Math 342
Modern Geometry II
Final Projects

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:

1. A written paper;
2. A ten minute in-class presentation; and
3. 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