Rivin's PhD thesis and a series of extensions characterized hyperbolic 3-dimensional
polyhedra in terms of their dihedral angles, resolving a long-standing open question of
Jakob Steiner on the inscribable combinatorial types. These, and some related results in convex geometry, have been used in 3-manifold topology, theoretical physics, computational geometry, and the recently developed field of
discrete differential geometry. Rivin has also made advances in counting geodesics on surfaces, the study of generic elements of discrete subgroups of Lie groups, and in the theory of dynamical systems. Rivin is also active in applied areas, having written large parts of the
Mathematica 2.0 kernel, and he developed a database of
hypothetical zeolites in collaboration with M. M. J. Treacy. Rivin is a frequent contributor to
MathOverflow. Igor Rivin is the co-creator, with economist Carlo Scevola, of Cryptocurrencies Index 30 (CCi30), an index of the top 30
cryptocurrencies weighted by
market capitalization. CCi30 is sometimes used by academic economists as a market index when comparing the cryptocurrency trading market as a whole with individual currencies. == Honors ==