Simplicial honeycomb

In geometry, the simplicial honeycomb (or $n$-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the $${\tilde{A}}_n$$ affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of $n + 1$ nodes with one node ringed. It is composed of $n$-simplex facets, along with all rectified $n$-simplices. It can be thought of as an $n$-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes $$x+y+\cdots\in\mathbb{Z}$$, then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an $n$-simplex honeycomb is an expanded $n$-simplex.

In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph, with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph, filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph, filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.

Projection by folding
The (2n-1)-simplex honeycombs and 2n-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:

Kissing number
These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the vertex figure. This represents the highest kissing number for 2 and 3 dimensions, but falls short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.