Syllabus (Spring 2006)

Mathematics 244 (Geometric Structures)

Instructor:

Joseph Malkevitch (Office: 2C07)

Phone: 718-262-2551 (Voicemail is available)

The best way to reach me is by electonic mail:

joeyc@cunymv.cuny.edu

Most of the assignments for this course will be available online at the following web page:

http://www.york.cuny.edu/~malk/mycoursematerials.html

(Prerequisite: Mathematics 122 or permission of the instructor)

3 hours, 3 credits

Objectives:

This course is designed to introduce students to as wide a variety of geometrical objects, tools, and proof methods as possible. Not only metrical ideas will be treated but discrete, combinatorial, and computational aspects of geometry will also be addressed. The relation between the geometry and physics of space will be treated. Both Euclidean and Non-Euclidean ideas will be discussed.

Topics:

1. Different kinds of geometry (Euclidean geometry, Bolyai-Lobachevsky geometry, projective geometry, etc.)

2. Geometrical transformations (translations, rotations, reflections, shears, homothetic mappings, projective transformations, applications to computer vision and robotics, Klein's Erlangen Program).

3. Symmetry (transformations that preserve symmetry, symmetry groups of strips, polyhedra and tilings).

4. Polygons (convex and non-convex polygons, simple polygons, space polygons, orthogonal polygons, visibility theory, art gallery theorems, Bolyai-Gerwien Theorem).

5. Polyhedra (convex polyhedra, regular and semi-regular polyhedra, symmetry issues, deltahedra, origami models, rod models, membrane models, nets, Steinitz's Theorem, Euler's Theorem).

6. Tilings (regular polygon tilings, polyominoe tilings, reptiles, symmetry properties of tilings).

7. Block designs (finite affine planes, finite projective planes, statistical applications).

8. Lattice point geometry (Pick's theorem, Sylvester's theorem)

8. Convexity geometry (Helly's theorem, curves of constant breadth, polyhedra, tilings, packing and covering problems.)

Note 1:

This course is of special interest to students planning to teach in secondary schools, and is required for those seeking NCATE certification.

Note 2:

This course will be a writing intensive course and will follow the guidelines for such courses.