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Mathematics and Turtles

Consider this problem: Can you cut a convex shape from a uniform sheet of plywood that will always end up sitting in only one position, regardless how it is initially placed? Sounds simple enough as stated, but it is a very difficult mathematical problem, as posed by the Natural History Magazine.

The latter magazine raises curiosity...what does it have to do with mathematics? The connection: The question is an abstraction of a known phenomena in nature. Any ideas?

Think about a turtle on its back. Withs legs waving randomly, it eventually rights itself. The secret lies in the mathematics of the shape of the turtle's shell.

The only stable configuration occurs when the turtle is upright on its feet. The new question: How can one model this mathematically?

Gábor Domokos, Budapest University of Technology and Economics, took on this challenge...and succeeded. Working with his graduate student Péter Várkonyi, he first observed and measured the shells of seventeen species of turtles.

Then, they used a spherical coordinate system to construct formulas modeling the turtle shells. As a result. they "slightly squashed a sphere, added a pair of extra flattened planes on the surface (... called "raceways"), and pressed one side into a sharp edge." When created physically, this shape satisfied the problem! For unknown reasons, the shape is called a Gömböc.

For more about the Gömböc and to see it in action, see this week's website review.

Source: http://www.naturalhistorymag.com/biomechanics/10309/the-living-g-mb-c