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In the 1960’s, John Leech found the tightest regular packing of spheres in 24 dimensions. [Priority may belong to Ernst Witt.] Three previously unknown «sporadic» groups—exceptional atoms of symmetry—were then found by John Conway, analysing Leech’s special arrangement.
Mark Ronan
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Maryna Viazovska has solved the densest packing problem in dimension eight. Subsequently, Maryna Viazovska with Henry Cohn, Steve Miller, Abhinav Kumar, and Danilo Radchenko solved the densest packing problem in 24 dimensions.
…Taking the quotient of M by w itself, we get the Leech lattice. In this lattice each sphere touches 196560 others, which is the most one can attain, and again this is the only way to get 196560 spheres to touch a central one in 24 dimensions. This should be obvious by visualizing it. :-)
John Baez, Week 20 of This Week’s Finds in Mathematical Physics
