Wednesday, May 12, 2021

[nikwyqhd] lattices with rows of spheres

in the face-centered cubic close packing of equal spheres, any pair of touching (kissing) spheres is part of an infinite straight row of kissing spheres.  there is a line through the row that goes through centers and tangent points.  through any sphere center there are 6 such lines or 12 rays.  each sphere touches 12 others.

the hexagonal close packing of spheres, while equally dense, does not have all these lines, so is (arguably) less pretty, probably objectively less symmetric.

what are the best (known) sphere packings in higher dimensions subject to this constraint that all kissing spheres must always be part of solid infinite rows?  do the famous E8 and Leech lattices meet this constraint?

Conway and Sloane, "What are all the best sphere packings in low dimensions?"

the densest lattice packings for dimensions up to 8 are known.

4D is the D4 lattice, the interleaving of 2 cubic (tesseract) lattices.  each sphere touches 24 others: 8 from its own cubic lattice along axis directions and 16 from the other lattice.  16 because each sphere is body-centered in a tesseract with 2^4 = 16 vertices from the other lattice.  this lattice seems to meet the solid-rows constraint.

No comments :