Cyclotruncated simplicial honeycomb explained

In geometry, the cyclotruncated simplicial honeycomb (or cyclotruncated n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the symmetry of the

{\tilde{A}} _{n} affine Coxeter group. It is given a Schläfli symbol t_0,1, 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 . In 3-dimensions it represents the quarter cubic honeycomb, with Coxeter graph filling space with alternately tetrahedral and truncated tetrahedral cells. In 4-dimensions it is called a cyclotruncated 5-cell honeycomb, with Coxeter graph, 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, 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, filling space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets.

n	{\tilde{A}} _{n}	Name Coxeter diagram	Vertex figure	Image and facets
1	{\tilde{A}} _{1}	Apeirogon		Yellow and cyan line segments
2	{\tilde{A}} _{2}	Trihexagonal tiling	Rectangle	With yellow and blue equilateral triangles, and red hexagons
3	{\tilde{A}} _{3}	quarter cubic honeycomb	Elongated triangular antiprism	With yellow and blue tetrahedra, and red and purple truncated tetrahedra
4	{\tilde{A}} _{4}	Cyclotruncated 5-cell honeycomb	Elongated tetrahedral antiprism	5-cell, truncated 5-cell, bitruncated 5-cell
5	{\tilde{A}} _{5}	Cyclotruncated 5-simplex honeycomb		5-simplex, truncated 5-simplex, bitruncated 5-simplex
6	{\tilde{A}} _{6}	Cyclotruncated 6-simplex honeycomb		6-simplex, truncated 6-simplex, bitruncated 6-simplex, tritruncated 6-simplex
7	{\tilde{A}} _{7}	Cyclotruncated 7-simplex honeycomb		7-simplex, truncated 7-simplex, bitruncated 7-simplex
8	{\tilde{A}} _{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:

{\tilde{A}} _{3}		{\tilde{A}} _{5}		{\tilde{A}} _{7}		{\tilde{A}} _{9}		{\tilde{A}} _{11}		...
{\tilde{A}} _{2}		{\tilde{A}} _{4}		{\tilde{A}} _{6}		{\tilde{A}} _{8}		{\tilde{A}} _{10}		...
		{\tilde{A}} _{3}		{\tilde{A}} _{5}		{\tilde{A}} _{7}		{\tilde{A}} _{9}		...
{\tilde{C}} _{1}		{\tilde{C}} _{2}		{\tilde{C}} _{3}		{\tilde{C}} _{4}		{\tilde{C}} _{5}		...

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,
• 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (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]