We classify all possible configurations of vectors in three-dimensional space with the property that any two of the vectors form an angle whose measure is a rational multiple of $\pi$. As a corollary, we find all tetrahedra whose six dihedral angles are all rational multiples of $\pi$. While these questions (and their answers) are of an elementary nature, their resolution will take us on a tour through cyclotomic number fields, computational algebraic geometry, and an amazing fact about the geometry of tetrahedra discovered by physicists in the 1960s. This is based on joint work with Sasha Kolpakov, Bjorn Poonen, and Michael Rubinstein.