# 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…

## 5 Citations

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. Since the 1970s, the complete classiﬁcation (up to isogeny) of abelian varieties over ﬁnite ﬁelds with trivial group of rational points has been known from results of Madan–Pal and Robinson; with…

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. Using factorization theorems for sparse polynomials, we compute the trace ﬁeld of Dehn ﬁllings of the Whitehead link, and (assuming Lehmer’s Conjecture) the minimal polynomial of the small…

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