# 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 ${\tilde {A}}_{n}$ 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 | ${\tilde {A}}_{n}$ | 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:
| ${\tilde {A}}_{3}$ | ${\tilde {A}}_{5}$ | ${\tilde {A}}_{7}$ | ${\tilde {A}}_{9}$ | ${\tilde {A}}_{11}$ | ... |
| ${\tilde {A}}_{2}$ | ${\tilde {A}}_{4}$ | ${\tilde {A}}_{6}$ | ${\tilde {A}}_{8}$ | ${\tilde {A}}_{10}$ | ... |
| | ${\tilde {A}}_{3}$ | ${\tilde {A}}_{5}$ | ${\tilde {A}}_{7}$ | ${\tilde {A}}_{9}$ | ... |
| ${\tilde {C}}_{1}$ | ${\tilde {C}}_{2}$ | ${\tilde {C}}_{3}$ | ${\tilde {C}}_{4}$ | ${\tilde {C}}_{5}$ | ... |