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Maths gets into shape

Is it a starfish? Is it an orchid? No, it's Superformula.
2 April 2003

JOHN WHITFIELD

For centuries scientists have sought to express natural forms in mathematical terms.
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One simple equation can generate a vast diversity of natural shapes, a Belgian biologist has discovered. The Superformula, as its creator Johan Gielis has christened it, produces everything from simple triangles and pentagons, to stars, spirals and petals.

"When I found the formula, all these beautiful shapes came rolling out of my computer," says Gielis, at University of Nijmegen, Holland. "It seemed too good to be true - I spent two years thinking 'What did I do wrong?' and 'How come no one else has discovered it?'" Having spoken to mathematicians, he reckons that he's found something new.

The Superformula is a modified version of the equation for a circle1. Changing one term in the formula varies the proportions of the shape - moving from a round circle to a long and skinny ellipse. Changing another varies the axes of symmetry - shifting from a circle to triangle, square, pentagon and so on.

Varying both proportion and symmetry together produces shapes with any number of sides, regular and irregular. It can also produce three-dimensional structures, and non-biological shapes such as snowflakes and crystals. "It's a new way of describing nature," says Gielis.

For centuries, scientists have sought to express natural forms - such as the spiral of a sheep's horn, the branching of a tree, or a bee's honeycomb - in mathematical terms.

"Describing form is one of the more intractable problems in biology," says botanist Karl Niklas of Cornell University in Ithaca, New York. Researchers have come up with many ways to describe leaves and shells, for example, but there is little unity: "Things have become cumbersome and idiosyncratic," he says.

The Superformula might provide a single, simple framework for analysing and comparing the shapes of life, believes Niklas. "This is an exciting development."

The Superformula produces regular and irregular shapes with any number of sides.
J. Gielis

Gielis has patented his discovery, and is developing computer software based on it. Using one formula to produce shapes will make graphics programs much more efficient, he says. It might also be useful in pattern recognition.

What's less clear is whether nature uses the formula to generate different shapes. "I'm not convinced this is significant, but it might turn out to be profound if it could be related to how things grow," says mathematician Ian Stewart of the University of Warwick, UK.

Other, more complicated, single equations can produce a similar diversity of shapes, says Stewart. He believes that the Superformula is more likely to provide a useful tool than an insight into how life actually works.

Gielis acknowledges that the formula describes nature's end product, not how it got there, but he hopes that time might prove the Superformula's profundity. "Description always precedes ideas about the real connection between maths and nature," he says.

References
  1. Gielis, J. A generic geometric transformation that unifies a wide range of natural and abstract shapes. American Journal of Botany, 90, 333 - 338, (2003).


Nature News Service / Macmillan Magazines Ltd 2003