Mark Ronan's website
The Leech Lattice
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In the mid-1960s, John Leech created a lattice that gives
the tightest lattice packing of spheres in 24 dimensions. In the late-1960s, John Conway analysed the symmetry of
this lattice in detail and discovered three previously unknown sporadic groups (exceptional 'symmetry atoms'). The
points of the Leech Lattice are the centres of spheres, each touching 196,560
others, which is the maximum possible in 24 dimensions. Each lattice point is
specified using 24 coordinates, and these coordinates can be determined using
the Witt design. This is 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 + 97,308 = 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, and they come
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.
This shows that all these 196,560 points are an equal distance
from the origin. |
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