Experiential

Math professor Dr. Andrew Parker offered an experiential math seminar on fractals. The class took a technological rather than natural approach to the subject, generating fractals using a video feedback loop instead of a computer, which is the more typical way to create manmade fractals.

Parker explains how the experiment came about: “We had studied the theory of iterated systems (a mathematical function system with a feedback loop), and the students and I had been writing computer programs to generate a class of fractals using different methods.” But Parker likes to have students put theory into practice—so he decided “to try to generate fractals using a video camera, projectors, and a simple feedback loop.”

With the help of Multimedia Services, the class created a video feedback loop by pointing a camera toward a projection of its own output. Specifically, three projectors were aimed at a wall, and a camera was set up on a tripod behind the projectors. The output from the camera was then routed to all three projectors simultaneously. Parker continues, “Now, if the camerais aimed at a flowerpot, you get threecopies of the image of the flowerpot onthe wall. But if it is aimed at the samewall at which the projectors are aimed,a video feedback loop is created. Inother words, you have created aniterated functionsystem: the outputof the systembecomes its owninput, and thus theprojection systemis iterated.”

What were the results? Seemingly infinite, elaborate, and yet predictable patterns. As Parker describes, “Remarkably intricate patterns can be created this way, and exactly which patterns are created can be predicted by the mathematics (the angle and size of the projections and camera, the number of projections, and so on).” That predictability was central to the lesson. “Part of our objective,” Parker explains, “was to verify the effect (which we had already determined in theory) that rotating the projectors or camera or both has on the eventual image. The students were able to recreate several fractals that we had studied on paper and in a computer lab, while discovering many new self-similar images in between.”

Asked to sum up the course as a whole, Parker comments on the nature of complexity: “Often the complexity we see or experience in the natural world can be modeled and explained very simply in the language of mathematics, and in particular using a self-referential, iterated system. Self-reference can be a driving force behind complexity.”

So where does the movie come in? “Much of our session was recorded using the camera itself,” Parker explains, “so we have a video of our experiment and our experience.”