What is made of 2040 compact discs, 3420 brass fasterners, all assembled on a spherical aluminum frame nine feet in diameter? The amazing Q-Ball, of course!
It all began one clement afternoon in Quincy High School's Creative Problem Solving Course (CPSC). Brainchild of Dr. Sandra Spalt Fulte, the CPSC was designed for high school seniors who had already taken calculus as juniors. The CPSC is now taught by Todd Klauser (QHS) (at the time the Amazing Q-Ball was constructed, it was team taught with Dr. Vince Matsko, now affiliated with the Illinois Mathematics and Science Academy). The text? Vince Matsko's Polyhedra and Geodesic Structures -- an innovative (if yet unpublished) text describing the design and construction of geodesic structures using spherical trigonometry.
Students were surfing the Internet looking for a class construction project, and stumbled upon the zonohedron pictured.
[Click here and go to Chapter 14.] A zonohedron is essentially a polyhedron whose edges come in parallel sets -- like a cube, whose edges come in three sets of four parallel edges. These sets are called zones.
George Hart designed this zonohedron using the 61 different directions a strut can be placed in a node of the geometrical construction kit, Zometools. As a result, it can actually be built with Zometools, but would collapse under its own weight. An alternative building strategy was needed.
Eventually a decision was made: use CDs, one for each vertex of the zonohedron. Drill holes in them corresponding to the edges adjacent to the vertices, and fasten them together with rivets. For stability, assemble the CDs around an aluminum frame. It sounded straightforward enough.
First was the design phase. Because each CD corresponded to a vertex of the zonohedron, the angles between the edges at each vertex had to be calculated. With seventeen different types of vertices, this was no small task. Students were aided by the fact that unknown angles between Zome struts could be incorporated into geometrical figures and calculated from previous or known results, as shown below.
As long as the relevant angles were present, the relative lengths of the struts used were unimportant. Vince, together with a few students, checked the results using spherical trigonometry. Discrepancies were not tolerated. At last, all results corroborated.
The astute reader sees a problem: the angles around a vertex of a convex polyhedron sum to less than 360 degrees, but the angles between holes drilled in a CD sum to exactly 360 degrees. Due to Descartes, the sum of these shortfalls over all the vertices of a convex polyhedron is 720 degrees, which in our case must be distributed over 2040 vertices. Thus, each CD was "too big" by an average of 1/3 degree, which we hoped was not significant enough to make a dramatic difference. It wasn't.
Using plastic templates as a guide, Todd drilled holes in 2040 of the 3000 CDs donated by Memorex. Panels of 34 = 2040/60 were assembled with rivets, but the resulting inflexibility resulted in too many cracked CDs. Inspired by the method of assembing plastic templates for the panels, brass fasteners were used to create a strong yet flexible configuration.
Next came the aluminum frame, with parts salvaged from old prom decorations. The trick was calculating the correct radius -- Todd's engineering genius did the rest.
Finally, the assembly! In less than two hours, the sixteen students had the Q-Ball built in the QHS library. All stood awestruck. We had done it!
A powerful message was communicated -- mathematics is powerful, beautiful, fun. Sure, with the CDs already drilled and detailed instructions provided, assembly wouldn't be so difficult. But our students understood the design of the Q-Ball, not merely its assmebly. Do we do a disservice to our brightest high school students if we offer them the opportunity to learn anything less?
Postscript:The Amazing Q-Ball was constructed in 2005. After CDs continued to crack under the stress of their own weight, Todd had to retire the Q-Ball. It awaits another group of enthuiastic builders!
