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Fibonacci Numbers in Nature

Fibonacci numbers in nature can be seen in pinecones

What do pine cones and the Parthenon have in common? How about an artichoke and a 3 x 5 index card? Consecutive numbers of the Fibonacci sequence are found in all of them. Before explaining, some background information is in order.

About Fibonacci Numbers

Leonardo Pisano, later known as Fibonacci, "son of Bonacci," was a brilliant mathematician, born in Italy about 1175 AD. He was the first to mention the following sequence of numbers that now bears his name: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ... Add the first two numbers together and they equal the third number. Add the second and third numbers and they equal the fourth, and so on, infinitely.

Fibonacci Numbers in Nature

Now, to explain the pine cones and Parthenon, artichoke and index card connection. Carefully look at a pine cone to notice that the scales grow in spirals. One set of spirals curves to the left, one set curves to the right. How many spirals are there? The cone I am looking at now has 13 spirals in one direction, 8 spirals in the other direction (both are Fibonacci numbers). The ratio of these two numbers is 13:8. Divided, these two numbers equal 1.62 which is very close to the "golden ratio" of 1.618. This same ratio is seen in the Parthenon, the dimensions of which form a "golden rectangle." Similar spirals and dimensions are seen in artichokes and index cards.

A golden rectangle is a rectangle with a shape that is particularly pleasing to the eye; it seems perfect, and it looks "just right." Golden rectangles always have dimensions that are Fibonacci numbers. They can be seen profusely in art and architecture: a beautiful painting, a well-proportioned car or building, or an especially nice looking piece of furniture. (A furniture maker that I met in Williamsburg said he uses golden rectangles in all of the pieces that he builds.)

It has been said that art imitates nature, and it appears that nature is where Fibonacci numbers have always been. In plants, the arrangement of seeds is frequently a Fibonacci number. We’ve already seen the pine cone and artichoke example; similar are pineapples. Growing in a comparable manner (with visible left and right spirals) are seeds on a sunflower, purple coneflower and daisy.

Plants are not the only example of this natural phenomenon; animals also exhibit Fibonacci numbers. To understand this fully we must first do some geometric constructions. Begin with a golden rectangle (width and length are Fibonacci numbers) and divide it into a perfect square and another golden rectangle. Keep dividing the rectangle into successively smaller squares and rectangles. Drawing an arc through each new square results in an equiangular spiral. Wild sheep horns, hurricane clouds, snails, and growing ferns all exhibit the equiangular spiral, which is seen abundantly in nature. This spiral has the distinctive characteristic of increasing in size while retaining the same shape. The need for this feature can best be seen in a growing mollusk, which grows larger but does not change shape.

Although not a rule of nature, Fibonacci numbers are an interesting tendency. Examples of these numbers and their corresponding ratios, rectangles and spirals are endless and are all around us. A closer look reveals them in unexpected places, from your home to the fields and forests of the Cleveland Metroparks.

For more information, read Fascinating Fibonaccis by Trudi Hammel Garland.

—Kathy Schmidt, Naturalist, Rocky River Nature Center