*Geometric Structures*

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College (CUNY)

Jamaica, New York

0. **What is Geometry?** (geometry as the study of space, shapes, and visual phenomena. The role of careful looking. Geometry as a branch of mathematics and as a branch of physics)

1. **Definitions, Axioms, and Models**. (Different kinds of geometry, Euclidean geometry, Bolyai-Lobachevsky geometry, projective geometry, affine geometry, and taxicab geometry. Axioms systems and rule systems in sports)

2. **Graph Theory** (basic graph theory will be used to unify a lot of the topics to be discussed)

3. **Geometrical transformations** (translations, rotations, reflections, shears, homothetic mappings, projective transformations, applications to computer vision and robotics, Klein's Erlangen Program. The role of distance functions in geometry.)

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

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

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

7. **Tilings** (regular polygon tilings, polyomino tilings, reptiles, symmetry properties of tilings).

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

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

**10. Geometry of surfaces** (basic topology of surfaces, folding and unfolding, Moebius strip, spheres with handles.)

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