4 - Toroidal Circle Planes

Summary

Flat Minkowski planes were first investigated by Schenkel [1980] in her dissertation. Later, Polster [1998b] studied the more general toroidal circle planes. For more information about general Minkowski planes and, in particular, finite Minkowski planes, we refer to the papers by Hartmann [1982a], Klein–Kroll [1989], Delandtsheer [1995], and the references given there.

A toroidal circle plane is a point–circle geometry whose point set is (homeomorphic to) the torus S¹ × S¹. The point set is equipped with two nontrivial parallelisms. The parallel classes of these parallelisms are the horizontals and verticals on the torus. The circles of the toroidal circle plane are graphs of homeomorphisms S¹ → S¹ that form a system of topological circles on the torus such that the Axiom of Joining B1 (see p. 7) is satisfied, that is, any three pairwise nonparallel points determine exactly one curve in the system. A toroidal circle plane is a flat Minkowski plane if it also satisfies the Axiom of Touching B2, that is, for each circle C and any two nonparallel points p, q with p ∈ C there is precisely one circle through p and q that touches C (geometrically) at p, that is, intersects C only at the point p or coincides with C.