Our Window on Nature

. . . exploring the world around us

Nature By The Numbers

Filed under: Others,Plants — Lowell Christie -- November 1, 2015 @ 6:55 pm

PineconeAs a child, did you play counting games as you plucked the petals off a flower? One – she loves me, two – she loves me not. Did you know that the probabilities are high that you’ll end on an odd number? In fact, if I can identify the plant, I can almost guarantee the outcome.

Perhaps the originators of this pastime were just lucky. Or maybe they actually knew about the regularity of numbers in nature. Most flowers have an odd number of petals. In fact, in the majority of flowers, the petal total equals one of the numbers in a series found throughout nature.

This series is associated with a 13th-century Italian mathematician named Leonardo Fibonacci, from the Italian town of Pisa. Right – the one with the listing tower. Fibonacci introduced both Arabic numerals (such as 1, 2, 3, 4…) and the decimal system to Europe. Without them, we would still be counting with Roman numerals. I – she loves me, II – she loves me not….

Fibonacci is best known, however, for a number sequence he explained in his book Liber Abacci (Book of Calculation). The sequence led to a bunch of bad mathematical jokes about how fast rabbits can multiply. Rather than tell that story, let’s look at the numbers themselves. The Fibonacci sequence or series starts with the number 1 followed by another number 1. The rest of the sequence results when you add together the two previous numbers: 1 1 2 3 5 8 13 21 34 55 89 144 and on to infinity.

Something that appears singly or comes in pairs would be considered normal. Let’s begin looking at flower petals starting with the number 3. Consider irises and some lilies.

Too easy? Let’s try the 5 petals displayed by the buttercup, wild rose, larkspur, and columbine.

The next number in our series is 8. So are the number of petals on delphiniums and coreopsis. Ragwort and marigolds have 13 petals, and the aster, black-eyed Susan, and chicory have 21.

You have to count higher with daisies. Some have 55 petals, some 89, and – horrors! – some display the Fibonacci number 34. Why the concern? Thirty-four – she loves me not! Relax; not all Fibonacci numbers are odd, but most of those related to flower petals are.

If the Fibonacci series related only to flowers, the story would be complete, but we need to study the series more closely. Mathematicians have discovered many surprising features of the series, but we need only explore a few to find more Fibonacci in the world around us.

Divide one number in the series by the number that precedes it, and you’ll get an answer very close to 1.618; the higher in the series, the closer the result. This relationship, or ratio, turns out to be one of the most pleasing proportions, both to the eye and to the ear. It is called the Golden Ratio. Look at how often it is used.

First, in art. Leonardo da Vinci used the Golden Ratio in much of his work. So did Albrecht Durer, Georges Seurat, and Piet Mondrian. Look back to the construction of the pyramid of Giza in Egypt or the structure of the Parthenon in Greece to see it in architecture. It also appears in mathematics; search for Fibonacci on the Internet, and at least half of your responses will take you to mathematics-related sites. A study of one of our favorite composers, Mozart, showed that a large portion of his sonatas divide into two parts expressed by the Golden Ratio. Did I mention that Mozart was fascinated by mathematics?

The relationship also has been found in the music of Beethoven, Bartok, and Bach. Stradivari employed the series to decide where to place the f-holes in his priceless violins. Furthermore, look at a piano’s keyboard. An octave is made up of eight white keys and five black keys. Add them together and you get 13 – all Fibonacci numbers.

These relationships also occur in the sounds of music, whether man-made or music of the universe. The radio signals generated by pulsars arrive using the Fibonacci sequence.

Now, if we had more space and a lot more time, we might explain how the ratio 1.618 can turn into a logarithmic spiral. But the logarithmic spiral is our entry to the rest of Fibonacci in nature.

Look at a pinecone, and you’ll find that the bracts arrange themselves in two spirals traveling in opposite directions. Guess what? The curves of the spirals will be Fibonacci numbers; in fact, they will be neighboring Fibonacci numbers from the series. If you don’t have a pinecone handy, check the refrigerator for a fresh pineapple. Same spirals, and same Fibonacci relationship. (According to one source, if the number isn’t from the Fibonacci series, the pinecone is probably diseased or deformed.)

After you’ve finished looking at the pinecone, look up the trunk of the tree it came from. Whether you recognize it or not, a Fibonacci relationship probably exists in the spacing of the branches projecting out from the tree. Working your way around the tree, you’ll notice that the number of branches between any particular branch and the next branch that falls directly above it will be a Fibonacci number. The number of times you have to walk around the tree to check this out also will be a Fibonacci number.

The scientific name for this is spiral phyllotaxis. It’s not that the tree can count; it’s that this positioning allows each branch to receive the maximum amount of sunlight and moisture. Various species have a specific phyllotaxis, or relationship between the number of branches and the number of trips around the tree. The same relationship occurs in many flowering plants.

Consider the sunflower. Not only does it have spiral phyllotaxis, but it builds its seed heads using logarithmic spirals – two of them moving in opposite directions with the curves consisting of Fibonacci numbers. It has been estimated that as many as 90 percent of all plants show this spiral phyllotaxis pattern in the placement of their leaves.

Look at the spirals in a cactus, or search the convolutions of a cauliflower. Slice a banana into sections and look at its shape. Or cut through the core (around the equator) the next time you eat an apple. Nature seems to like Fibonacci numbers.


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