The sphere is a familiar object, but its geometry is less well known than plane geometry. Most of us know a lot more about straight lines or triangles on the plane than on the sphere. Spherical geometry has lots of applications. It is the geometry for getting around the surface of the Earth or navigating with the stars or designing geodesic domes. Spherical geometry is a beautiful subject in its own right, but it is even more interesting when its features are connected to the symmetries of polyhedra or contrasted with the geometry of the plane. Spherical geometry is an example of a two-dimensional non-Euclidean geometry. It can be used as a jumping-off place for exploring the geometry of curved surfaces and ultimately the famous non-Euclidean geometry whose discovery stood the world of nineteenth century math and science on its ear.
In this course we will study the sphere intensively and also get an introduction to these other non-Euclidean geometries. The course will include experiments and visualizations with models, drawings, maps and computer software. It will also include numerical and symbolic problems and examples, with applications to the earth and the sky and spherical structures. Some spherical trigonometry will be introduced. The course will also include big conceptual ideas and history, with geometric problem solving, reasoning and proof. We will learn about transformations and symmetries of the sphere and use coordinates and vectors to solve problems and to draw and calculate with computers or calculators.
It is hoped that student will emerge from the course with some expertise about spherical geometry and a wider appreciation and enjoyment of geometry in general. While this course is a college course, it should provide many examples, ideas and resources that can be used in the secondary school mathematics classroom.
The prerequisites for the course are a good understanding of informal geometry and trigonometry, with some knowledge of vectors and matrices and some ideas from calculus (mainly velocity and area). Also, a degree of comfort with mathematical reasoning will be needed. There will be some opportunity to review to fill in gaps in background.
Individual homework problems will be assigned each week. Also problems will be discussed in small groups during class time. The Web will be used for class discussions. There will be some quizzes and tests. A paper or major problem portfolio on some special topic will be due at the end of the quarter. Since this is a three-credit course, a typical student should expect to work about six hours each week outside of class.
Grades will be based on written assignments, tests, and classroom participation (regular attendance is necessary). This course should satisfy the Washington math certification requirement in non-Euclidean geometry and will prepare teachers to teach the spherical geometry topics on the state Essential Learnings.
For additional information check the course web page or send e-mail to < king@math.washington.edu>.