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Jul 11 at 20:43 comment added Steven Landsburg See here also: mathoverflow.net/a/128881/10503
Jul 11 at 16:24 comment added André Henriques for N=4, your procedure produces a beautiful, dense, sphere packing of R^4, with each sphere toughing 24 other spheres. [The procedure I'm referring to is the following: place spheres centred at the 0-cells of the standard cubulation of R^4 by 4-cubes + spheres at centred at the centers of the 0-cells]. The resulting lattice (formed by the centers of the spheres) is called the $D_4$ lattice, and has quite a larger symmetry group than the group initially visible from the construction I provided.
Jul 11 at 13:38 history answered Gavin Wraith CC BY-SA 4.0