Archimedes' Limit is a piece I designed and built with my students in the Math Crew for the Science Museum of Minnesota's calculus exhibit. The piece demonstrates the concept of a limit by using a green laser and a circular mirror to create a triangle, then a square, a pentagon, and eventually a circle.

When the museum first asked for pieces, I sought to understand what makes calculus "tick." I came to believe that its heart and soul are found in the limit concept. To me, this is what makes calculus both beautiful, and difficult.

The most accessible illustration of this concept I could come up with is the example of regular polygons with n sides. The simplest member of this family is an equilateral triangle (n=3). Next comes a square, with n=4. As the process proceeds (n goes to infinity), almost everyone (including very young kids) can predict that the shape "at the limit" is a circle. Thinking of a circle as a regular polygon, with infinitely many, infinitely short sides gets at the heart of calculus. And while Archimedes was not using calculus when he used a very similar idea to try to calculate the exact value of Pi, his strategy of comparing the areas of inscribed and circumscribed polygons, certainly points toward the limit concept--indeed some recent research suggests he was the first to describe calculus.

After presenting the Math Crew with this example, and challenging them to come up with a way of presenting it as an exhibit, a 15 year old member of the Crew presented his idea two days later: put a laser inside of a cylindrical mirror. With the beam pointed at various angles, it will trace out a triangle, square, etc. When pointed just along the mirror surface, it will form a circle.

We tried out a few versions of this idea on a small scale, using a plastic gallon jug and some Mylar. It works!

Archimedes' Limit is currently installed in the Science Museum of Minnesota in the Experiments Gallery.

Some pictures of the Math Crew and me working on the piece:

 drilling a hole for the tubing filling the tank our first polygon! gluing the mirror onto the frame