# 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 ${\displaystyle {\tilde {A}}_{n}}$ 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 . 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 ${\displaystyle {\tilde {A}}_{n}}$ Name
Coxeter diagram
Vertex figure Image and facets
1 ${\displaystyle {\tilde {A}}_{1}}$ Apeirogon

Yellow and cyan line segments
2 ${\displaystyle {\tilde {A}}_{2}}$ Trihexagonal tiling

Rectangle

With yellow and blue equilateral triangles,
and red hexagons
3 ${\displaystyle {\tilde {A}}_{3}}$ quarter cubic honeycomb

Elongated
triangular antiprism

With yellow and blue tetrahedra,
and red and purple truncated tetrahedra
4 ${\displaystyle {\tilde {A}}_{4}}$ Cyclotruncated 5-cell honeycomb

Elongated
tetrahedral antiprism
5-cell, truncated 5-cell,
bitruncated 5-cell
5 ${\displaystyle {\tilde {A}}_{5}}$ Cyclotruncated 5-simplex honeycomb
5-simplex, truncated 5-simplex,
bitruncated 5-simplex
6 ${\displaystyle {\tilde {A}}_{6}}$ Cyclotruncated 6-simplex honeycomb
6-simplex, truncated 6-simplex,
bitruncated 6-simplex, tritruncated 6-simplex
7 ${\displaystyle {\tilde {A}}_{7}}$ Cyclotruncated 7-simplex honeycomb
7-simplex, truncated 7-simplex,
bitruncated 7-simplex
8 ${\displaystyle {\tilde {A}}_{8}}$ Cyclotruncated 8-simplex honeycomb
8-simplex, truncated 8-simplex,
bitruncated 8-simplex, tritruncated 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:

${\displaystyle {\tilde {A}}_{3}}$ ... ... ... ...

## 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 ${\displaystyle {\tilde {A}}_{n-1}}$ ${\displaystyle {\tilde {C}}_{n-1}}$ ${\displaystyle {\tilde {B}}_{n-1}}$ ${\displaystyle {\tilde {D}}_{n-1}}$ ${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21