![]() 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. All are fractions with fibonacci numbers, at least.P. Different plants have favored fractions, but they evidently don't read the books because I just computed fractions of 1/3 and 3/8 on a single apple stem, which is supposed to have a fraction of 2/5. So if the stems made three full circles to get a bud back where it started and generated eight buds getting there, the fraction is 3/8, with each bud 3/8 of a turn off its neighbor upstairs or downstairs. You can determine the fraction on your dormant stem by finding a bud directly above another one, then counting the number of full circles the stem went through to get there while generating buds in between. Eureka, the numbers in those fractions are fibonacci numbers! The amount of spiraling varies from plant to plant, with new leaves developing in some fraction-such as 2/5, 3/5, 3/8 or 8/13-of a spiral. The buds range up the stem in a spiral pattern, which kept each leaf out of the shadow of leaves just above it. To confirm this, bring in a leafless stem from some tree or shrub and look at its buds, where leaves were attached. Scales and bracts are modified leaves, and the spiral arrangements in pine cones and pineapples reflect the spiral growth habit of stems. Count the number of spirals and you'll find eight gradual, 13 moderate and 21 steeply rising ones. One set rises gradually, another moderately and the third steeply. Focus on one of the hexagonal scales near the fruit's midriff and you can pick out three spirals, each aligned to a different pair of opposing sides of the hexagon. I just counted 5 parallel spirals going in one direction and 8 parallel spirals going in the opposite direction on a Norway spruce cone. The number of spirals in either direction is a fibonacci number. Actually two spirals, running in opposite directions, with one rising steeply and the other gradually from the cone's base to its tip.Ĭount the number of spirals in each direction-a job made easier by dabbing the bracts along one line of each spiral with a colored marker. ![]() Look carefully and you'll notice that the bracts that make up the cone are arranged in a spiral. To see how it works in nature, go outside and find an intact pine cone (or any other cone). So the sequence, early on, is 1, 2, 3, 5, 8, 13, 21 and so on. Better known by his pen name, Fibonacci, he came up with a number sequence that keeps popping up throughout the plant kingdom, and the art world too.Ī fibonacci sequence is simple enough to generate: Starting with the number one, you merely add the previous two numbers in the sequence to generate the next one.
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