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The topology and geometry of infinite periodic surfaces

The use of space group stereohedra derived by Bashkirov as free boundaries result in a multiplicity of intersection-free (as well as self-intersecting) translationally periodic minimal surfaces of that space group. Surface solutions have differing tunnel networks and curvature, distinguished by their topology. The exact geometry of these surfaces can be calculated by constructing a function which depends only on the surface normal vectors at flat points on the surface, and solving the Weierstrass equations. Translationally periodic surfaces of constant average curvature can be determined from the periodic minimal surface solutions. Some examples of the various techniques for calculating the geometry and topology of these periodic surfaces are given