Erlangen program
The Erlangen Program describes the scientific program (1872) presented by Felix Klein when he entered the University of Erlangen . In this, he developed the concept of a systematic classification of geometric sub-disciplines, which is based on the idea that geometry examines the properties of figures that are retained in the event of changes in position and therefore a classification using the possible changes in position considered in each case, i.e. the permitted geometric transformations, aspires to.
Details of the geometrical research program
Felix Klein sketched a geometry beyond Euclidean geometry , namely the hyperbolic geometry according to Lobachevsky , which later became important for the theory of relativity in physics , as well as elliptic geometry . These two non-Euclidean geometries soon became important in differential geometry .
With each of the resulting geometries , the associated transformations form a group with regard to their execution one after the other, the transformation group of the geometry under consideration. The properties examined in the relevant geometry remain invariant with regard to all transformations of the transformation group.
The elementary Euclidean geometry or congruence geometry is the geometry of the visual space, the transformation group of which is the group of movements (i.e. translations , rotations or reflections ) which are all length and angle true images.
- If one waives the length fidelity in the permitted transformations and also allows point stretching, one obtains the equiform group of transformations that characterize the similarity or equiform geometry.
- If one also waives the correct angle, one arrives at the transformation group of the linear transformations in the coordinate representation, i. H. of the collineations that receive the division ratio of three points. They characterize the affine geometry .
- Finally, if one adds infinitely distant or improper points as intersections of parallels to the visual space, then the collineations in this space leave the double ratio of four points invariant and form the group of projective transformations, the associated geometry of which is the projective geometry.
In addition to the classical geometries mentioned here, which all result from the restriction of the transformation group from projective geometry, one can also get from projective geometry to elliptical and hyperbolic geometry in this way ; these non-Euclidean geometries can also be classified according to the Erlangen program. However, the Erlangen program is not sufficient for a complete classification of all geometries: for example, the Riemannian geometry on which the general theory of relativity is based cannot be covered by this classification ( Lie groups ).
literature
- Renate Tobies : Felix Klein. Teubner, Leipzig 1981.
- Renate Tobies: Felix Klein in Erlangen and Munich. In: Amphora: Festschrift for Hans Wussing on his 65th birthday. Birkhäuser, 1992.
- David J. Rowe , John McCleary (Eds.): Klein, Lie, and the Geometric Background of the Erlangen Program. In: The History of Modern Mathematics. Ideas and their reception. Academic Press, Boston 1989, Vol. 1, pp. 209-273.
- Lizhen Ji, Athanase Papadopoulos (eds.): Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics. IRMA Lectures in Mathematics and Theoretical Physics 23, European Mathematical Society Publishing House, Zurich 2015.