Space vectors forming rational angles

Abstract

We classify all sets of nonzero vectors in $\mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $π$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples of $π$, solving a 1976 problem of Conway and Jones: there are $2$ one-parameter families and $59$ sporadic tetrahedra, all but three of which are related to either the icosidodecahedron or the $B_3$ root lattice. The proof requires the solution in roots of unity of a $W(D_6)$-symmetric polynomial equation with $105$ monomials (the previous record was $12$ monomials).