5 simplex 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 A ~ 5 Coxeter group. It is the 5-dimensional case of a simplectic honeycomb.

The A2
5 lattice is the union of two A₅ lattices:

∪

The A3
5 is the union of three A₅ lattices:

∪ ∪ .

The A*
5 lattice (also called A6
5) 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

Related polytopes and honeycombs

This honeycomb is one of 12 unique uniform honeycombs constructed by the A ~ 5 Coxeter group. The extended symmetry of the hexagonal diagram of the A ~ 5 Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:

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: