Once formed, the spirals tend to reinforce each other and suppress any other arrangement. Newell says the stable configurations have precisely three families of spiral waves, with spirals from all three families intersecting at each sticker. The pair then computed the buckling patterns that are most stable. To test the idea, mathematicians Patrick Shipman and Alan Newell of the University of Arizona in Tucson created a mathematical model of cactus growth that takes into account the elastic properties and stresses on the plant’s growing tip.

Although biologists have some experimental evidence for the theory, no one has shown exactly how a plant’s internal forces could generate the patterns. On a cactus, these hills become the locations for stickers. Where different sets of ridges intersect, they generate hills and valleys as they reinforce one another and cancel each other out. As the plant grows, the theory goes, the shell grows faster than the core, so spiral ridges form in the shell to accommodate the extra surface area, just as wrinkles form on skin when there is more skin than the flesh below requires. New leaves on a plant emerge from a rounded growing tip that consists of an outer shell covering a squishy core. One theory for these patterns is that they are driven by mechanics.

Other cacti, sunflowers, and pinecones display this or other triples of Fibonacci numbers. These are three consecutive numbers from the Fibonacci sequence. The round head of a cactus is covered with small bumps, each containing one pointy spike, or “sticker.” For some cacti, you can start at the center and “connect the dots” from each sticker to a nearest neighbor to create a spiral pattern containing 3, 5, or 8 branches. Now a mathematical model published in the 23 April PRL suggests that these spiral patterns, and the Fibonacci relationships among the spirals, arise out of simple mechanical forces acting on a growing plant. The intricate spiral patterns displayed in cacti, pinecones, sunflowers, and other plants often encode the famous Fibonacci sequence of numbers: 1, 1, 2, 3, 5, 8, …, in which each element is the sum of the two preceding numbers. The three sets of spirals have 3 branches (red), 5 branches (yellow), and 8 branches (brown)–three numbers that form a so-called Fibonacci triple. Computer simulations (bottom) can reproduce the spiral patterns in a cactus (top) by calculating the forces in the growing plant and finding the most stable arrangement.

Fibonacci spiral face