Course Goals: To help students to develop logical reasoning skills and knowledge in understanding the two-dimensional geometry with a focus on the axiomatic development of geometry.
Course Objectives:
- Explore the content of Euclid's Elements.
- Introduce Hilbert's axiom system.
- Introduce other Euclidean geometries (e.g. analytic geometry, transformational geometry, etc.)
- Explore variations of Euclid's Parallel Postulate.
- Relate the angle sum of a triangle to curvature and non-Euclidean geometries. (Gauss-Bonnet theorem)
- Review trigonometry and explore in terms of Euclidean and non-Euclidean geometry.
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.)