Bjorn Poonen: Tetrahedra with rational dihedral angles (NTWS 026)
Channel: Number Theory Web Seminar
Date: 2023-05-28

Abstract: 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, 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 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.

Link to slides: https://drive.google.com/file/d/1sD7ofVHaRPkN388EVYMv-nvPWt3lzjN6

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, August 6, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 7, 2020 (1am AEST, 3am NZST)