Leech grid

In mathematics , the Leech grid , named after John Leech , is a 24-dimensional grid that is used, among other things, for the construction of particularly efficient spherical packings in 24- dimensional space .

construction

The nodes of the leech lattice are the vectors of the shape

{\ displaystyle {\ frac {1} {2 {\ sqrt {2}}}} (a_ {1}, \ ldots, a_ {24}) \ in \ mathbb {R} ^ {24}}

with whole numbers for which ${\ displaystyle a_ {1}, \ ldots, a_ {24}}$

{\ displaystyle a_ {1} + a_ {2} + \ cdots + a_ {24} \ equiv 4a_ {1} \ equiv 4a_ {2} \ equiv \ cdots \ equiv 4a_ {24} {\ pmod {8}}}

should apply.

properties

Apart from isomorphism, the Leech lattice is the only lattice in the with the following properties: ${\ displaystyle \ mathbb {R} ^ {24}}$

It is unimodular , i.e. That is, the lattice is whole, i.e. it has an integral Gram matrix , and the determinant of this Gram matrix is the same . ${\ displaystyle 1}$
It is straight , i.e. that is, the square of the norm of each node is an even integer.
The norm of any non-zero node is at least . ${\ displaystyle 2}$

Ball packing

The spheres with a radius around the nodes of the leech lattice form a spherical packing in which each sphere touches exactly 196,560 other spheres. ${\ displaystyle {\ sqrt {2}}}$

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Maryna Viazovska proved in 2016 that the leech lattice is the optimal 24-dimensional spherical packing.

Symmetry group

The symmetry group of the Leech lattice is the Conway group , it has 8 315 553 613 086 720 000 elements. ${\ displaystyle C_ {0}}$

The Leech grating has no mirror symmetries.

literature

John Leech : Notes on sphere packings. Canad. J. Math. 19, 1967, 251-267.
John Conway , Neil Sloane : Sphere packings, lattices and groups. Third edition. With additional contributions by E. Bannai, RE Borcherds, J. Leech, SP Norton, AM Odlyzko, RA Parker, L. Queen and BB Venkov. Basic Teachings of Mathematical Sciences, 290. Springer-Verlag, New York, 1999. ISBN 0-387-98585-9

Popular science :

George Szpiro : The Kepler Conjecture. How mathematicians solved a 400-year-old puzzle. Translated from English by Manfred Stern. Berlin: Springer (2011). ISBN 978-3-642-12740-3 / hbk; 978-3-642-12741-0 / ebook
Marcus du Sautoy : Finding moonshine. A mathematician's journey through symmetry. London: Fourth Estate (2008). ISBN 978-0-00-721461-7 / hbk
Thomas Thompson: From Error-Correcting Codes through Sphere Packings to Simple Groups , Carus Mathematical Monographs, Cambridge University Press 2004

Individual evidence

^ Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska: The sphere packing problem in dimension 24 . online (pdf)

[1] Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska: The sphere packing problem in dimension 24 . online (pdf)