Leech Lattices

Elicia WilliamsApril 24, 2008

• Background

• Brief Definition

• Lattices– 2-Dimensions– 3-Dimensions– 24-Dimensions– Higher Dimensions

• Applications

Leech: the man

John LeechBorn: July 21, 1926

Died: September 28, 1992

Educated at Trent College. Received B.A. from King’s College Cambridge in

1950.

August 1969 Computers in Number Theory Conference

Oxford, England

In 1964 Leech published a paper on sphere packing in eight or more

dimensions. It contained a lattice packing in 24 dimensions.

In 1965 he submitted a supplement to the paper giving a packing in 24

dimensions.

He did not have the group theory skills necessary to prove his

conjectures of the symmetry of the group, so he sought the help of John

Conway.

Leech: the lattice

A lattice with no elements of length equal to 2, thus making it the tightest

lattice packing of spheres in 24 dimensions.

Okay, but what is a lattice?

Lattices

A discrete subgroup of Rn which spans the real vector space Rn. Every

lattice Rn can be generated from a basis for the vector space by forming all linear combinations with integral

coefficients. The elements of a lattice are regularly spaced.

2-Dimensions

A typical 2-dimensional lattice is given by the vertices (or centers) of a tiling by square tiles.

Another 2-dimensional lattice is given by the centers of tiling by hexagons.

This lattice is the most symmetrical, 2-dimensional lattice.

3-Dimensions

In 3-dimensions there are two common regular lattices that achieve the highest average density. They are the face-centered cubic

(FCC), and the hexagonal close-packed (HCP).

The FCC

The HCP

24-Dimensions

The Leech lattice is the unique lattice in R24 with the following list of properties:

It is unimodular

It is even

The shortest length of any non-zero vector in is 2

The points of the Leech lattice are the centers of spheres,each touching 196,560 others.

Each lattice point is specified using 24 coordinates.

Take one sphere centered at the origin, so the coordinatesof its center are all zero.

The centers of the 196,560 neighboring spheres splitnaturally into three subsets of sizes:

97,152 + 1,104 + 97,308 = 196,560

The subset of size 97,152 = 27 x 759

There are 759 octads and for each one there are 27 lattice points. The coordinates of each

point are plus or minus 2 in the positions of an octad, and zero elsewhere; the number of minus signs is even.

The subset of size 1,104 = 22 x 276

There are 276 ways of choosing two coordinates fromEach 24: each of these two coordinates is plus orMinus 4 and the other 22 coordinates are zero.

The subset of size 98,304 = 212 x 24

One coordinate is plus or minus 3. The others are plus orminus 1.

A representation of a Leech lattice:

The 50 nodes in the drawing depicts 50 point withinthe Leech lattice.

Higher Dimensions

The densest known packings are nonlattice.

Applications

The Leech lattice is unusually symmetrical and efficient in its packing and covering

of the 24-dimensional space.

John Conway analyzed the symmetry of the lattices that provide the densest

packings and the thinnest coverings of their spaces.

The study of the symmetry of the Leech lattice led Conway to discover three new sporadic groups: Co1, Co2, Co3.

Co1 is the largest of the Conway groups, of order

4,157,776,806,543,360,000

It is obtained as the quotient of Co0 (or 0) by its center.

Co0 is the group of automorphisms of the Leech lattice in 24-dimensions space R24.

It contains as subquotients 12 exceptional simple groups.

Co2, of order 32, 305,421,312,000, and Co3 , of order 495,766,656,000, consists of the automorphisms of fixing a

lattice vector of type 2 and a vector of type 3 respectively.

The Leech lattice also led to the construction of the largest of the

sporadic simple groups.

The Friendly Giant or Fischer-Griess “Monster” simple group

of order:

2463205976112133171923293141475971 =

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

The 26-dimensional spacetime in string theory.

Start with an even unimodular lattice in 26-dimensional Minkowski space.

Then look at M, the set of vectors in the lattice perpendicular to the vector w = (70,1,2,3,…,24). This is the null vector.

Taking the quotient of M by w by itself yields the Leech lattice.

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of California Riverside. April 2008. <http://math.ucr.edu/home/baez/week20.html>.

Borcherds, Richard E. “The Leech Lattice.” 1985. University of Cambridge. April 2008. <http://math.berkeley.edu/~reb/papers/leech/leech.pdf>.

Conway, J.H. and Sloane, N.J.A. Sphere Packings, Lattices, and Groups. New York, NY: Springer-Verlag, 1999.

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