Explaining Plant Configurations Mathematically

August 23, 2007

Remarkable for their complexity and beauty, plants display biological-growth mechanisms and consistent mathematical patterns that scientists are striving to understand through the use of mathematical models. The cover story of the July 21 issue of Science News, called "Mathematical Lives of Plants," by Julie J. Rehmeyer, surveys the reasons why plants grow in geometrically curious ways.

"I feel the biological models are so complicated that some of the beauty of phyllotaxis is lost in the process. It becomes purely computational," botanist Jacques Dumais of Harvard University told Science News. "The mathematical models are so simple and yet so powerful, and they explain so much of what we see."

It turns out that while the seeds of a sunflower, the spines of a cactus, and the bracts of a pine cone grow in spiral patterns, a surprising number of these plants have patterns in which each leaf, seed, or other structure follows the next at a particular angle related to the golden ratio. This "golden angle" is about 137.5º. Two radii of a circle C form the golden angle if they divide the circle into areas A and B such that A/B = B/C.

Mathematician Scott Hotton of Harvard has come up with a model that shows the forces underlying golden-angle spirals. Hotton's work, however, also supports the idea that patterns other than golden-angle spirals can form, such as those closely related to Lucas sequences or even odder ones that have a series of angles in a cycle. Further, Hotton and others are working on more general models explaining growth patterns. One, called the "coin model," would even supplant models based on the golden angle.

Source: Science News, July 21, 2007.