Space vectors forming rational angles
In 1895, Hill discovered a $1$-parameter family of tetrahedra whose dihedral angles are all rational multiples of $\pi$. In 1976, Conway and Jones related the problem of finding all such tetrahedra to solving a polynomial equation in roots of unity. Many previous authors have solved polynomial equations in roots of unity with $\leq 12$ monomials, but the Conway-Jones polynomial has $105$ monomials! I will explain the method we use to solve it and our discovery that the full classification consists of two $1$-parameter families and an explicit list of $59$ sporadic tetrahedra.
Building on this work, we classify all configurations of vectors in $\mathbb R^3$ such that the angle between each pair is a rational multiple of $\pi$. Sample result: Ignoring trivial families and scalar multiples, any configuration with more than $9$ vectors is contained in a particular $15$-vector configuration.
This is joint work with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein.