# Mathematics Colloquia and Seminars

We classify all sets of nonzero vectors in $R^3$ 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 J. H. Conway and A. 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 only 12 monomials). This is a joint work with Kiran S. Kedlaya, Bjorn Poonen, and Michael Rubinstein.