Modern Geometry II
For your final project, pick a geometric topic that we have not covered in Math 341 or 342 and explore it on your own.
Your project will consist of three parts:
- A written paper;
- A ten minute in-class presentation; and
- A model of some kind related to your topic. Your model can be a physical model or a computer model.
You can look at any topic, as long as I approve it. Each project must include some proof(s) of some kind. People working on the same or related topics may work together, but a group of two people will be expected to cover twice as much material as a single person working alone.
Some possible topics:
- Triangle congruence theorems: ASA (6.5), SSS(9.1), ASS (9.2), SAA (9.3), AAA (9.4)
- Parallel Postulates: 10.2, 10.3, 10.4
- Patterns: 11.5, 11.6
- Spherical Projections and map making: 14.1 – 14.4. (14.1 + 14.2 together count as one problem, and only for a group also doing 14.3 and/or 14.4)
- Upper half-plane and Poincarré Disk models:17.1-17.6
- 2-Manifolds (Flat torus, Klein bottle, etc.): (18.1+18.2) (flat), 18.3 (spherical), 18.4 (hyperbolic), 18.5 (Euler Number), 18.6 (2D Shape of Space)
- Geometric algebra: Euclid, Book II.
- Disection Theory: Ch. 12, 13.
- Geometric solutions of Quadratic/Cubic Equations: (19.1-19.4)
- Trigonomitry and Duality: 20.1 (intrinsic circumference of a circle), 20.2 (spherical law of cosines), 20.3 (spherical law of sines), (20.4 + 20.5)(Duality and the laws of sines/cosines) or (20.4 + 20.6) (Duality + Projective plane) (relates to perspective drawing and 18.3)
- Mechanisms 21.1, 21.2 (Four bar linkages); 21.3 (Reuleaux triangles)
- 4D space: 22.1, only if in a group doing other problems from this chapter; 22.2 (3-Spheres), 22.3-22.6
- Polyhedra: 23.1-23.4 (Spherical angles and tetrahedra congruence theorems)
- Demi-regular tesselations, symmetries of tesselations, etc.
- Tesselations of 3space by polyhedra
- Questions related to astronomy: astrolabe, calendars, sundial, orrery
- Golden Ratio