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Space vectors forming rational angles
Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, Michael Rubinstein
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We classify all sets of nonzero vectors in ℝ³ 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 B₃ root lattice. The proof requires the solution in roots of unity of a W(D₆)-symmetric polynomial equation with $105$ monomials (the previous record was $12$ monomials).
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