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Hyperbolic polyhedra: volume and scissors congruence


Yana Zilberberg Mohanty

In this dissertation we solve several problems relating to volumes of hyperbolic polyhedra and scissors congruence. Regarding volumes, we derive a new formula for the volume of a general hyperbolic tetrahedron in three dimensions. This formula is a sum of the Lobachevsky function applied to functions of the dihedral angles of the tetrahedron, and it was developed by using the hyperboloid model of hyperbolic space and an exterior algebra on vectors in Minkowski space.

We then make use of Gregory Leibon's formula for the volume of a hyperbolic tetrahedron to solve a problem posed by Justin Roberts of whether the Regge symmetry is a scissors congruence in hyperbolic space by producing a constructive proof. This proof involves permuting certain components of Leibon's construction.

Finally, we show several geometric proofs of the Kubert identities in the cases n = 2,3,4. Some of these are extensions of the geometric proof of the cyclotomic identities. Others involve exposing the geometry underlying Dupont and Sah's proof of the generalized Kubert identities in [5].

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