Professor Banchoff has already assigned two of the most relevant exercises for chapter 1: reading Flatland and drawing and thinking about hypercubes. These assignments provide a jumping off point for other projects:
1. The natural environment of an amoeba is presumably not a microscope slide. Is an amoeba's habitat a (virtually) two-dimensional world, such as the surface of water? If not, how does the membrane protect the amoeba's nucleus?
I don't know the full physiology of amoebae, but my understanding is that the being is rather like a flat sac, like a flattened balloon, with a membrane surrounding a liquid interior. One could imagine the membrane as consisting of three parts, a lower disc in the water, an upper disc exposed to the air, and a sort of rim separating these two pieces at the interface of air and water. If so, it is only the rim that would be sensed by any other amoeba inhabiting the surface of the same pond. Whether or not there is any cell differentiation in different parts of the membrane I don't know. It is a biological question of course, not a mathematical one, but it does provide a model. It was Professor Heywood in the biology division with whom I discussed the wording of that example, and he helped me choose some non-technical description which would not be totally alien to the true situation. Of course amoebae are not really two-dimensional, but they come pretty close to functioning as if they were.
Cellular slime molds might be better candidates, since they really do slide across solid surfaces in a very thin state. And the square bacteria in the example Jeremy Kahn came up with the first day are also good to investigate. A good exercise would be to discuss the dimensionality of the day-to-day existince of various life forms. Grazing animals are pretty two-dimensional, on a flat plain anyway. Giraffes are different. Seals are quite different, as are eagles. A possible source of discussions at many levels.
2. The Penguin Science Fiction edition of Flatland (and others, I believe) attributes the book to "A. Square", as if this were a first initial and last name. But the original edition is by "A Square" - an anonymous description, not a name. When and how did the erroneous period appear?
The original is definitely by "A Square", where the "A" is either honorific or part of the name. None of the other squares are identified by name, for example the narrator's brother. Perhaps he is "B Square" or "T Square"? In any case, the "A" is not an initial--it is not short for anything. The mistake is common though. My feeling is that "Edwin Abbott Abbott = EAA = EA^2". In the prepublication copy I own, Abbott signs the book to his friend William Stedman Aldis with the words "from the Square", which might allude either to the book at hand or to some standing schoolboy joke.
3. How do William Stringham's illustrations represent four dimensional figures?
Later in the book in Chapter 6 there is a discussion of projections of figures from three-space to two-space, and an analogous description for projections from four to three dimensions. Unfortunately I do not have the drawing tools I can use to show a three-to-two example in this text-only document. Perhaps someone should look up the original Stringham paper in the American Journal of Mathematics of 1880. We have it in the SciLi and it isn't too long. It is a good candidate for inclusion in our data base in any case.