D4 Lattice
Its vertex arrangement is called the D4 lattice or F4 lattice. The vertices of this lattice are the centers of the 3-spheres in the densest possible packing of equal spheres in 4-space; its kissing number is 24, which is also the highest possible in 4-space.
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The D4+ lattice (also called D42) can be constructed by the union of two 4-demicubic lattices, and is identical to the tesseractic honeycomb:
- + = =
This packing is only a lattice for even dimensions. The kissing number is 23=8, (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).
The D4* lattice (also called D44 and C42) can be constructed by the union of all four 5-demicubic honeycombs, but it is identical to the D4 lattice: It is also the 4-dimensional body centered cubic, the union of two 4-cube honeycombs in dual positions.
- + + + = = = + .
The kissing number of the D4* lattice (and D4 lattice) is 24 and its Voronoi tessellation is a 24-cell honeycomb, containing all rectified 16-cells (24-cell) Voronoi cells, or .
Read more about this topic: 16-cell Honeycomb