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Leech lattice

In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech (Canad. J. Math. 16 (1964), 657--682). It is the unique lattice with the following list of properties:

The last condition means that unit balls centered at the points of Λ do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls which can simultaneously touch a single unit ball (compare with 6 in dimension 2, as the maximum number of pennies which can touch a central penny). It seems to be expected that this configuration also gives the densest packing of balls in 24-dimensional space, but this is still open.

The Leech lattice can be explicitly constructed as the set of vectors of the form 2−3/2(a1, a2, ..., a24) where the ai are integers such that

and the set of coordinates i such that ai belongs to any fixed residue class (mod 4) is a word in the binary Golay code.

The Leech lattice is highly symmetrical. Its automorphism group is the double cover of the Conway group Co1; its order is approximately 8.3(10)18.