Chapter 1: Pictures of the Projective Plane

Günter Törner started his mathematical career with a doctoral dissertation on Hjelmslev planes. These planes were named after the Danish mathematician Johannes Hjelmslev (1873–1950), who defined them as a sort of “natural geometries” meaning that distinct lines may meet in more than one point, like it may happen in a real drawing. There is a natural homomorphism H → P from a Hjelmslev-plane H onto an ordinary projective plane. The geometric axioms for H are in the “desarguesian case” such that H may be co-ordinatized by a ring that has a unique chain of ideals. Toener went on to study these rings in detail, which today is the main topic of his mathematical research beside his activities in mathematical education and his service to the Deutsche Mathematiker Vereinigung. On top of all this he is interested in the relations of mathematics and art and has even organized meetings on this topic. That is why I have decided to offer him some remarks on the pictorial history of the ordinary (real) projective plane P. I am sorry that I do not know of any nice pictures of Hjelmslev planes or their uniserial rings of coordinates.