# Fibonacci and the Mathematics of Nature

By James R. Coffey, 9th Jun 2011 | Follow this author
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In the late 12th century, Leonardo Fibonacci encountered the Hindu-Arabic numeral system and introduced it to Europe. In doing so, he discovered a sequence of numbers that occurs so frequently in nature that it is often referred to as a "law of nature.” We know it today as the Fibonacci Sequence.

- Leonardo Fibonacci and Hindu-Arabic numerals
- The Fibonacci number and nature
- The sequence . . . and the "golden number"
- The birds and bees and uncurling ferns
- Bees do it . . .
- Even daisies do it
- Buttercups, lillies, and black-eyed Susans
- Pinecones, sunflowers, and pineapples

## Leonardo Fibonacci and Hindu-Arabic numerals

Leonardo Fibonacci was born around 1170 to Guglielmo Fibonacci, a wealthy Italian merchant who directed a trading post in Bugia, a port east of Algiers in North Africa (now Bejaia, Algeria). As a young boy, Leonardo traveled with his father during his business ventures, subsequently encountering the Hindu-Arabic numeral system.

Recognizing that performing arithmetic with Hindu-Arabic numerals--which was based on the digits 0 through 9--is infinitely simpler and more efficient than working with Roman numerals, Fibonacci spent the next several years traveling throughout the Mediterranean studying under several prominent Arab mathematicians of the time.

Returning to Italy around the year 1200 at the age of about thirty, he published what he had learned in a text called *Liber Abaci * (Book of Abacus or Book of Calculation) about two years later, thus popularizing Hindu-Arabic numerals in Europe.

In the *Liber Abaci *, Fibonacci introduced the so-called *modus Indorum* (method of the Indians), and showed the practicality of this new numeral system, using lattice multiplication and Egyptian fractions, applying it to commercial bookkeeping, conversion of weights and measures, the calculation of interest, money-changing, and other everyday applications. Subsequently, *Liber Abaci * was well received throughout educated Europe and went on to have a profound impact on European thought. But Fibonacci discovered some other interesting applications of this new math.

## The Fibonacci number and nature

In *Liber Abaci*, Fibonacci posed and solved a specially chosen problem involving the hypothetical growth of a population of rabbits based on “idealized assumptions” (which, for the sake of the problem, include rabbits that never die), posing the following scenario: a newly-born pair of rabbits, one male, one female, are put into a field. In that rabbits are able to reproduce at the age of about one month, at the end of the first month, they mate, but there is still only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. So, the question becomes, "How many rabbits will there be at the end of a year? Or ten years?"

Following this progression indefinitely, we see that at the end of the “nth” month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month. This is the nth * Fibonacci number*.

## The sequence . . . and the "golden number"

Known to East Indian mathematicians as early as the 6th century CE, the first two * Fibonacci numbers * are 0 and 1, with each subsequent number the sum of the previous two.

Thus, this sequence begins 0, 1, 1--and then continues--2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 . . . In mathematical terms, this sequence of numbers is defined by this recurring numerical relationship.

But where it gets interesting is that the higher up in the sequence the numbers go, the closer two consecutive "

*" when divided by each other approaches what is known as “the golden ratio” or “golden number” (approximately 1 : 1.618 or 0.618 : 1).*

**Fibonacci numbers**Okay, a pretty cool concept for mathematicians. But how does this apply to nature in general?

## The birds and bees and uncurling ferns

As Fibonacci discovered, this sequence of number appears in countless biological natural settings such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of an artichoke, an uncurling fern, and the petal arrangement of a pine cone. And along with the breeding of rabbits, the spirals of shells, and the curve of waves, even the family tree of honeybees follows the pattern of * Fibonacci numbers*.

## Bees do it . . .

If an egg is laid by an unmated female bee, it hatches a male or drone bee. If, however, an egg was fertilized by a male, it hatches a female. Thus, a male bee will always have one parent, while a female bee will always have two. Thus, if one traces the ancestry of any male bee we see that he has 1 female parent (1 bee). Any female bee had 2 parents, a male and a female (2 bees). The female had two parents, a male and a female, and the male had one female (3 bees). Those two females each had two parents, and the male had one (5 bees). This sequence of numbers of parents begins the * Fibonacci sequence*.

Likewise, if we examine the close-packed arrangement of tiny florets in the core of a common daisy blossom, we see the

*phenomenon: twenty-one florets going counterclockwise, and thirty-four logarithmic or equiangular spirals going counterclockwise. In any daisy, the combination of counterclockwise and clockwise spirals generally consists of successive terms of the Fibonacci sequence.*

**Fibonacci sequence**## Even daisies do it

Likewise, if we examine the close-packed arrangement of tiny florets in the core of a common daisy blossom, we see the * Fibonacci sequence* phenomenon: twenty-one florets going counterclockwise, and thirty-four logarithmic or equiangular spirals going counterclockwise. In any daisy, the combination of counterclockwise and clockwise spirals generally consists of successive terms of the Fibonacci sequence.

## Buttercups, lillies, and black-eyed Susans

With further observation, we see that with many plants the number of petals is also a * Fibonacci number*: buttercups have 5 petals; lilies and iris have 3 petals; many delphiniums have 8; ragwort, corn marigold, cineraria, and some daisies have 13 petals; many aster, black-eyed Susan, and chicory have 21, daisies can be found with 34, 55 or even 89 petals, plantain and pyrethrum have 34 petals, michaelmas daisies and the asteraceae family all have 55 or 89 petals--Fibonacci numbers.

## Pinecones, sunflowers, and pineapples

With the scale patterns of pinecones, the seed patterns of sunflowers, and even the bumps on pineapples, we have something rather different--but still in keeping with this concept. The seed-bearing scales of a pinecone are really modified leaves, crowded together and in contact with a short stem. Here we do not find phyllotaxis (leaf arrangement) as it occurs with true leaves, but we do find two prominent arrangements of ascending spirals growing outward from the point where they are attached to the branch; 8 spirals can be seen ascending up the cone in a clockwise direction, while 13 spirals ascend more steeply in a counterclockwise direction.

Similarly, pineapple scales are also patterned into spirals, and because they are roughly hexagonal in shape, 3 distinct sets of spirals may be observed: one set of 5 spirals ascends at a shallow angle to the right, a second set of 8 spirals ascends more steeply to the left, and the third set of 13 spirals ascends very steeply to the right.

This sequence occurs so frequently in nature that it is often referred to as a "law of nature.” There have been thousands of documented examlpes of * Fabonacci's sequence *discovered in nature. It's actually a bit unnerving when you consider the probability of such a common, predictable occurrence in what often appears to be patternless chaos.

*:*

**References**http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits

http://mathworld.wolfram.com/FibonacciNumber.html

http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

http://www1.math.american.edu/newstudents/shared/puzzles/fibbee.html

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## Comments

Mark Gordon Brown

9th Jun 2011 (#)

best, most interesting, math lesson I have ever had.

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James R. Coffey

9th Jun 2011 (#)

Cool! Glad I could provide it!

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The Phoinix Φοινιξ

9th Jun 2011 (#)

THANKS FOR SHARING :)

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Delicia Powers

9th Jun 2011 (#)

James, I failed math, twice, but even I can see the beauty within this article and the pattern of life that you explain so well, well done...!!! thank you.

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James R. Coffey

9th Jun 2011 (#)

You're quite welcome, Delicia.

And you're quite welcome as well, Phoinix.

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Retired

12th Jun 2011 (#)

A first-rate article, both fascinating and marvelously entertaining. Thanks!

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James R. Coffey

12th Jun 2011 (#)

And thank you. Mike!

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Retired

13th Jun 2011 (#)

as always great to read, wonderful to learn and you've made it fun, I can't thank you enough.

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James R. Coffey

13th Jun 2011 (#)

My pleasure as always, Rebecca.

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Tranquilpen

17th Jun 2011 (#)

Hello James. An excellent article thank you my friend, Im always fascinated by your stuff. P.s even a tough/tuff little critter living on the bottom of the Indian Ocean,at a depth of 2400 meters, called a scaly- foot snail, does it, regards: Andre'

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James R. Coffey

17th Jun 2011 (#)

Indeed, Andre. Thanks!

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Retired

28th Jun 2011 (#)

Superb and very interesting. Super photos too. Bravo for another star!

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James R. Coffey

29th Jun 2011 (#)

Thanks! Nature loves to pose!

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Ammu

25th Mar 2013 (#)

it's unbelievable that nature also involved in math.......

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Johnny Knox

8th Jul 2013 (#)

Magnificent article and read! Thanks for sharing James!

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