Geometric Structures

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York


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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.)

Applications of geometry and unsolved problems will be discussed throughout the course.


Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene F. Krause. Originally published by Addison-Wesley, 1975, but now available as a Dover Publication reprint.

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