Geometry, Kinematics, and Rigid Body Mechanics in Cayley-Klein Geometries

The 19 century witnessed a dramatic freeing of the human intellect from its naive assumptions. This development is particularly striking in mathematics. New geometries were discovered which challenged and expanded the familiar space of euclid, notably projective geometry and non-euclidean metric geometries. Developments in algebra led to similarly earth-shaking discoveries. This expansion of consciousness was accompanied by a desire to find a higher unity in the growing diversity. Hermann Grassmann made a bold, early attempt to bring the two domains together in his ”Science of extension” of 1844. Later in the century, Cayley and Klein contributed to this process by their discovery that metric geometry could be constructed atop projective geometry. William Clifford introduced a geometric product combining the metric inner product with Grassmann’s exterior product; the resulting structure has become known as a Clifford or geometric algebra. A completely satisfactory formulation of geometric algebra had to await the rigorous clarity of 20 century mathematics. The work presented here (the author’s Ph. D. thesis from the Technical University Berlin, 2011) represents an attempt to bring this stream of 19 century mathematics into the aforementioned rigorous, modern form, and to apply the resulting thought forms to understand kinematics and rigid body mechanics in the spaces under consideration (including euclidean, hyperbolic and elliptic space). One guiding light in the treatment is the reliance on a projective geometric foundation throughout, particularly a consequent application of the principle of duality. This projective approach guarantees that the result is metric-neutral – the results are stated in such a way that they apply equally well to the two non-euclidean geometries as well as euclidean geometry. And a consequent consideration of duality leads to the recognition that this classical triad must be extended by a fourth geometry, dual euclidean geometry. Readers of the MPK will not be surprised to learn that dual euclidean geometry has a close affinity with counter-space [in German, Gegenraum], a concept which Rudolf Steiner introduced and developed, particularly in his natural science lectures. That such a connection exists in the work presented here is not surprising, as the initial impulse for this thesis was provided by the later works of George Adams (for example, see [Ada59]), to whom the present publication is dedicated. One of Adams’ great gifts was to show how the results of modern science, particularly mathematical, when grasped with enlivened thinking, reveal deep, often unexpected, connections to the world and the human being – in a manner reminiscent of Goethe’s reading the “open secrets” of nature. Such insights can indicate fruitful directions for further research. In this spirit is the current publication offered, with special thanks to Peter Gschwind for making it possible.