  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.
