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Concept
5-simplex honeycomb
Summary
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.
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 Coxeter group. It is the 5-dimensional case of a simplectic honeycomb.
The A_2 lattice is the union of two A5 lattices:
∪
The A_3 is the union of three A5 lattices:
∪ ∪ .
The A_* lattice (also called A_6) 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.
Official source
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