Syllabus

Fall 2005, Clark University

Dept. Math. & Comp. Sci.

D Joyce, PB 322, 793-7421.

- What is geometry?
- 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

- Euclidean plane geometry.
Euclid's
*Elements*- The Pythagorean theorem
- Axiomatic structure of the
*Elements.*Definitions, postulates, common notions, propositions - Survey of Book I of the
*Elements* - Compass geometry

- Möbius 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

- Elliptic geometry
- Absolute geometry. Interpret the three geometries—Euclidean, hyperbolic, and elliptic— as special cases of Möbius geometry
- Projective geometry
- The real projective plane
- Projective transformations

- Axiomatic systems
- Hilber's axioms for Euclidean plane geometry
- Bachmann's axioms basing geometry on group theory