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Space vectors forming rational angles
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abstract
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 unity of a $W(D_6)$-symmetric polynomial equation with $105$ monomials (the previous record was $12$ monomials).
Forward citations
Cited by 2 Pith papers
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On structured cosine sums and applications
Structured cosine sums on cyclic groups are classified through vanishing sums of roots of unity, yielding a small-weight Fourier rigidity theorem and eigenvalue multiplicity bounds for cyclic Cayley graphs.
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Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles
Rational-angle a^4b pentagonal sphere tilings are exactly three families: a 12-tile tetrahedral subdivision, a 4m-tile symmetric family with flips, and a 20-tile non-symmetric case.
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