Non-Euclidean Tetrahedra and Rational Elliptic Surfaces

I will explain how to construct a rational elliptic surface out of every non-Euclidean tetrahedra. This surface "remembers" the trigonometry of the tetrahedron: the edge lengths, the dihedral angles and the volume can be naturally computed in terms of the surface. The main property of this construction is self-duality: the surfaces obtained from the tetrahedron and its dual coincide. This leads to some unexpected relations between angles and edges of the tetrahedron. For instance, the cross-ratio of the exponents of the spherical angles coincides with the cross-ratio of the exponents of the perimeters of its faces. I will explain two proofs of these facts. The first proof uses classical birational geometry technics. The second one is based on relating mixed Hodge structures, associated to the tetrahedron and the corresponding surface.