We start this short note by introducing two remarkable mathematical
objects: the root lattice in 8-dimensional Euclidean
space and the Leech lattice in 24-dimensional space. These two
lattices stand out among their lattice sisters for several reasons.
The first reason is that these both lattices are related to other unique and exceptional mathematical objects. The lattice is the root lattice of the semisimple exceptional Lie algebra . The quotient of by a suitable sublattice is isomorphic to the Hamming binary code of dimension 8 and minimum distance 4, which in its turn is an optimal error-correcting binary code with these parameters. The Leech lattice is famously connected to the exceptional finite simple groups, monstrous moonshine  and the monster vertex algebra .
Another reason is that and are solutions to a number of optimization problems. The and Leech lattice provide optimal sphere packings in their respective dimensions [5, 23]. Also both lattices are universally optimal, which means that among all point configurations of the same density, the and have the smallest possible Gaussian energy .
The third reason for our interest in these lattices is less obvious. The optimality of the and Leech lattices can be proven in a rather short way, while the solutions of analogous problems in other dimensions, even dimensions much smaller than 8 and 24, is still wide open. Finally, this last property seems to be inherited by other geometric objects obtained from and , such as Hamming code, Golay code and the sets of shortest vectors of both lattices.
Cite this article
Maryna Viazovska, Almost impossible and Leech lattices. Eur. Math. Soc. Mag. 121 (2021), pp. 4–8DOI 10.4171/MAG/47