Math 128, Modern Geometry
Fall 2005, Clark University
Dept. Math. & Comp. Sci.
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.
Groups of transformations and the Erlanger Programme
- Euclidean plane geometry.
- Möbius geometry
- The inversive plane.
- Steiner circles.
Elliptic, hyperbolic loxodromic, and parabolic transformations
- Hyperbolic geometry.
Models for hyperbolic geometry,
Tiling the hyperbolic plane
- Hyperbolic length
- Hyperbolic area
- Elliptic geometry
- Absolute geometry.
Interpret the three geometriesEuclidean, 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
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