• Corpus ID: 227227595

Space vectors forming rational angles.

@article{Kedlaya2020SpaceVF,
  title={Space vectors forming rational angles.},
  author={Kiran S. Kedlaya and Alexander Kolpakov and Bjorn Poonen and Michael O. 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… 

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