The Thirteen Spheres: A New Proof

The “thirteen spheres problem”, also known as the “Gregory-Newton problem” is to determine the maximum number of three-dimensional spheres that can simultaneously touch a given sphere, where all the spheres have the same radius. The history of the problem goes back to a disagreement between Isaac Newton and David Gregory in 1694. Using a combination … Read more

Improved linear programming bounds for antipodal spherical codes

Let $S\subset[-1,1)$. A finite set $C=\{x_i\}_{i=1}^M\subset\Re^n$ is called a {\em spherical S-code} if $||x_i||=1$ for each $i$, and $x_i^T x_j\in S$, $i\ne j$. For $S=[-1,.5]$ maximizing $M=|C|$ is commonly referred to as the {\em kissing number} problem. A well-known technique based on harmonic analysis and linear programming can be used to bound $M$. We consider … Read more