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Spectacular Mathematical Bubble Design at the Olympics

August 8, 2008

C:\Users\rmiller\AppData\Local\Temp\aWaterCubeExt_468x312.jpgThe spectacular design of the National Aquatics Center in Beijing—the swimming venue for the 2008 Olympics—was inspired by the mathematics (and physics) of foam.

The mathematics originated with a search by William Thomson (Lord Kelvin) in the 19th century for a 14-sided polyhedron that packs together into a structure with minimal surface area. Fourteen-sided polyhedra are known as tetradecahedra (or tetrakaidecahedra).

Kelvin's tetrakaidecahedron was a kind of octahedron, with each vertex sliced off and the edges and faces slightly curved. Kelvin surmised that, when packed together, these shapes form a foam with least surface area for a particular bubble size. However, he couldn't prove that this structure is optimal.

Kelvin's work prompted a search for a foam structure that has a smaller surface area than that of the one he had found. A century later, Denis Weaire and Robert Phelan of Trinity College, in Dublin, noted that a family of natural cagelike compounds called clathrates form lattice structures with many of the properties of a good foam. By examining nature's repertoire of clathrates, they identified a foam structure that does better that Kelvin's assemblage, a structure made from two different shapes instead of one. One shape is a slightly curved dodecahedron and the other is a 14-sided shape with two opposite hexagonal faces and 12 pentagonal faces. Both components have the same volume.

Weaire and Phelan haven't yet proved that their foam has the least surface area. At this point, Weaire told Science News, a proof would be difficult. "I'm not holding my breath." he said.

The walls, roof, and ceiling of the Olympic "Water Cube" are based on the Weaire-Phelan foam. Made of a plastic known as ethylene tetrafluoroethylene and filled with air, the bubbles are attached to a steel framework outlining the bubble edges. Surface tension holds the bubbles together and tends to pull them into a structure with least surface area.

The building "really looks like nothing else in the world," Tristam Carfrae told the New York Times. "It's a box made of bubbles." Carfrae is the structural engineer who designed the center.

Carfrae's one significant modification to the Weaire-Phelan foam in his design. He used a diagonal section through the structure, as if cutting a block of foam at a 60-degree angle. "That's what gives it its random appearance," Weaire said. "Only if you look carefully do you see that it's a repeating pattern."

The resulting structure is so strong, engineers say, that the building could be turned on its side without collapsing.

The structure also contains no triangles. Because triangles can't be deformed without changing the lengths of their sides, they commonly give a structure stiffness and strength. The Weaire-Phelan foam has the flexibility necessary to make the structure better able to withstand earthquakes.

Source: Science News, July 19, 2008; New York Times, Aug. 5, 2008.

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Friday, August 8, 2008

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