D5 Lattice
The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 of this lattice.
The D5+ packing (also called D52) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).
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The D5* lattice (also called D54 and C52) can be constructed by the union of all four 5-demicubic lattices: It is also the 5-dimensional body centered cubic, the union of two 5-cube honeycombs in dual positions.
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The kissing number of the D5* lattice is 10 (2n for n≥5) and it Voronoi tessellation is a tritruncated 5-cubic honeycomb, containing all with bitruncated 5-orthoplex, Voronoi cells.
Read more about this topic: 5-demicubic Honeycomb