# Geometrical Concepts from Constructions, Models, and Investigations Summary

## Monday, July 7, 2003

As we started the 2nd week, we began with a discussion of topics for our final projects. The topics being investigated include modeling from middle school through calculus and explorations in non-Euclidean space. The focus of our final products is to create work which not only enriches our own teaching, but that will serve as a resource to the broader community.

We then moved on to a comparison of Euclidean and spherical geometries. Melissa shared some of the work which she did over the weekend, using the Adventures on the Lenart Sphere book from Key Curriculum Press. Beginning with a discussion of the definitions of point, line and plane in Euclidean space, we developed analogous definitions for the sphere. While we did not address it at this time, we left open the further comparison to hyperbolic space and other non-Euclidean surfaces.

To clarify our terminology, Jim told us that within research mathematics "sphere" refers to the boundary surface of a ball; ball is the volume. He also clarified "geodesic" to mean shortest path; in "nice" geometries the geodesic is also straight, but not in taxi-cab geometry.

Once the point, line and plane had been defined along with their analogous concepts on the sphere, we began to explore the ways that lines do and do not intersect on the sphere versus the way they behave on the Euclidean plane. Circles and poles were also described on the sphere, and we found that a bigon, a polygon with two sides, exists on the sphere. It's called a lune.

The conclusion of today's time was focused on creating orthogonal lines, and we were left with the following question: What is the relationship between the area of a triangle on a sphere to the measure of its angles?

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
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