If you visit Arizona you can hardly fail to miss the impressive saguaro cacti that march up the foothills of the mountains near Tucson. So it's not totally surprising to find that a bunch of Arizona mathematicians have turned their minds to the beautiful patterns in cacti. In particular, how do the striking patterns of ribs arise? Patrick Shipman and Alan Newell, at the University of Arizona in Tucson, have provided an important part of the answer. It all depends on elasticity.
Since ancient times, people have noticed strange numerical patterns in plants. Marigolds have 13 petals, asters have 21, daisies have 34, 55, or 89, and sunflowers have 55, 89, or 144. These numbers have been familiar to mathematicians since 1202, when the Italian mathematician Leonardo of Pisa posed a problem about rabbits. His somewhat unrealistic rabbit population increased according to a fascinating sequence of numbers: 1 1 2 3 5 8 13 21 34 55 89 144 233 ... Each number is obtained by adding the previous two. Leonardo was the son of Bonaccio, and he later acquired the nickname Fibonacci. His numbers became known as Fibonacci numbers, and they turn up all over the plant kingdom. For instance, the hexagonal segments of pineapples form two interlocking families of spirals. One family winds anticlockwise and contains 8 spirals; the other winds clockwise and contains 13. The scales of pine-cones are similar, and so are the seeds in the head of a ripe sunflower. 19th Century botanists and mathematicians described the geometry of plants and figured out why Fibonacci numbers arise from it - but they couldn't explain the geometry itself. Moreover, some plants break the rules; for instance, fuchsias have four petals.
Botanical experiments showed that plant numerology is determined by small clumps of cells called primordia, which appear successively at the tip of a growing shoot. They space themselves along a tightly wound spiral at angles separated by about 137.5˚, called the golden angle. It has been known since the 1960s that this pattern packs the primordia together efficiently, and that it leads to Fibonacci numbers. But why do the primordia arrange themselves in this manner?
In 1992 Stéphane Douady and Yves Couder traced this to the dynamics of the growing shoot: the existing primordia push each new one into the appropriate position, and this makes the spacing efficient. Their model also explains the occasional occurrence of non-Fibonacci numbers, such as the four petals of the fuchsia. These come from a different solution of the dynamical equations, and follow another sequence, Lucas numbers: 1 3 4 7 11 18 29 47 76 123 ... Some cacti exhibit a pattern with 4 spirals in one direction and 7 in the other, or 11 in one direction and 18 in the other. A species of echinocactus has 29 ribs.
‘Push' suggests that forces are acting. So Shipman and Newell investigated the forces that push primordia around, using elasticity theory. Here, forces often cause materials to buckle into regular patterns. For example, if an elastic rod is compressed, it buckles into a series of equally spaced waves. They modeled the growth of primordia as a kind of buckling of the surface of the tip of the shoot.
A central principle in elasticity theory is that systems minimize their elastic energy. In this model it turns out that minimum-energy configurations are superposed patterns of parallel waves. Each pattern is characterized by its wave number, which determines the spacing between successive waves. In the cactus model, three such waves interact, and the wave number for the third wave must be the sum of the other two wave numbers. So the mathematical rule defining Fibonacci numbers corresponds directly to the mechanics of the buckling tip.
However, this is still not the full story. The appearance of primordia is not just a buckling effect: it is triggered by a hormone called auxin, which in turn depends on the genetics of the plant. Newell's group has now shown that similar wave patterns arise in the auxin distribution. So an explanation of the strange numerology of the plant kingdom involves a combination of biochemistry, mechanical forces, elastic buckling, and geometry.