Los Angeles Math Circle

10/2 -- LAMC High School: Introduction to Geometry, or Is your World Flat? (Part II) -- Dr. Oleg Gleizer

In this 6-hours-long (3 meetings) mini-course, we shall compare a few 2-dimensional (2D) surfaces, the Euclidean plane, a cylinder, a Moebius plane (a.k.a. an unbounded Moebius strip), a sphere, and, if time permits, the Lobachevsky plane.

The following are well-known features of the Euclidean plane.

1. There exists one and only one straight line passing through two different points.
2. The shortest path between two different points is a segment of the straight line connecting them.
3. Two straight lines intersect at no more than one point. If the lines do not intersect, they are called parallel.
4. For any straight line and a point away from it, there exists exactly one straight line parallel to the original one that passes through the point. (This is the modern version of the Euclid's fifth postulate, considered by many as the most controversial scientific statement of all times.
5. For any triangle in the Euclidean plane, the sum of its angles equals to pi.

In the course, we shall see if these properties hold for the 2D surfaces mentioned above. Our point of view will be that of a tiny bug living on a surface. Our ultimate goal will be to see, if our world is flat (a.k.a. Euclidean). Our study will be hands-on, so please bring the tools, a few letter-size sheets, a few pencils, an eraser, a compass, and a ruler (straightedge) to every class.

Handouts: