Ivars Peterson's MathTrek

May 24, 1999

# Möbius in the Playground

Many people are familiar with the remarkable one-sided surface known as the Möbius band. You can easily make one by gluing or taping together the two ends of a long strip of paper after giving one end a half twist (see Recycling Topology, Sept. 30, 1996).

Finding a Möbius strip amid the slides and swings of a playground is a much more unusual occurrence. At the Sugar Sand Science Playground in Boca Raton, Florida, children can crawl and scramble along a three-dimensional variant of a Möbius strip.

The Möbius climber was designed by Gerald Harnett, a mathematics professor at Florida Atlantic University in Boca Raton, with the assistance of Jerome Schwartz and Sean Powers.

It consists of 64 triangles linked together and mounted so that, at any one point, the twisted structure appears to have four sides, but overall turns out to have just two.

Perspective diagram of the Möbius Climber.

Schwartz came up with the original idea for the climber. When the playground designers first called for suggestions several years ago, he had constructed a little model of his concept from a length of foam rubber with a square cross section. After rotating one end through 180 degrees, he had fastened the ends together to form a twisted loop--like a thickened Möbius strip. He had also drawn some triangles on it to indicate the locations of struts.

Schwartz, however, didn't know how to go beyond his model and asked Harnett to help him define the structure more precisely and demonstrate its feasibility.

Harnett faced a number of challenges and constraints. The playground layout determined the climber's dimensions. Safety concerns limited its height above the ground and other features. They were also paramount in the decision to shift from open triangular struts to full triangular faces.

"In early computer sketches, I chose to use 64 triangles in all simply because it looked good," Harnett says. "That was an aesthetic constraint, and the choice worked out well in the end."

The first step in the design involved specifying a pair of "skeleton curves"--two, entwined double loops--to which the triangle vertices would be attached.

"Rendering the skeleton was just the beginning," Harnett says. "A great deal of work was needed to produce a complete description of each of the 64 different triangles." That description included the configuration and dimensions of the boards making up each triangle and the location of bolt holes in each corner.

"In addition, we needed diagrams showing the configuration of the triangles so that volunteers could put them together in the field and diagrams for the locations and heights of the supporting pipes," he remarks. "Mathematically, all of this was a massive exercise in trigonometry and analytic geometry in two and three dimensions."

Top view of the Möbius Climber.

With the help of the computer program Mathematica, Harnett created a set of construction plans providing specifications and templates of all the triangles. "The hardest part. . .was determining the locations of the bolt holes for the brackets," he recalls.

The moment of truth came in the playground. Would the last triangle actually link up with the first to close up the climber?

"Until that moment, I don't think anyone was certain that it would work," Harnett says. He himself had tiny concerns about the accuracy of his numbers and the possibility that the weight and limited angular freedom of the supporting brackets would somehow cause a problem.

Constructing the Möbius Climber.

It worked! The climber is now one of the playground's established features. "Kids just like to climb on it and through it," Harnett comments.

The Möbius Climber isn't the only mathematical structure that can be found in a playground. If you're interested in plans for building the PlayDome, which consists of used automobile tires bolted together to form more than half of a truncated iscosahedron (or buckyball), check out http://buzzard.ups.edu/playdome.html.

There's more than one way to experience the playful side of mathematics.

References:

Peterson, I., and N. Henderson. In press. Math Trek: Adventures in the MathZone. New York: Wiley. For children ages 10 and up, this activity book brings readers to a mathematical amusement park.

Peterson, I. 1999. Chasing arrows. Muse 3(January):27.

______. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W.H. Freeman.

Gerald Harnett (harnett@bwn.net) provided the Mathematica code for generating the illustrations of his Möbius Climber. Additional information can be found at http://www.wolfram.com/discovery/mobius.html.

Information about building the buckyball (truncated icosahedron) PlayDome out of used automobile tires is available at http://buzzard.ups.edu/playdome.html.