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. This project might include reading "The Planiverse", by A.K. Dewdney, which treats these issues very carefully in the setting of a story.
Some group might want to write a play or dance 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 pantomime.
The idea of dimension has appeared in painting a sculpture in a significant way, particularly in twentieth century art. Several examples appear in Beyond the Third Dimension. It would be interesting to compare the use of dimension by various artists, and when possible, read some of their writings on the subject. Those with a creative bent might want to try making some of their own dimensional artwork.
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 explanatory text (perhaps as a web site) would make a very nice project.
In the movie The Shape Of Space, we saw that a 2D universe can be finite but unbounded if it is in the shape of a torus or a sphere, for example. The video asks us to consider what the shape of our own universe is, and this is a current subject of research in astronomy. An interesting project would be to describe how this question is being addressed by scientists using recently launched satellites.
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.
In class, we determined how the volume changes when the size of an object doubles. There are lots of measurements that can be made within the n-cubes. For example, what is the length of the diameter of the n-cube? What is the volume of the n-sphere?
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).
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 truly lovely project. Electronic models and movies are also a possibility for this, though physical models are the nicest.