5-simplex honeycomb

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.

A5 lattice
This vertex arrangement is called the A5 lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the $${\tilde{A}}_5$$ Coxeter group. It is the 5-dimensional case of a simplectic honeycomb.

The A$$ lattice is the union of two A5 lattices: ∪

The A$$ is the union of three A5 lattices: ∪ ∪.

The A$$ lattice (also called A$$) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex. ∪ ∪ ∪ ∪ ∪ = dual of

Projection by folding
The 5-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement: