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Module MA232A: Euclidean and Non-Euclidean Geometry

Credit weighting (ECTS)

5 credits

Semester/term taught

Michaelmas term 2017-18

Contact Hours

11 weeks, 3 lectures per week

Lecturer

Prof. David Wilkins

Learning Outcomes

On successful completion of this module, students will be able to:

• justify with reasoned logical argument basic properties of triangles, circles and polygons in the Euclidean plane;
• describe measures of curvature, both extrinsic and intrinsic, applicable to smooth surfaces in three-dimensional Euclidean space;
• identify, and justify with reasoned logical argument, significant geometric features and properties of the hyperbolic plane, and the corresponding features and properties of the disk that represent them in the Poincaré disk model of the hyperbolic plane.

Module Content

Aims to introduce complex variable theory and reach the residue theorem, applications of that to integral evaluation.

• Euclidean geometry: an exploration of Euclid's Elements of Geometry, based on editions freely available online, with detailed discussion of the definitions, axioms and propositions of Book 1, followed by discussion of a selection of significant results contained in Books 2—6.
• Curvature of surfaces: a discussion of some of the principal results of C.F. Gauss's General Investigations of Curved Surfaces.
• The hyperbolic plane: the Poincaré disk model of the hyperbolic plane, geodesics and curvature, homogeneity, the representation of isometries as Möbius transformations.