In the course of talking to people at the Geometry Center and observing their work, I have noticed many are trying to visualize four dimensional space. I noticed this common element when talking to Eduardo Tabacman, Richard McGehee, Bruce Peckham (who visited in the fall), and most recently Brad Barber. In addition to research, there is talk of higher dimensional graphics software. The Center's graphics program Geomview has four dimensional capabilities; both arbitrary projections and slices are possible. Last week there was a meeting to discuss the possibilities for software to accommodate k dimensional manifolds in n dimensional spaces. (There were no concrete conclusions as to how to actually do this, but the meeting resulted in a lengthy wish list.)
When I asked people how well their work made them understand 4D, I got the following interesting responses: Tabacman and McGehee said they still didn't have the intuitive sense that one gets for 3D, and therefore did not feel they understood 4D. Barber felt he could visualize 4D, as it corresponded to theorems he knew. These answers made me curious for two reasons; first, they show a difference in opinion on what it means to visualize and understand. Second, it makes me wonder how much the understanding depends on the mathematical context.
Inspired by these questions, I decided to approach the subject directly; namely, I have interviewed anyone here who looked at four dimensions in their research who was willing to discuss it. I am making these interviews into a series of articles, trying to specifically focus on 4D visualization. The content of the articles varies quite a bit since the answer to the second question above is that the understanding and even what it means to understand varies significantly with mathematical context.
As a general guideline for the interviews, I asked the following list of questions. I would be interested to hear any comments that non Geometry Center people have on the subject. I hope that perhaps by linking these articles by a common theme I can inspire some discussion?
Questions (meant as guidelines only):
In what context did you use 4D space?
Do you feel that you can understand or visualize 4D?
If so, what did it take to make you feel that you understood? What are the mathematical ideas that help you understand it?
If not, what would it take to make you feel that you understood?