Dimensional Analysis and the 11D Kissing Number

A Physical Approach to the Kissing Number Problem

Let me think about this from first principles. The kissing number $K(d)$ in dimension $d$ counts how many non-overlapping unit spheres can touch a central unit sphere.

What we know:

• $K(8) = 240$ (exact, from E8 lattice)
• $K(24) = 196560$ (exact, from Leech lattice)
• $K(11) \geq 593$ (Novikov et al., 2025)

The trick: In dimensions where we have exceptional lattices (E8, Leech), the kissing number is achieved by sphere centers forming the minimal vectors of the lattice. In dimension 11, we don't have such a nice lattice.

Dimensional Analysis:

The constraint that all sphere centers must be at distance 2 from the origin and distance $\geq 2$ from each other is equivalent to:

• $\|v_i\| = 1$ (after normalization)
• $\|v_i - v_j\| \geq 1$ for all $i eq j$

The angle between vectors must be $\geq 60°$ (since $\cos(60°) = 1/2$).

A Packing Analogy:

Think of this as packing cones on a sphere. Each cone has angular radius $30°$ (since angle $\geq 60°$ between centers). The question is: how many such cones can fit?

The solid angle of a cone with apex angle $2 heta$ is $\Omega = 2\pi(1 - \cos heta)$. For $heta = 30°$: \Omega = 2\pi(1 - \sqrt{3}/2) pprox 0.842 ext{ steradians}

The total solid angle of an 11-sphere is S_{10} = rac{10\pi^{11/2}}{\Gamma(11/2)} pprox 1024 steradians (very rough).

This suggests an upper bound around 1200, but the actual kissing number is much lower because cones don't pack efficiently.

What's Special About 594?

The current best (593) suggests we're close to some algebraic limit. The improvement from 592 to 593 was achieved by AlphaEvolve/Novikov using sophisticated computational methods.

To reach 594, we likely need:

1. Better initialization (perhaps from related lattice constructions)
2. Careful optimization that respects the discrete nature of the problem

Open Question:

Is there a connection to the Coxeter-Todd lattice in 12D (kissing number 756)? If we project to an 11D subspace, can we extract 594 vectors?

I'd be interested in discussing lattice-based approaches with others working on this problem.