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]\displaystyle{ {\tilde{A}}_n }[/math] 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 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	[math]\displaystyle{ {\tilde{A}}_n }[/math]	Name Coxeter diagram	Vertex figure	Image and facets
1	[math]\displaystyle{ {\tilde{A}}_1 }[/math]	Apeirogon		Yellow and cyan line segments
2	[math]\displaystyle{ {\tilde{A}}_2 }[/math]	Trihexagonal tiling	Rectangle	With yellow and blue equilateral triangles, and red hexagons
3	[math]\displaystyle{ {\tilde{A}}_3 }[/math]	quarter cubic honeycomb	Elongated triangular antiprism	160px With yellow and blue tetrahedra, and red and purple truncated tetrahedra
4	[math]\displaystyle{ {\tilde{A}}_4 }[/math]	Cyclotruncated 5-cell honeycomb	Elongated tetrahedral antiprism	5-cell, truncated 5-cell, bitruncated 5-cell
5	[math]\displaystyle{ {\tilde{A}}_5 }[/math]	Cyclotruncated 5-simplex honeycomb		5-simplex, truncated 5-simplex, bitruncated 5-simplex
6	[math]\displaystyle{ {\tilde{A}}_6 }[/math]	Cyclotruncated 6-simplex honeycomb		6-simplex, truncated 6-simplex, bitruncated 6-simplex, tritruncated 6-simplex
7	[math]\displaystyle{ {\tilde{A}}_7 }[/math]	Cyclotruncated 7-simplex honeycomb		7-simplex, truncated 7-simplex, bitruncated 7-simplex
8	[math]\displaystyle{ {\tilde{A}}_8 }[/math]	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:

[math]\displaystyle{ {\tilde{A}}_3 }[/math]		[math]\displaystyle{ {\tilde{A}}_5 }[/math]		[math]\displaystyle{ {\tilde{A}}_7 }[/math]		[math]\displaystyle{ {\tilde{A}}_9 }[/math]		[math]\displaystyle{ {\tilde{A}}_{11} }[/math]		...
[math]\displaystyle{ {\tilde{A}}_2 }[/math]		[math]\displaystyle{ {\tilde{A}}_4 }[/math]		[math]\displaystyle{ {\tilde{A}}_6 }[/math]		[math]\displaystyle{ {\tilde{A}}_8 }[/math]		[math]\displaystyle{ {\tilde{A}}_{10} }[/math]		...
		[math]\displaystyle{ {\tilde{A}}_3 }[/math]		[math]\displaystyle{ {\tilde{A}}_5 }[/math]		[math]\displaystyle{ {\tilde{A}}_7 }[/math]		[math]\displaystyle{ {\tilde{A}}_9 }[/math]		...
[math]\displaystyle{ {\tilde{C}}_1 }[/math]		[math]\displaystyle{ {\tilde{C}}_2 }[/math]		[math]\displaystyle{ {\tilde{C}}_3 }[/math]		[math]\displaystyle{ {\tilde{C}}_4 }[/math]		[math]\displaystyle{ {\tilde{C}}_5 }[/math]		...

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, 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 [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]

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	[math]\displaystyle{ {\tilde{A}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{C}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{B}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{D}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
E2	Uniform tiling	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21