Periodic Hyperbolic Surfaces and Subdivision of 3-Space

Abstract

The paper is concerned with the subject of generating continuous periodic hyperbolic surfaces which subdivide space between two identical (dual, complementary, reciprocal) periodic space lattices, into two congruent sub-spaces; all that with an exhaustive search m mind.

A search methodology is developed, which combines insights and manipulative power on the “atomistic”, Elementary Periodic Region (E.P.R) level and the certainty of the crystallographic symmetry groups domain.

The search discloses and proves the existence of the exclusive seven topologically different (already familiar) “primary” periodic hyperbolic surfaces and opens up a way to generate many more self- dual periodic lattices and the associated hyperbolic partition surfaces in between.