Abstract: Inspired by the rich theory of triply periodic minimal surfaces, we will study a family of triply periodic polyhedral surfaces that have several conformal representations. In particular, these surfaces are embedded surfaces in $\mathbb{R}^3,$ and they are invariant under a rank-three lattice, hence ``triply periodic". When quotiented by the lattice, we get a compact Riemann surface. We will study specific examples of surfaces that answer questions in minimal surface theory, Teichmüller dynamics, and physics.
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