• Models of geometric spaces. Mathematical and extramathematical models, uniform and nonuniform geometries

• Synthetic vs. analytic geometry

• Elements of geometry. Points, lines, planes, distances, angles, areas

• Geometric Transformations. Isometries, similarities, particular transformations. Wallpaper groups of transformations. Inversion in a circle. Stereographic projection

• Mathematical tools for analytic geometry. Complex numbers. Groups of transformations and the Erlanger Programme

• The Pythagorean theorem

• Axiomatic structure of the Elements. Definitions, postulates, common notions, propositions

• Survey of Book I of the Elements

• Compass geometry

• The inversive plane. Möbius transformations

• Steiner circles. Elliptic, hyperbolic loxodromic, and parabolic transformations

• Hyperbolic geometry. Models for hyperbolic geometry, Tiling the hyperbolic plane

• Cycles

• Hyperbolic length

• Hyperbolic area

• The real projective plane

• Projective transformations

• Hilber's axioms for Euclidean plane geometry

• Bachmann's axioms basing geometry on group theory

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