Space vectors forming rational angles.
@article{Kedlaya2020SpaceVF, title={Space vectors forming rational angles.}, author={Kiran S. Kedlaya and A. Kolpakov and B. Poonen and M. Rubinstein}, journal={arXiv: Metric Geometry}, year={2020} }
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… CONTINUE READING
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