In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.
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A6 lattice
This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the
The A*
6 lattice (also called A7
6) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.
∪ ∪ ∪ ∪ ∪ ∪ = dual of
Related polytopes and honeycombs
This honeycomb is one of 17 unique uniform honeycombs constructed by the
Projection by folding
The 6-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: