Geometry
Conway, Parker & Sloane (1982) showed that the covering radius of the Leech lattice is ; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space. The points at distance at least from all lattice points are called the deep holes of the Leech lattice. There are 23 orbits of them under the automorphism group of the Leech lattice, and these orbits correspond to the 23 Niemeier lattices other than the Leech lattice: the set of vertices of deep hole is isometric to the affine Dynkin diagram of the corresponding Niemeier lattice.
The Leech lattice has a density of, correct to six decimal places. Cohn and Kumar showed that it gives the densest lattice packing of balls in 24-dimensional space. Their results suggest, but do not prove, that this configuration also gives the densest among all packings of balls in 24-dimensional space. No arrangement of 24-dimensional spheres can be denser than the Leech lattice by a factor of more than 1.65×10−30, and it is highly probable that the Leech lattice is indeed optimal.
The 196560 minimal vectors are of three different varieties, known as shapes:
- 1104 vectors of shape (42,022), for all permutations and sign choices;
- 97152 vectors of shape (28,016), where the '2's correspond to octads in the Golay code, and there an even number of minus signs;
- 98304 vectors of shape (3,123), where the signs come from the Golay code, and the '3' can appear in any position.
The ternary Golay code, binary Golay code and Leech lattice give very efficient 24-dimensional spherical codes of 729, 4096 and 196560 points, respectively. Spherical codes are higher-dimensional analogues of Tammes problem, which arose as an attempt to explain the distribution of pores on pollen grains. These are distributed as to maximise the minimal angle between them. In two dimensions, the problem is trivial, but in three dimensions and higher it is not. An example of a spherical code in three dimensions is the set of the 12 vertices of the regular icosahedron.
Read more about this topic: Leech Lattice
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