In 1895, Hill discovered a 1-parameter family of tetrahedra whose dihedral angles are all rational multiples of π. 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, but never with more than 12 monomials, and 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 finite list of sporadic tetrahedra. Building on this work, we classify all configurations of vectors in ℝ3 such that the angle between each pair is a rational multiple of π. Sample result: Ignoring trivial families and scalar multiples, any configuration with more than nine vectors is contained in a particular 15-vector configuration.
This is joint work with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein.
This video is part of the Number Theory Web Seminar series.
