  In classical differential geometry, a central question has been whether
abstract surfaces with given geometric features can be realized as surfaces in
Euclidean space. Inspired by the rich theory of embedded triply periodic
minimal surfaces, we seek examples of triply periodic polyhedral surfaces that
have an identifiable conformal structure. In particular, we are interested in
explicit cone metrics on compact Riemann surfaces that have a realization as
the quotient of a triply periodic polyhedral surface. This is important as
Riemann surfaces where one has equivalent descriptions are rare. We construct
periodic surfaces using graph theory as an attempt to make Schoen's heuristic
concept of a dual graph rigorous. We then apply the theory of cyclically
branched coverings to identify the conformal type of such surfaces.

Differential Geometry