Math and Computer Science Honors Theses

Isometry Groups of Geometric Spaces

Date of Creation

5-13-2011

Document Type

Departmental Honors Thesis - Restricted Access

Department

Mathematics

First Advisor

Thomas E. Cecil

Abstract

This thesis is based primarily on the book, Euclidean and Non-Euclidean Geometry: An Analytic Approach, Cambridge University Press, 1986, by Patrick J. Ryan. With attempts to prove the parallel postulate in Euclidean geometry, ideas emerged about the existence of non-Euclidean geometries such as spherical geometry and hyperbolic geometry. Professor Cecil and I studied the isometry groups in the Euclidean plane, the sphere, and the hyperbolic plane. In order to understand the group of isometries, we first studied how geometry on the Euclidean plane is defined using linear algebra. Then we studied the transformations that can be obtained as a product of a finite number of reflections in lines: the identity transformation, reflections, translations, rotations, and glide reflections. We proved that in the Euclidean plane, every isometry, i.e., an onto mapping T from the Euclidean plane to itself that preserves distance, must be one of these types of transformations. We continued our research by analyzing spherical geometry. Using our knowledge of the Euclidean plane we were able to study transformations on the sphere S2 as well. Similar transformations to those in the Euclidean space a rc defined on the sphere. We concluded our research with the hyperbolic plane H2. Again, we found similarities to both the Euclidean plane and the sphere. However , in studying hyperbolic geometry, we needed to use a different inner product than in the Euclidean and spherical cases. Ultimately, we found the isometry groups of the Euclidean plane, the sphere, and the hyperbolic plane.

Comments

Reader: Andrew D. Hwang

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