Do plants know math? (Horticulture).

For more than three centuries, botanists and mathematicians have marveled at the complex and beautiful spiral patterns that form as plants develop. As they generate leaves around a stem, or seeds or flowers in a blossom, plants as diverse as broccoli, pinecones, artichokes, and water lilies create intricate spirals that follow a well-known mathematical sequence of numbers.

"A prominent theme in science today, whether in biology, math, or computer science, is the generation of complex patterns through simple rules," explains Chris Gole, professor of mathematics, Smith College, Northampton, Mass. "Plants have been succeeding at this throughout their evolution, with great consistency and visual beauty."

To highlight the mathematical underpinnings of phyllotaxis, which refers to the arrangement of leaves or other botanical elements around a stem, Gole and Pau Atela, associate professor of mathematics, teamed up with Michael Marcotrigiano and Madelaine Zadik of the Smith College Botanic Garden to produce an exhibition, "Plant Spirals: Beauty You Can Count On" (www.math.smith.edu/phyllo/expo), that depicts with rare beauty and clarity the geometry and biology of plant spiral formation.

Through a computer animation, scanning electron micrographs, and large-scale panels featuring vivid color images and historical contexts, the viewer is guided from the natural phenomena to a recently developed simple mathematical model that reproduces the spiral patterns seen in plants. The model is based on a branch of mathematics known as dynamical systems, which includes chaos theory.

Gole notes that plant spirals often form according to the Fibonacci Sequence (1, 1, 2, 3, 5, 8, 13 ...), in which each digit is the sum of the previous two. The spiral helixes visible in plants usually come in two sets winding in opposite directions. The numbers of spirals are most often two consecutive Fibonacci numbers. The flower of an English daisy, pictured in the exhibition, consists of 21 spirals clockwise and 34 counterclockwise. A pinecone pictured has eight spirals in one direction and 13 in the other, eight and 13 being successors in the Fibonacci Sequences.

To understand why Fibonacci numbers predominate in spiral plants, Gole and Atela started with the theories of 19th-century botanist Wilhelm Hofmeister, who observed that a plant's leaves emerge at the least-crowded spot around a circular meristem, or growing tip. They are then radially displaced from the center. The dynamical systems model developed by Atela, Gole, and Scott Hotton of Miami University, based on the recent work of French physicists Stephane Douady and Yves Couder, suggests that those simple geometric rules are enough to produce the spiral patterns with Fibonacci numbers that appear in nature.

"Whether plants 'know' mathematics or not," Atela remarks, "they are clearly programmed to follow a common set of developmental rules, which would suggest that these patterns confer evolutionary advantage."

"That the intricate patterns in nature, such as a leaf arrangement, leopard spots, and butterfly wings, may all be regulated by simple and related mechanisms is an intriguing concept," Marcotrigiano adds.