  We classify all sets of nonzero vectors in $\mathbb{R}^3$ such that the angle
formed by each pair is a rational multiple of $\pi$. The special case of
four-element subsets lets us classify all tetrahedra whose dihedral angles are
multiples of $\pi$, 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).
