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# The Geometry of the Sphere

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The material on these pages was the text for part of the Advanced Mathematics course in the High School Teachers Program at the IAS/Park City Mathematics Institute at the Institute for Advanced Study during July of 1996.

Teachers are requested to make their own contributions to this page. These can be in the form of comments or lesson plans that they have used based on this material. Please send email to the author at polking@rice.edu to inquire. Pages can be kept at Rice or on your own server, with a link to this page.

Putting mathematics onto a web page still presents a significant challenge. Much of the effort in making the following pages as nice as they are is due to Dennis Donovan. Boyd Hemphill added two nice appendices. Susan Boone helped construct the Table of Contents. All of them are teachers and members of the Rice University Site of the IAS/Park City Mathematics Institute.

## Introduction

We are interested here in the geometry of an ordinary sphere. In plane geometry we study points, lines, triangles, polygons, etc. On the sphere we have points, but there are no straight lines --- at least not in the usual sense. However, straight lines in the plane are characterized by the fact that they are the shortest paths between points. The curves on the sphere with the same property are the great circles. Therefore it is natural to use great circles as replacements for lines. Then we can talk about triangles and polygons and other geometrical objects. In these notes we will do this, and at the same time we will continuously look back to the plane to compare the spherical results with the planar results.

We will study the incidence relations between great circles, the notion of angle on the sphere, and the areas of certain fundamental regions on the sphere, culminating with the area of spherical triangles. Our ultimate goal is two very nice results. First we will prove Girard's Theorem, which gives a formula for the sum of the angles in a spherical triangle. Then we will use Girard's Theorem to prove Euler's Theorem that says that in any convex, three dimensional polyhedron we have V - E + F =  2, where V is the number of vertices, E is the number of edges, and F is the number of faces.

 The next section contains a discussion of the basic properties of the sphere.

The orthographic projection of the earth at the beginning of this page, and the earth icon used throughout these pages were produced with the program xearth, written by Kirk Johnson of the University of Colorado.
John C. Polking <polking@rice.edu>