The structures are called tensegrities -- for tension with integrity -- that form interesting geometric shapes, like dodecahedra (made from 12 regular pentagons). Connelly, who builds such toys based on these principles, described them to an audience at the annual meeting of the American Association for the Advancement of Science today (Feb. 14) in Seattle. His talk, titled "Highly Symmetric Tensegrity Structures: From Molecules to Monstrous Models," was in a morning session on "Art, Bubbles, Crystals, Domes: Geometry in the World Around You."
Tensegrity structures that are highly symmetrical are plentiful, Connelly said. A good example: buckminsterfullerenes, the soccer-ball-looking molecule made of a series of hexagons and pentagons, such as the famed, Nobel Prize-winning buckyballs.
Such structures can be modeled as a tensegrity. While there may be tensegrities with just a few vertices -- say, six vertices, three struts and three cables -- there can be arbitrarily many vertices in such a structure. In the case of the child's toy, the struts are rigid beams, the cables are elastic, and the vertices are round where more than one strut may come together.
"The mathematical model for this is a simplified version of these buckminster molecules," Connelly said, noting that the mathematics could be important in molecular studies for certain-shaped molecules. "The stability of the molecule could be related to its geometric structure. Some structures are super-stable in the sense that they are stable under a wide range of physical conditions."
That means that they will, like the squishy toys, always fall back to the same structure when perturbed. "You build it, it will hold its shape. I can guarantee it," said Connelly, whose Cornell business card reads, "Geometry, Toys, Tensegrities." The mathematician is in the midst of creating a catalogue of stable tensegrity structures. "You will be able to look them up and see what you like," he said.
It's no easy task. A high-performance workstation at Cornell's national supercomputer center, the Cornell Theory Center, was required to make the images, along with a graphics work station in Cornell's Math Laboratory. A display of these computer-generated pictures of stable tensegrity structures was held at the Cathedral of St. John the Divine, New York City, sponsored by the Pratt Institute (Nov. 1995 through Jan. 1996).
"The basic problem is simple to understand. Some pairs of points cannot get further apart. In others, the points cannot get any closer. The question I am looking at is when there are no other configurations, other than congruent ones, that satisfy those constraints."
EDITORS: Robert Connelly can be reached at the AAAS meeting Feb. 13-18 at the Crowne Plaza, (206) 464-1980. After Feb. 18, he can be reached at (607) 255-4301. Larry Bernard of the Cornell News Service can be reached in the AAAS newsroom or at the Sheraton Seattle, (206) 621-9000, Feb. 13-18, or at (607) 255-3651 after Feb. 18.