Course Goals: To help students to develop logical reasoning skills and knowledge in understanding two-dimensional geometry with a focus on an analytic development of geometry.
- Explore the content of Euclid’s Elements.
- Use analytic geometry to gain an understanding of hyperbolic, elliptic, and Euclidean geometry, and how they are related.
- Use the concept of curvature to unify the various geometries and extend to non-uniform spaces.
- Relate the angle sum of a triangle to curvature and non-Euclidean geometries. (Gauss-Bonnet theorem)
- Review trigonometry and explore in terms of non-Euclidean geometry.
- Explore Descartes’ exterior angle theorem, and its relation to curvature and Euler’s theorem.
- Explore topologies of closed surfaces.
Student Learning Outcomes: Students will be able to:
- Cite basic definitions.
- Prove typical geometry proofs. (Pythagorean theorem, alternate interior angles, propositions in the Elements, etc.)
- Compute geometric quantities in Euclidean and non-Euclidean spaces.
- Prove geometric theorems in non-Euclidean spaces (e.g. angle sum theorem on the sphere.)
- Prove basic manifold theorems (e.g. Euler characteristic of a torus is zero.)