### 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 $\mathbb{R}^{24}$.

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

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

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 $\textrm{cos}\phi = 0.5$. 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.

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

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

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 $\textrm{cos}\phi = 0.5$. 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.

## 3 Comments:

Once again this hints at some underlying order, like the hexagons seen repeatedly in nature.

I found, by accident, two articles that may relate to this topic:

1 - Nature v446, 26 April 2007, p992, Bernard Chazelle, 'The security of Knowing Nothing'

discusses the work of Boaz Barak & Amit Sahai. They use complicated cyptographic techniques to do 'zero-knowledge proofs' to resolve NP-complete problems enhancing online security [with an example]

2 - An arxive paper by Zur Izhakian, 'Duality of Tropical Curves' has figures 1-4 illustrating how to project a corner locus, subdivisions, conic duals and find a compatible Newton polytype.

Some 2D plots resemble the hexagon diagrams previously dicussed then relate them to 3D projections.

Tropical Algebra is an extension of Max-Plus Algebra that I have only recently learned of and do not yet understand

http://arxiv.org/PS_cache/math/pdf/0503/0503691v2.pdf

Cool, Doug! That's great. Yes, we all have a lot to learn. Personally I am willing to relinquish security to know just a little.

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