Numbers and Nature

banana2.jpgDid you know that bananas have five sides?  Not sure?  Pick up a banana and count the sides.  Bet you there are five.

How many one- or two-petalled flowers have you ever seen?  Probably not many.  They are relatively rare in nature.  Flowers with three petals are more common, those with five petals more common still.  But flowers with four or six petals are few and far between.

What’s the deal?  The answer lies with Fibonacci.

Leonardo Fibonacci was born in Pisa, Italy, around 1170 and spent several years in Algeria with his father, a wealthy merchant.  Roman culture had spread widely in Europe by the Middle Ages and the Roman numeral system was commonly used for arithmetic.  While addition and subtraction are relatively easy with the system, anything more advanced — even multiplication or division — is difficult; the lack of zero poses a particular problem.  In Algeria, Fibonacci learned of the Hindu-Arabic numeral system and recognized the simplicity and efficiency of mathematics in this system compared to the Roman system.  He traveled throughout the Mediterranean, studying under many leading Arab mathematicians, and returned to Pisa around 1200.  The publication of his book Liber Abaci (Book of Calculation) two years later helped to popularize the Hindu-Arabic numeral system in Europe, becoming the numeral system we still use today.

In Liber Abaci, Fibonacci introduced a number sequence that solved a problem relating to the growth of a population of rabbits generation by generation assuming some idealized constraints.  This number sequence had been known to Indian mathematicians since the 6th century, but after publication of his book, it became known as the “Fibonacci sequence.”

In the Fibonacci sequence, each number is the sum of the two preceding numbers, starting with 0 and 1:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …

(In mathematical terms, we can write this as Fn = Fn-1 + Fn-2, with F0 = 0 and F1 = 1.)

The amazing thing about the Fibonacci sequence (aside from rabbit populations, of course) is that numbers in the sequence occur regularly in nature.  Look at your banana again.  Five sides – a Fibonacci number.  The most common flowers have 3, 5, 13, 21 petals – again, Fibonacci numbers.

In other cases, pairs of consecutive Fibonacci numbers determine the pattern of seeds in a sunflower, fruitlets on a pineapple, or scales on a pinecone.  Let’s look at the chamomile flower as an example.


To get the most compact arrangement, the florets are arranged in a spiral pattern, and — surprise! — the number of spirals corresponds to Fibonacci numbers!  Highlighted in turquoise in the picture are the florets spiraling counterclockwise.  Count the spirals, and you get 13.  Now count the number of spirals circling in the opposite direction.  Another Fibonacci number!

The next time you have a pinecone or pineapple in hand, look for the Fibonacci numbers.  (Hint: the pineapple has three.)  Where else do you find the Fibonacci numbers in nature?  We’d love to hear from you!

4 Responses to “Numbers and Nature”

  1. Bob Estes says:

    Nice article. But why are the Fibonacci numbers so prevalent in nature? I assume it has to do with the temporal details of how these organic units develop, but if I ever knew, I’ve forgotten. How about more on this?

  2. geo geller says:

    bob it might have something to do with the rotation of the earth – deb, in a conversation with benoit mandelbrot in snow storm documentary he talks about seeing equations geometrically – i thought that was so interesting i asked him if i could film him in a snow storm and few years later on april 1 2003 i think it was in a rare snow storm i filmed him talking about how he saw himself and how lucky he was too – food for imagination and maybe he secretly was sleeping with fibonacci –
    deb keep this coming – i love it –
    secret admirer

  3. Cuffy says:

    Fascinating and nicely done! Also, what Bob said: why?

  4. Debbie says:

    Fibonacci sequences happen in many seed and flower heads in nature. The reason, some suggest, is that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage of growth. All the seeds being the same size, no crowding in the centre and not too sparse at the edges. It seems to be like an “optimization” process to maximize the space for each leaf, or the average amount of light falling on each one. For leaves in cabbages, for example, availability of space is crucial for their development so it is natural that they find the optimzal arrangement.

    Nature isn’t trying to use the Fibonacci numbers: they appear as a by-product of a deeper physical process. The plant is responding to physical constraints. The explanation is linked to another famous number, the golden mean. For more information and references please feel free to check this out:

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