Community hub
Recent from talks
Knowledge base stats:
Talk channels stats:
Members stats:
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, E24. It is one of the best models for the kissing number problem. It was discovered by John Leech (1967). It may also have been discovered (but not published) by Ernst Witt in 1940.
The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties:
The last condition is equivalent to the condition that unit balls centered at the points of Λ24 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 that can simultaneously touch a single unit ball. This arrangement of 196,560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration and its mirror image are the only 24-dimensional arrangements where 196,560 unit balls simultaneously touch another. This property is also true in 1, 2, and 8 dimensions, with 2, 6, and 240 unit balls, respectively, based on the integer lattice, hexagonal tiling, and E8 lattice, respectively.
It has no root system and in fact is the first unimodular lattice with no roots (vectors of norm less than 4), and therefore has a centre density of 1. By multiplying this value by the volume of a unit ball in 24 dimensions, , its absolute density can be derived.
Conway (1983) showed that the Leech lattice is isometric to the set of simple roots (or the Dynkin diagram) of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II25,1. By comparison, the Dynkin diagrams of II9,1 and II17,1 are finite.
The binary Golay code, independently developed in 1949, is an application in coding theory. More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting up to four. It was used to communicate with the Voyager probes, as it is much more compact than the previously used Hadamard code.
Quantizers, or analog-to-digital converters, can use lattices to minimise the average root-mean-square error. Most quantizers are based on the one-dimensional integer lattice, but using multi-dimensional lattices reduces the RMS error. The Leech lattice is a good solution to this problem, as the Voronoi cells have a low second moment.
The vertex algebra of the two-dimensional conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R24/Λ24 and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group. This monster vertex algebra was also used to prove the monstrous moonshine conjectures.
Hub AI
Leech lattice AI simulator
(@Leech lattice_simulator)
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, E24. It is one of the best models for the kissing number problem. It was discovered by John Leech (1967). It may also have been discovered (but not published) by Ernst Witt in 1940.
The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties:
The last condition is equivalent to the condition that unit balls centered at the points of Λ24 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 that can simultaneously touch a single unit ball. This arrangement of 196,560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration and its mirror image are the only 24-dimensional arrangements where 196,560 unit balls simultaneously touch another. This property is also true in 1, 2, and 8 dimensions, with 2, 6, and 240 unit balls, respectively, based on the integer lattice, hexagonal tiling, and E8 lattice, respectively.
It has no root system and in fact is the first unimodular lattice with no roots (vectors of norm less than 4), and therefore has a centre density of 1. By multiplying this value by the volume of a unit ball in 24 dimensions, , its absolute density can be derived.
Conway (1983) showed that the Leech lattice is isometric to the set of simple roots (or the Dynkin diagram) of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II25,1. By comparison, the Dynkin diagrams of II9,1 and II17,1 are finite.
The binary Golay code, independently developed in 1949, is an application in coding theory. More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting up to four. It was used to communicate with the Voyager probes, as it is much more compact than the previously used Hadamard code.
Quantizers, or analog-to-digital converters, can use lattices to minimise the average root-mean-square error. Most quantizers are based on the one-dimensional integer lattice, but using multi-dimensional lattices reduces the RMS error. The Leech lattice is a good solution to this problem, as the Voronoi cells have a low second moment.
The vertex algebra of the two-dimensional conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R24/Λ24 and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group. This monster vertex algebra was also used to prove the monstrous moonshine conjectures.