Università Roma Tre

1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.

R. Trudeau: "La rivoluzione non Euclidea" Bollati Boringhieri
V. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.