Flat Möbius planes were first investigated by Wölk [1966] and Strambach [1967c]. Later, Strambach [1970d], [1972], [1973], [1974a], [1974b] studied the more general spherical circle planes. For more information about Möbius planes and, in particular, finite Möbius planes, we refer to Dembowski [1968], Delandtsheer [1995], Hering [1965], Mäurer [1967], Wilker [1981] and the references given there.

A spherical circle plane is a point–circle geometry whose point set is (homeomorphic to) S2 and whose circles are topological circles on S2. Furthermore, the Axiom of Joining B1 (see p. 7) is satisfied, that is, any three distinct points are contained in exactly one of the circles. A spherical circle plane is a flat Möbius plane if, in addition, the Axiom of Touching B2 is satisfied, that is, for each circle C and any two distinct points p, q with p ∈ C there is precisely one circle through p and q that touches C at p geometrically, that is, intersects C only at the point p or coincides with C.

Models of the Classical Flat Möbius Plane

In this first section we describe a number of models of the classical flat Möbius plane. For detailed information about most of these models see Benz [1973].