Rational Elliptic Surfaces and the Trigonometry of Tetrahedra

We study the trigonometry of non-Euclidean tetrahedra using tools from algebraic geometry. We establish a bijection between non-Euclidean tetrahedra and certain rational elliptic surfaces. We interpret the edge lengths and the dihedral angles of a tetrahedron as values of period maps for the corresponding surface. As a corollary we show that the cross-ratio of the exponents of the solid angles of a tetrahedron is equal to the cross-ratio of the exponents of the perimeters of its faces. The Regge symmetries of a tetrahedron are related to the action of the Weyl group $W(D_6)$ on the Picard lattice of the corresponding surface.