Translation covers of some triply periodic Platonic surfaces

We study translation covers of several triply periodic polyhedral surfaces that are intrinsically Platonic. describe their affine symmetry groups and compute the quadratic asymptotics for counting saddle connections cylinders, including count cylinders weighted by area. The mathematical was initiated Novikov, motivated electron transport. we consider particular interest as they admit different explicit geometric algebraic descriptions, described, example, in second author’s thesis.