Parametrization of triply periodic minimal surfaces. I. Mathematical basis of the construction algorithm for the regular class

An explicit parametrization algorithm is reported for the simplest class of triply periodic minimal surfaces (the `regular' class) for which the Weierstrass function specifying the complex plane representation has a simple product form. As the Gauss map links triply periodic minimal surfaces with spherical tessellations, the set of Schwarz triangular tilings of the sphere is used as the basis of an exhaustive listing of all such possible branch-point distributions, and hence surfaces, in this class. The symmetry and geometry of the resulting surfaces are determined by the locations and orders of these branch points.