Models of the Real Projective Plane: Computergraphics of Steiner and Boy Surfaces

Preface by Egbert Brieskorn If you feel attracted by the beautiful figure on the cover of this book, your feeling is something you share with all geometers. "It is the delight in figures in a higher sense which distinguishes the geometer". This is a well known statement of the algebraic geometer Alfred Clebsch. Throughout the history of our science, great men like Plato and Kepler and Poincare were inspired by the beauty and harmony of such figures. Whether we see them as elements or representations of some universal harmony or simply look at them as a structure accessible to our mind by its very harmony, there is no doubt that the contemplation and the creation of beautiful geometric figures has been and will be an integral part of creative mathematical thought. Perception of beauty is not the work of the intellect alone - it needs sensitivity. In fact, it does need sensual experience. To seperate the ideas from the appearances, the structure from the surface, analytical thinking from holistic perception and science from art does not correspond to the reality of our mind and is simply wrong if it is meant as an absolute distinction. It would deprive our creative thought of one of its deepest sources of inspi- ration. I remember a visit of the blind mathematician B. Morin in G6ttingen. He gave us a lecture on the eversion of the sphere, the process which he had discovered for turning the sphere inside out by a continuous family of immersions. At that time, he had no analytic des- cription of this very complicated process, but he had a very precise qualitative picture of its geometry in his mind, and he had pictures of models made according to his instruction. I was deeply impressed by the beauty of these figures, and I was moved by the fact that he had been able to see all this beautiful and complex geometry which we could visualize only with great difficulty even after it was shown to us. In G6ttingen we had a very nice collection of mathematical models dating froin the days of Felix Klein and David Hilbert. You can find photographs of SOine of then1 in a very nice book of Gerd Fischer entitled HMathematical n10dels", which was published by Vieweg. Among these models there is one made in 1903 showing the surface of Boy. This is a surface obtained by an immersion of the real projective plane in 3-dimensional space. At the time when it was made it was known that the projective plane cannot be embed- ded as a smooth surface in 3-space. This is so because any smooth closed surface in 3-space divides the space into an interior part and an exterior. This implies that it is orientable: If you stand in the exterior with your feet on the surface, you know what it means to turn left. But the real projective plane is not orientable. This was a very remarkable dis- covery of Felix Klein in 1874, and it implies that the projective plane IP 2 cannot be embedded in the 3 -dimensional space IR.3 . So if we wan t to visualize JP2 by n1eans of a surface in IR 3 , we have to be con ten ted with less than a sinooth surface. F or instance we should admit that different parts of the surface penetrate each other.