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 ${\tilde{A}}_n$ 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 . 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 an 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 simplectic 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, 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]