Math 53 Project Suggestions:

1. Some of you may be interested in making more detailed 2D creatures and their environments than we did during the first weeks of class. You could work out carefully their internal body structures, and their machines, and other such items, and then explain them carefully in a written report. Someone with a more literary bent might want to write a story about people in such a world and how they interact with each other and their surroundings.

2. Some group might want to write a play or dance along the lines of Flatland that expresses life in a two-dimensional world and how a flat creature can understand three dimensions. A more daring choreographer could try to show us four-dimensional objects through dance or pantemime.

3. Others may want to use the computer to make a series of movies like the ones that I've been showing you. There are lots of sequences that could be used to illustrate the 2D-3D and 3D-4D analogs. A careful study of several of this together with movies and clear explanitory text (perhaps as a web site) would make a very nice project.

4. Those with more mathematical background might want to look at the way that coordinates and functions work in four dimensions. For example, for a surface in four-space, can we compute tangent directions and normal directions like we do in calculus? Some computer graphics would make a nice addition to this project as well.

5. In class, we calculated the diameters of cubes in various dimensions (we say the long diagonal of the n-cube was of length sqrt(n)). There are other measurements we could compute as well. What is the volume of the n-sphere, for example?

6. For those interested in model building, it would be nice to have a good set of views of the hypercube with various of the cubes highlighted. We have seen several views of the hypercube and will see several more next week. A carefully made set of models would make a great project. There are also objects other than the hypercube that could be made (such as the analogs to the triangle and other shapes).

7. We used slicing to understand the hypercube, but so far we have only sliced it in one direction. Later this week or next week we will be slicing the hypercube in other directions, and will find some very interesting shapes. A set of models showing the slicing sequences in various directions would be a truely lovely project. Electronic models and movies are also a possibility for this, though physical models are the nicest.