# 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 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 ${\tilde {A}}_{n}$ 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,
${\tilde {A}}_{3}$ ... ... ... ...