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 ${\displaystyle {\tilde {A}}_{n}}$ 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 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, quadritruncated 8-simplex