Cyclotruncated simplectic honeycomb
In geometry, the cyclotruncated simplectic honeycomb (or cyclotruncated n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the symmetry of the affine Coxeter group. It is given a Schläfli symbol t_{0,1}{3^{[n+1]}}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with two adjacent nodes ringed. It is composed of n-simplex facets, along with all truncated n-simplices.
It is also called a Kagome lattice in two and three dimensions, although it is not a lattice.
In n-dimensions, each can be seen as a set of n+1 sets of parallel hyperplanes that divide space. Each hyperplane contains the same honeycomb of one dimension lower.
In 1-dimension, the honeycomb represents an apeirogon, with alternately colored line segments. In 2-dimensions, the honeycomb represents the trihexagonal tiling, with Coxeter graph
n | Name Coxeter diagram |
Vertex figure | Image and facets | |
---|---|---|---|---|
1 | Apeirogon |
Yellow and cyan line segments | ||
2 | Trihexagonal tiling |
Rectangle |
With yellow and blue equilateral triangles, and red hexagons | |
3 | quarter cubic honeycomb |
Elongated triangular antiprism |
With yellow and blue tetrahedra, and red and purple truncated tetrahedra | |
4 | Cyclotruncated 5-cell honeycomb |
Elongated tetrahedral antiprism |
5-cell, truncated 5-cell, bitruncated 5-cell | |
5 | Cyclotruncated 5-simplex honeycomb |
5-simplex, truncated 5-simplex, bitruncated 5-simplex | ||
6 | Cyclotruncated 6-simplex honeycomb |
6-simplex, truncated 6-simplex, bitruncated 6-simplex, tritruncated 6-simplex | ||
7 | Cyclotruncated 7-simplex honeycomb |
7-simplex, truncated 7-simplex, bitruncated 7-simplex | ||
8 | Cyclotruncated 8-simplex honeycomb |
8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, quadritruncated 8-simplex |
Projection by folding
The cyclotruncated (2n+1)- and 2n-simplex honeycombs and (2n-1)-simplex honeycombs can be projected into the n-dimensional 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} |