Università Roma Tre

1. Euclidean Geometry
Rudiments of Greek mathematics history. Ruler and compass constructions. Classical problems. The Elements. Axioms, definitions and postulates of Book I. Theorems I-XXVIII with proofs. Theorems XXIX, XXX, XXXI, XXXII: the role of V Postulate.
2. The question of V Postulate
The attempt by Posidonio. Equivalent propositions: Playfair, Wallis, transitivity of parallelism. Saccheri's quadrilateral. Quadrilateri di Saccheri. Saccheri-Lagrange theorem and the exclusion of the obtuse angle hypothesis. The non-euclidean geometries of Bolyai and Lobachevski.
3. Isometries of the plane
Even and odd isometries. Characterisation of an isometry by the image of three points not on a line. Chasles' Theorem. Products of reflections. Discrete groups of isometries. Finite groups, friezes, crystals. The theorem of addition of the angle. Leonardo's Theorem and the characterisation of finite groups.
Sketch of proof of the theorem of classification of frieze groups. Crystallographic restriction Theorem and the classification of wallpaper groups.
4. The geometry after Gauss
The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Moebius strip and the Klein bottle. Classification of uniformly discontinuous groups Sketch of the proof of the Theorem of Classification of locally euclidean geometries.
5. Geometries on the Torus and the Hyperbolic geometry
Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré Half plane model. Lines and distance. What is repugnant for Saccheri, but not for Aristotle

R. Trudeau: La Rivoluzione non euclidea. Bollati Boringhieri ed, 1991

V. Nikulin, I. Shafarevich: Geometries and groups. Springer ed, 1987