The goal of this course is to gain an understanding of the techniques involved in understanding objects in four dimensions and higher. The primary tools (in addition to a good imagination) are slicing, projection and unfolding, and we will treat each in detail. Video tapes and computer graphics will help us to compute some of these, but you will also be asked to think hard about how these processes work, and to explore them by hand computations as well.
For example, the following pictures (each linked to an MPEG movie) represents something occurring in four-space:
By the end of the course, you should be able to understand and explain what is going on in each.
Throughout the course, we will use the philosophy that the best way to understand four dimensions is to understand two and three dimensions very well. In particular, knowing what things change when you move from two to three dimension (and how they change) will suggest the changes that occur when we move from three to four. This "building up" of dimensions is a basic theme of the course.
The methodology of the course will be experimental and hands-on, whenever possible. We will try to develop the ideas together (rather than simply having me lecture), so class participation will be important to the success of the course. We will build models, analyze slicing sequences, and draw projections of four-dimensional objects in class. In place of a final exam, we will be doing group projects, and the final week of the course will be devoted to project presentations. See the the course outline for more details.
All of this is intended to help build your intuition concerning the way that dimensions work, and in particular, how to describe and visualize the fourth dimension.