M Theory Lesson 46

Henry Cohn studies sphere packing in different dimensions. In 2004, along with A. Kumar, he proved that the Leech lattice is the unique densest lattice in ${ℝ}^{24}$.

If $V$ is the volume of a fundamental polytope for a lattice, and $r$ is the minimal length of a basis vector for the lattice, then with spheres of diameter $r$ the packing density in dimension $n$ is

$\rho =\frac{{\pi }^k}{2^n\left(k\right)!}{V}^{-1}{r}^n$

where $k$ is $\frac{n}{2}$ and for odd $n$, $k!=\Gamma (k+1)$. Cohn and Kumar solve the Leech problem by finding an $r=2$ basis under the normalisation $V=1$, which saturates a known upper bound on $\rho$. It turns out that the Leech lattice has 196560 vectors of minimal length equal to 2. The next smallest length for vectors is about $\sqrt{6}$.

One scales the minimal vectors to fit on a unit sphere $S^{23}$. The minimal angle satisfies $\text{cos}\phi =0.5$. Looking at points on spheres is something one does in coding theory. The connection with coding theory is a good way to look at energy minimisation problems. Think of the selected points as satisfying some potential. Cohn and Kumar have a concept of universally optimal distribution for points on spheres.