Introducing Hyperbolicity via Piecewise Euclidean Complexes

Abstract

Many topics in a sophomore level geometiy class can be presented in a very tangible fashion. For example, spherical geometiy can be introduced by bringing string, stick-um, and a few large rubber balls to class. Students can use these manipulatives to experiment with the angle sums of spherical triangles, building intuition that leads to a proof that the area of a spherical triangle is directly related to its angular excess. (For other examples of exploratoiy activities in elementary geometiy see [2] and [3].) While spherical geometiy is relatively easy to introduce through such in-class exercises, hyperbolic geometiy is much more difficult. You can bring potato chips to class to indicate the local geometiy of the hyperbolic plane, but it isn't so easy to use them to explore geodesics and triangles. Because they are small and brittle, it's hard to construct geodesics on potato chips. Also, it's not clear how to measure angles on such surfaces. One can estimate angles between geodesics on a sphere by placing a sheet of paper (= tangent plane) on the sphere, and then measuring the angle between the tangent lines to the geodesics. You can't do this with a physical model of the hyperbolic plane since the hyperbolic plane is negatively curved, and any tangent plane you make will cut through the surface. If you are willing to let go of the local hyperbolic structure in favor of a model that mimics hyperbolic geometiy on a large scale, then Thurston paper does the trick. Thurston paper is briefly described in [5] and [6] and is the subject of exercise 2.1.4 in Thurston's book [4]. None of these sources discuss the topic we present here, that the geometiy of geodesics in Thurston paper nicely approximates the geometiy of geodesics in the hyperbolic plane. That this approach to hyperbolic geometiy is accessible to students is evidenced by the fact that four of the five authors wrote the first draft of this paper as part of a student research project! The Euclidean plane can be subdivided into equilateral triangles, where six triangles are joined at eveiy vertex. Thurston paper is constructed by joining together seven equilateral triangles at every vertex. The addition of the extra triangle causes the system to bend and twist, instead of lying flat like the Euclidean plane.