Arcadian Functor

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Marni D. Sheppeard

Thursday, May 03, 2007

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.


Blogger L. Riofrio said...

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

May 04, 2007 5:58 AM  
Anonymous Doug said...

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

May 04, 2007 10:10 AM  
Blogger Kea said...

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.

May 04, 2007 10:22 AM  

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