Until the middle of the nineteenth century the word “geometry” in the singular seemed to have lost its meaning: alongside Euclid’s geometry, projective geometry, affine geometry and the non-Euclidean geometries had all been born. Each had its own theorems and its own rules, and it was not clear what held them together. The answer came in 1872, when the twenty-three-year-old Felix Klein, on taking up his chair at the University of Erlangen, presented a short programmatic text that went down in history as the Erlangen program (in German Erlanger Programm).
The central idea: geometry = invariants of a group
Klein’s insight is that a geometry is defined not by the objects it studies, but by the transformations it is willing to treat as “harmless”. Once we fix a set of plane transformations that is closed under composition and inverse — that is, a group — the corresponding geometry studies only those properties of figures that stay unchanged under all the transformations of the group (Boyer). Change the group and you change the geometry.
The isometries you study in this chapter are the simplest example. Translations, rotations, symmetries and their compositions form a group: composing two isometries gives another one, every isometry has an inverse (itself an isometry), and the identity transformation plays the role of neutral element. The invariants of this group are the distance between two points, the measure of angles and the area: this is “metric” geometry, the one in which two figures are considered equal when they are congruent.
A hierarchy of geometries
Enlarging the group, less is preserved and the geometry becomes “coarser”:
- Isometries — distances, angles and areas are preserved (congruent figures);
- Similarities — uniform dilations are added: distances are no longer preserved, but the ratios of distances and the angles are (similar figures);
- Affinities — parallelism and ratios of segments on the same line are preserved, but not angles: a circle and an ellipse become “the same figure”;
- Projectivities — even less is preserved (the characteristic invariant is the cross-ratio of four collinear points); this is the geometry of shadows and perspective.
Each group is contained in the next, and the larger the group the fewer the invariant properties: in this way one passes, in an orderly fashion, from one geometry to another.
The legacy
The Erlangen program grew out of Klein’s dialogue with the Norwegian mathematician Sophus Lie, with whom he had studied the theory of continuous groups (Kline). That idea — describing a structure through the group of transformations that leave it unchanged — has become one of the guiding threads of modern mathematics and physics: we find it again in the theory of relativity, where the physical laws must stay invariant under a precise group of transformations, and more generally whenever a symmetry becomes a tool of investigation (Katz).
Links
Topics: Plane transformations Concepts: Plane transformation · Isometry · Composition of transformations People: Felix Klein · Sophus Lie