed by the Coxeter group. The extended symmetry of the hexagonal diagram of the Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:
A5 honeycombs | ||||
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Hexagon symmetry |
Extended symmetry |
Extended diagram |
Extended group |
Honeycomb diagrams |
a1 |
[3^{[6]}] | |||
d2 |
<[3^{[6]}]> | ×2_{1} | ||
p2 |
[[3^{[6]}]] | ×2_{2} | ||
i4 |
[<[3^{[6]}]>] | ×2_{1}×2_{2} | ||
d6 |
<3[3^{[6]}]> | ×6_{1} | ||
r12 |
[6[3^{[6]}]] | ×12 |
See also
Regular and uniform honeycombs in 5-space:
Notes
- mathworld: Necklace, OEIS sequence A000029 13-1 cases, skipping one with zero marks
References
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- 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 | ||||||
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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} |