In the mid-1960s, John Leech created a lattice that gives the tightest lattice packing of spheres in 24 dimensions. Then in the late-1960s, John Conway analysed the symmetry of this lattice and discovered three previously unknown sporadic groups (exceptional ‘symmetry atoms’), as described in my book Symmetry and the Monster.
The points of the Leech Lattice are the centres of spheres of equal diameter each of which touches 196,560 others — the maximum possible in 24 dimensions. Since there are 24 dimensions, each lattice point is specified using 24 coordinates, which can be determined using the Witt design, a pattern using a set of 24 symbols, in which certain subsets of 8 symbols, called ‘octads’, play an important role.
Take one sphere centred at the origin, so the coordinates of its centre are all zero. The centres of the 196,560 neighbouring spheres split naturally into three subsets of sizes
97,152 + 1,104 + 98,304 = 196,560.
The subset of size 97,152. This number is 27×759. There are 759 octads in Witt’s design, as explained on another page, 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. This number is 22×276. There are 276 ways of choosing two coordinates from twenty-four: each of these two coordinates is plus or minus 4, and the other twenty-two coordinates are zero.
The subset of size 98,304. This number is 212×24. One coordinate is plus or minus 3, the others are plus or minus 1. There are 212 sign choices, all coming from the Golay code.
The distance of a point from the origin, when squared, is the sum of the squares of its coordinates—this is Pythagoras’s theorem generalized to n dimensions. For each of the 196,560 points specified above, the sum of the squares of its coordinates is 32.
|In the first subset||22+22+22+22+22+22+22+22 = 32|
|In the second subset||42+42 = 32|
|In the third subset||32+12+12+ . . . +12 = 32|
This shows that all these 196,560 points are equidistant from the origin.