The Tammes Problem

For each integer \(N \geq 2\), place \(N\) points on the unit sphere so as to maximise the minimal angular separation between distinct points. Determine this optimum — the Tammes problem, equivalently the optimal spherical code — for every \(N\). Exact answers are known only for scattered small \(N\).

Posed by the Dutch botanist Pieter Tammes in 1930 from pollen-grain microscopy: how evenly can pores distribute on a sphere? The problem became the theory of spherical codes (Delsarte, Goethals and Seidel, 1970s), with exact optima known only for scattered values of \(N\). It sits beside the kissing-number problem — \(N = 12\) is the Newton–Gregory twelve — but asks for every \(N\), where most answers remain unknown. Fejes Tóth's work and later computational bounds form the partial record.