Strings and springs net mechanical surprise.

Intuition can sometimes lead one astray. Consider a weight hanging from a spring, which in turn is suspended by a piece of string from an identical spring attached to the ceiling. Cutting the connecting string would send the weight and the lower spring plummeting to the floor.

Now add two "safety" strings to the original arrangement. One string joins the upper end of the lower spring to the ceiling. An identical string joins the lower end of the upper spring to the weight. Both safety strings initially hang limply.

When the taut string in the middle is cut, the safety strings prevent the weight from plunging all the way to the floor. Intuition suggests that, given the safety strings' slack, the weight will end up hanging somewhat lower than before. However, for certain combinations of springs, string lengths and weights, the opposite is true.

In the Aug. 22 NATURE, applied mathematician Joel E. Cohen of Rockefeller University in New York City and physicist Paul Horowitz of Harvard University argue that under a broad range of conditions, cutting the linking string and letting the safety strings carry the load actually pulls the weight above its initial position and closer to the ceiling.

The idea for this startling demonstration arose out of Cohen's long-standing interest in mathematical models of biological competition, especially models that produce counterintuitive outcomes. One model involving traffic flow, discovered in 1968 and now known as Braess' paradox, demonstrates that adding extra roads to a congested transportation network may actually increase the amount of congestion rather than alleviate it.

As a step toward learning whether the same kind of surprising result could occur in a biological system, Cohen started by looking for a mechanical analog of the traffic paradox, and he came up with the string-spring arrangement described above. He then turned to Horowitz for an electrical version of the same situation.

"That turned out to be straighforward," Horowitz says. He designed an electrical circuit in which appropriate resistors replaced the springs, and devices known as Zener diodes replaced the strings.

"The result is quite surprising," Horowitz says. "When you add extra current-carrying paths, less current flows." The same paradoxical behavior occurs in a hydraulic system in which appropriate lengths of tubing and pressure-relief valves replace strings and strings.

These theoretical results were intriguing enough to prompt Richard L. Garwin of the IBM Thomas J. Watson Research Center in Yorktown Heights, N.Y., to construct a working model made up of strings, rubber bands and a plastic jug partially filled with water as a weight. "It's realy easy to do," Garwin says. "When people see [the effect], they just don't believe it."

The more general lesson, Cohen and Horowitz says, is that physical networks may not necessarily behave as expected when paths or components are added.
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