Английская Википедия:Cyclotruncated simplicial honeycomb
In geometry, the cyclotruncated simplicial honeycomb (or cyclotruncated n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the symmetry of the <math>{\tilde{A}}_n</math> affine Coxeter group. It is given a Schläfli symbol t0,1{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 Шаблон:CDD. In 3-dimensions it represents the quarter cubic honeycomb, with Coxeter graph Шаблон:CDD filling space with alternately tetrahedral and truncated tetrahedral cells. In 4-dimensions it is called a cyclotruncated 5-cell honeycomb, with Coxeter graph Шаблон:CDD, with 5-cell, truncated 5-cell, and bitruncated 5-cell facets. In 5-dimensions it is called a cyclotruncated 5-simplex honeycomb, with Coxeter graph Шаблон:CDD, filling space by 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets. In 6-dimensions it is called a cyclotruncated 6-simplex honeycomb, with Coxeter graph Шаблон:CDD, filling space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets.
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:
| <math>{\tilde{A}}_3</math> | Шаблон:CDD | <math>{\tilde{A}}_5</math> | Шаблон:CDD | <math>{\tilde{A}}_7</math> | Шаблон:CDD | <math>{\tilde{A}}_9</math> | Шаблон:CDD | <math>{\tilde{A}}_{11}</math> | Шаблон:CDD | ... |
|---|---|---|---|---|---|---|---|---|---|---|
| <math>{\tilde{A}}_2</math> | Шаблон:CDD | <math>{\tilde{A}}_4</math> | Шаблон:CDD | <math>{\tilde{A}}_6</math> | Шаблон:CDD | <math>{\tilde{A}}_8</math> | Шаблон:CDD | <math>{\tilde{A}}_{10}</math> | Шаблон:CDD | ... |
| <math>{\tilde{A}}_3</math> | Шаблон:CDD | <math>{\tilde{A}}_5</math> | Шаблон:CDD | <math>{\tilde{A}}_7</math> | Шаблон:CDD | <math>{\tilde{A}}_9</math> | Шаблон:CDD | ... | ||
| <math>{\tilde{C}}_1</math> | Шаблон:CDD | <math>{\tilde{C}}_2</math> | Шаблон:CDD | <math>{\tilde{C}}_3</math> | Шаблон:CDD | <math>{\tilde{C}}_4</math> | Шаблон:CDD | <math>{\tilde{C}}_5</math> | Шаблон:CDD | ... |
See also
- Hypercubic honeycomb
- Alternated hypercubic honeycomb
- Quarter hypercubic honeycomb
- Simplectic honeycomb
- Omnitruncated simplicial honeycomb
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, Шаблон:ISBN
- 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, Шаблон:ISBN [1]
- (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]
