1. Leech Lattices Elicia Williams April 24, 2008

2. Background Brief Definition Lattices –2-Dimensions –3-Dimensions –24-Dimensions –Higher Dimensions Applications

3. Leech: the man John Leech Born: July 21, 1926 Died: September 28, 1992 Educated at Trent College. Received B.A. from King’s College Cambridge in 1950.

4. August 1969 Computers in Number Theory Conference Oxford, England

5. 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.

6. 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?

7. Lattices A discrete subgroup of R n which spans the real vector space R n. Every lattice R n 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.

8. 2-Dimensions

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

10. Another 2-dimensional lattice is given by the centers of tiling by hexagons. This lattice is the most symmetrical, 2-dimensional lattice.

11. 3-Dimensions

12. 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).

13. The FCC

14. The HCP

15. 24-Dimensions

16. The Leech lattice  is the unique lattice in R 24 with the following list of properties: It is unimodular It is even The shortest length of any non-zero vector in  is 2

17. The points of the Leech lattice are the centers of spheres, each touching 196,560 others. Each lattice point is specified using 24 coordinates.

18. Take one sphere centered at the origin, so the coordinates of its center are all zero. The centers of the 196,560 neighboring spheres split naturally into three subsets of sizes: 97,152 + 1,104 + 97,308 = 196,560

19. The subset of size 97,152 = 2 7 x 759 There are 759 octads and for each one there are 2 7 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.

20. The subset of size 1,104 = 2 2 x 276 There are 276 ways of choosing two coordinates from Each 24: each of these two coordinates is plus or Minus 4 and the other 22 coordinates are zero.

21. The subset of size 98,304 = 2 12 x 24 One coordinate is plus or minus 3. The others are plus or minus 1.

22. A representation of a Leech lattice: The 50 nodes in the drawing depicts 50 point within the Leech lattice.

23. Higher Dimensions

24. The densest known packings are nonlattice.

25. Applications

26. 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.

27. The study of the symmetry of the Leech lattice led Conway to discover three new sporadic groups: Co 1, Co 2, Co 3.

28. Co 1 is the largest of the Conway groups, of order 4,157,776,806,543,360,000 It is obtained as the quotient of Co 0 (or  0) by its center. Co 0 is the group of automorphisms of the Leech lattice in 24- dimensions space R 24. It contains as subquotients 12 exceptional simple groups.

29. Co 2, of order 32, 305,421,312,000, and Co 3, 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.

30. The Leech lattice also led to the construction of the largest of the sporadic simple groups.

31. The Friendly Giant or Fischer-Griess “Monster” simple group of order: 2 46  3 20  5 9  7 6  11 2  13 3  17  19  23  29  31  41  47  59  71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,3 68,000,000,000

32. 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.

33. Bibliography Baez, John. “This Week’s Finds in Mathematical Phyisc.” 1993. University of California Riverside. April 2008.. Borcherds, Richard E. “The Leech Lattice.” 1985. University of Cambridge. April 2008.. Conway, J.H. and Sloane, N.J.A. Sphere Packings, Lattices, and Groups. New York, NY: Springer- Verlag, 1999. Motl, Lubos. “Monstrous Symmetry of Black Holes: Beauty and the Beast.: April 2008.. O’Connor, J.J. and Robertson, E.F. “John Leech.” University of St. Andrews School of Mathematics and Statistics. April 2008.. Wisstein, Eric W. “Hypsersphere Packing.” Mathworld--A Wolfram Web Resource. April 2008.. “The Leech Lattice.” Community College of Philadelphia Mathematics Department. April 2008.. “The Leech Lattice.” University of Illinois. April 2008..