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:
It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1.
It is even; i.e., the square of the length of any vector in Λ is an even integer.
The shortest length of any non-zero vector in Λ is 2.
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; see kissing number). 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.
Conway, J. H.; Sloane, N. J. A. (1999). Sphere packings, lattices and groups. (3rd ed.) With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften, 290. New York: Springer-Verlag. ISBN 0-387-98585-9.
A {\it lattice packing} in ${\Bbb R}^{n}$ is then a sphere packing where the centers of spheres are placed at the points of a lattice $L\subset {\Bbb R}^{n}$, the radius of each sphere being half the length of the shortest non-zero vectors in $L$.
The Leechlattice appears in several places in `Moonshine' which is a term first coined by J. Conway and S. Norton in 1979 to describe the mysterious connections between finite sporadic simple groups and modular functions.
An investigation of the techniques of Cohn and Elkies, and of Cohn and Kumar, which are conjectured to give rise to a proof that the $E_8$ root lattice and the Leechlattice give the densest sphere packings in ${\Bbb R}^{8}$ and ${\Bbb R}^{24}$ respectively.
In mathematics, the Leechlattice is a lattice Λ in R
There are 23 orbits of them, and they correspond to the 23 Niemeier lattices other than the Leechlattice.
Conway showed that the Leechlattice is isometric to the Dynkin diagram of the reflection group of the 26-dimensional even Lorentzian unimodularlattice II
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