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Spiraling Into Form

3 May 2004 (All day)
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Fibonacci cactus. Energy minimization may explain why the red, yellow, and brown spirals on this cactus make a 3, 5, 8 Fibonacci sequence.

There's something special about spirals. Take a close look at plants and they pop out all over the place. Sunflower petals, pinecone spears, cacti spines, and tree leaves all form whorls in sets of spirals. Now, researchers explain how the laws of physics force spirals to develop in triples with special mathematical relationships.

Spirals are built into a plant from its very beginning. The tender tip of a growing plant is capped by a thin outer shell. As the plant cells inside the shell grow and divide, they create stress that can deform the shell. The easiest way for the shell to relieve the stress is to buckle into ridges that form the arms of a spiral, centered at the stem. Sometimes, the stress pushes the plant to add a second or third spiral. Although spirals differ among plants, they show particular kinds of patterns.

To investigate why, mathematicians Patrick Shipman and Alan Newell of the University of Arizona in Tucson calculated the stresses on the growing tips of plants, assuming that the plant wrinkles to minimize the total stress on its skin. They found that the first two spirals strain and pull on each other, sometimes making a third spiral. Crucially, the number of arms in the third spiral always equals the sum of the arms of the first two in order to minimize stress, they found. This is the hallmark of the famous Fibonacci series, in which each number is the sum of the preceding two numbers. The series seems to show up with surprising regularity in nature, describing patterns of population growth, the proportions of seashells, and other phenomena.

When Shipman and Newell graphed the solutions to their stress calculations, they saw spiral patterns with Fibonacci-like relationships that looked very similar to those in real cacti. Their analysis appears in the 23 April issue of Physical Review Letters.

"The key nice part about what Patrick and Alan have done is they do some excellent mathematics and link it to what the physics might be," says Neil Mendelson, a molecular biologist at University of Arizona. Shipman says, "For this sort of work to be useful to biologists, we have to know what can actually be predicted."

Related Sites
Examples of Fibonacci numbers in nature
A page with many pictures of plants exhibiting Fibonacci-like spiral patterns

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