Non-Euclidean geometry

The main purpose of the course is to develop a unique approach to the study of different branches of modern geometry such as Euclidean, spherical, hyperbolic, elliptic, and projective geometries. This approach is based on the fundamental idea going back to F. Klein, who considered geometry as a set endowed with a group action satisfying some additional conditions. This point of view will enable us to study uniformly not only the aforementioned geometries but also such beautiful objects like tilings and kaleidoscopes as well as the Platonic bodies and their higher dimensional analogs.


  1. The Erlangen program: Klein geometry, elements of group theory, examples of Klein geometries: toy geometries, discrete geometries, tilings and kaleidoscopes.
  2. Finite geometries: the Platonic bodies and their classification, regular polyhedra in higher dimensions.
  3. Non-Euclidean geometry I: hyperbolic plane and hyperbolic space, Poincare and Cayley-Klein models, hyperbolic trigonometry.
  4. Non-Euclidean geometry II: spherical geometry, Riemann’s elliptic geometry, models of elliptic geometry.
  5. Projective geometry: projective duality on the plane and in the space, the Desargues, Pappus, and Pascal theorems, projective geometry is all geometry.
  6. Geometries over finite fields: projective and affine geometries over finite fields.


  • А.B. Sossinsky. Geometries.
  • М. Berger. Geometry.
  • H.S.М. Coxeter. Introduction to Geometry.