Nature at times can seem chaotic in its choice of shape and form, so we often use artistic principles to model organic objects.  However, as visual effects artists, we can also use math to help us recreate nature.

Have you ever taken a look at the bottom of a pinecone?  Or the center of the sunflower?  Or the top of a Romanesco cauliflower?  Looking at the placement of the scales, seeds, or florets in each of these plants, you should begin to see a series of spirals.

These spirals are not a coincidence.  It is simply the plant’s way of optimally packing new growth in a limited amount of space.  In 1868, the German botanist Wilhelm Hofmeister performed a study on shoot apical meristems, the tips of stems where new growth takes place.  Hofmeister found that these new growths, or primordia, appeared around the circular meristem one at a time in the least crowded spot.  In addition, the primordia were radially pushed out from the meristem as they matured.  As more and more primordia grew around the meristem, the angle between one new primordium and the next converged to the Golden Angle (about 137.5 degrees).  This pattern of placing objects radially from the center at 137.5 degree intervals creates the optical illusion of criss-crossing spirals. (http://www.math.smith.edu/phyllo//About/math.html)

137.5 seems like a rather arbitrary number, but it, in fact, has a geometric significance.  Given a circle, divide the circumference into two unequal parts such that the ratio between the length of the larger arc and the smaller arc is equal to the ratio between the circumference of the circle and the length of the larger arc.  The angle created by the smaller arc is the Golden Angle.  This ratio can also be described by the Fibonacci sequence.  Each number in the Fibonacci sequence is the sum of the previous two numbers (1, 1, 2, 3, 5, 8, 13…).  As we divide each number by next (5/8, 8/13,…), the ratio will converge to approximately 0.618, which in turn leads us back to the Golden Angle ([1-0.618]*360=137.5).  This is why the study of these types of spirals in plants is called Fibonacci phyllotaxis.

By studying some of the math and science behind the growth of plants, we can accurately model the naturally occurring patterns in pinecones, sunflowers, and Romanesco cauliflowers.  The next time you are tasked with modeling plants for film, television, or commercials, spend a little time researching the plant in addition to collecting reference photos.  You’ll never know what pattern you might find.

About:  Sarah McGee is a Pipeline Tools Developer here at Zoic Studios Los Angeles. She holds a Master’s degree in Entertainment Technology from Carnegie Mellon University, where she specialized in animation and visual effects. Previously, she earned her B.S. in Electrical and Computer Engineering from Rice University.