Archimedean Fountain in Copper
Some History
During my junior year I decided that I should do more art. In particular, I really like fountains, and wanted to make one. It turns out that this is easier said than done, but I really wanted to try, so I contacted Norm Hines, Pomona College's sculpture professor. He suggested that I enroll in his Issues in Contemporary Sculpture course, which was basically a bunch of senior studio art majors making the art that they wanted to make. It was a little intimidating, but I figured that the worst that could happen was I would be forced to learn a lot very quickly.
After some research and much brainstorming with help from Norm, I came up with an idea. The idea can be relatively simply explained, but its execution required a startling amount of multi-variable calculus, and the questions it raised can only be solved (to the best of my knowledge) via numerical methods.
The Basic Idea
The basic description is:
Water flowing from a deep pool at the top of a cone, to an almost-as-deep pool adjacent to it, and so on in a spiral down the cone.This description, while accurate, is also pretty opaque. For a real explanation, check out the diagrams to your left.
I knew I wanted (as viewed from above) an archimedean spiral, and I also knew that, when viewed from the side, I wanted the profile to be a flat sided cone rather than rounded hyperbola. These demands meant that i had to dive back into some continuous math that I hadn't dealt with since freshman year.
Mathematical Necessities
I knew how much copper I had (one 8'x3' sheet), and I knew what I
wanted. This sheet needed to provide all of the copper for all of my
fountain, including the base, so I couldn't just start cutting
willy-nilly. I needed to figure out how much space I could put
between successive spirals without making the fountain look too
pointy, but also with a sufficient number of turns of the spiral to
look good and to point out that it really was a
well-defined mathematical shape. After much hand wringing on my part,
it turned out that, for the overhead view, the equation I wanted was
R=4inches*theta/(2*pi). Using this, I knew exactly how
many turns I would get (a little more than 2 full rotations), but I
was still unsure about how to cut the copper to maintain a straight
side profile.
Had I not cared about the side profile, simply cutting a straight diagonal would have been the easiest. It would have resulted in a rounded profile, however, and I wanted straight sides. Curved from one view, but straight and angular from another was an idea that I found interesting.
It would be an ideal extra-credit question for a multi-variable final:
Given the restrictions above, how should he cut the copper?The problem was quickly solved by Prof. Su who noted that my restrictions implied that all heights at a particular angle were collinear, and quickly derived that I should therefore cut it in a parabola. The constants for the parabola were left as an exercise for the student, who had neglected to bring measurements with him.
Using Professor Su's insight I was able to draw the curve on the copper. I then began discretizing the curve by starting 2 inches below the curve and moving horizontally until I was 2 inches above the curve, then dropping 4 inches and repeating the process.
Construction
Construction took a long time. I had to learn to use a plasma cutter (a totally awesome tool) and I had to get better at sheet metal work and soldering. Suffice to say, it took the better part of a semester, and I worked pretty hard the entire time.
The Final Product
When I was finally done, the fountain was put on display at the student art show. It currently resides at my parents' house in Arlington, VA.
Mathematical Diversions
So, after I had constructed the thing, I took a long look at it, and a question occurred to me: do all of the reservoirs have the same volume? It certainly looks like it, but the [ahem] error bars on my construction are non-trivial, so the fountain I made can't really be used as an argument either way, except to say "yup, they look pretty close".
Further exploration is left as an exercise for the reader, who hopefully enjoys numerical methods more than I do.
Peter Boothe
10/28/01