How Are Fibonacci Numbers Expressed in Nature?

By: Robert Lamb & Jesslyn Shields  | 

Fibonacci sequence
The Fibonacci sequence floats over the Atlantic coastline under our home spiral galaxy, the Milky Way, to the South. shaunl/Getty Images

Is there a magic equation to the universe? Probably not, but there are some pretty common ones that we find over and over in the natural world. Take, for instance, the Fibonacci numbers — a sequence of numbers and a corresponding ratio that reflects various patterns found in nature, from the swirl of a pinecone's seeds to the curve of a nautilus shell to the twist of a hurricane.

Humans have probably known about this numerical sequence for millennia — it can be found in ancient Sanskrit texts — but in modern times we have associated it with one medieval man's obsession with rabbits.

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In 1202, Italian mathematician Leonardo Pisano (also known as Fibonacci, meaning "son of Bonacci") pondered the question: Given optimal conditions, how many pairs of rabbits can be produced from a single pair of rabbits in one year? This thought experiment dictates that the female rabbits always give birth to pairs, and each pair consists of one male and one female [source: Ghose].

Think about it: Two newborn rabbits are placed in a fenced-in yard and left to, well, breed like rabbits. Rabbits can't reproduce until they are at least 1 month old, so for the first month, only one pair remains. At the end of the second month, the female gives birth, leaving two pairs of rabbits. When month three rolls around, the original pair of rabbits produce yet another pair of newborns while their earlier offspring grow to adulthood. This leaves three pairs of rabbit, two of which will give birth to two more pairs the following month.

The order goes as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and on to infinity. The equation that describes it looks like this: Xn+2= Xn+1 + Xn. Basically, number is the sum of the previous two. This series of numbers is known as the Fibonacci numbers or the Fibonacci sequence. The ratio between the numbers in the Fibonacci sequence (1.6180339887498948482...) is frequently called the golden ratio or golden number.

Want to see how these fascinating numbers are expressed in nature? No need to visit your local pet store; all you have to do is look around you.

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The Golden Ratio in Nature

Romanesco cauliflower
Take a good look at this Roman cauliflower. Its spiral follows the Fibonacci sequence. Tuomas A. Lehtinen/Getty Images

While some plant seeds, petals and branches, etc. follow the Fibonacci sequence, it certainly doesn't reflect how all things grow in the natural world. And just because a series of numbers can be applied to an object, that doesn't necessarily imply there's any correlation between figures and reality. As with numerological superstitions such as famous people dying in sets of three, sometimes a coincidence is just a coincidence.

But while some would argue that the prevalence of the Fibonacci numbers in nature are exaggerated, they appear often enough to prove that they reflect some naturally occurring patterns. You can commonly spot these by studying the manner in which various plants grow. Here are a few examples:

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Seed heads, pinecones, fruits and vegetables: Look at the array of seeds in the center of a sunflower and you'll notice what looks like spiral patterns curving left and right. Amazingly, if you count these spirals, your total will be a Fibonacci number. Divide the spirals into those pointed left and right and you'll get two consecutive Fibonacci numbers. You can decipher spiral patterns in pinecones, pineapples and cauliflower that also reflect the Fibonacci sequence in this manner [source: Knott].

Flowers and branches: Some plants express the Fibonacci sequence in their growth points, the places where tree branches form or split. One trunk grows until it produces a branch, resulting in two growth points. The main trunk then produces another branch, resulting in three growth points. Then the trunk and the first branch produce two more growth points, bringing the total to five. This pattern continues, following the Fibonacci numbers. Additionally, if you count the number of petals on a flower, you'll often find the total to be one of the numbers in the Fibonacci sequence. For example, lilies and irises have three petals, buttercups and wild roses have five, delphiniums have eight petals and so on.

Honeybees: A honeybee colony consists of a queen, a few drones and lots of workers. The female bees (queens and workers) all have two parents, a drone and a queen. Drones, on the other hand, hatch from unfertilized eggs. This means they have only one parent. Therefore, Fibonacci numbers express a drone's family tree in that he has one parent, two grandparents, three great-grandparents and so forth [source: Knott].

fibonacci illustration
The golden ratio is expressed in spiraling shells. In the above illustration, areas of the shell's growth are mapped out in squares. If the two smallest squares have a width and height of 1, then the box below has measurements of 2. The other boxes measure 3, 5, 8, 13, 21, etc.
José Miguel Hernández/Getty Images

Storms: Storm systems like hurricanes and tornados often follow the Fibonacci sequence. Next time you see a hurricane spiraling on the weather radar, check out the unmistakable Fibonacci proportions of the spiral of clouds on the screen.

The human body: Take a good look at yourself in the mirror. You'll notice that most of your body parts follow the numbers one, two, three and five. You have one nose, two eyes, three segments to each limb and five fingers on each hand. The proportions and measurements of the human body can also be divided up in terms of the golden ratio. DNA molecules follow this sequence, measuring 34 angstroms long and 21 angstroms wide for each full cycle of the double helix.

Why do so many natural patterns reflect the Fibonacci sequence? Scientists have pondered the question for centuries. In some cases, the correlation may just be coincidence. In other situations, the ratio exists because that particular growth pattern evolved as the most effective. In plants, this may mean maximum exposure for light-hungry leaves or maximum seed arrangement.

Where there is less agreement is whether the Fibonacci sequence is expressed in art and architecture. Although some books say that the Great Pyramid and the Parthenon (as well as some of Leonardo da Vinci's paintings) were designed using the golden ratio, when this is tested, it's found to not be true [source: Markowsky].

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Originally Published: Jun 24, 2008

Lots More Information

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  • Anderson, Matt, et al. "The Fibonacci Series." 1999. (June 14, 2008) http://library.thinkquest.org/27890/mainIndex.html
  • "Fibonacci numbers." Britannica Online Encyclopedia. 2008. (June 14, 2008) http://www.britannica.com/eb/article-9034168/Fibonacci-numbers
  • "Fibonacci Numbers in Nature." World Mysteries. (June 14, 2008) http://www.world-mysteries.com/sci_17.htm
  • Caldwell, Chris. "Fibonacci Numbers." The Top Twenty. (June 14, 2008) http://primes.utm.edu/top20/page.php?id=39
  • Ghose, Tia. "What Is the Fibonacci Sequence?" Oct. 24, 2018 (Aug. 31, 2021) https://www.livescience.com/37470-fibonacci-sequence.html
  • Grist, Stan. "The Hidden Structure and Fibonacci Mathematics." StanGrist.com. 2001. (June 14, 2008) http://www.stangrist.com/fibonacci.htm
  • Knott, Ron. "Fibonacci Numbers in Nature." Ron Knott's Web Pages on Mathematics. March 28, 2008. (June 14, 2008) http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
  • Markowsky, George. "Misconceptions about the Golden Ratio." The College Mathematics Journal, Vol. 23, No. 1. Jan., 1992. (Aug. 31, 2021) https://www.goldennumber.net/wp-content/uploads/George-Markowsky-Golden-Ratio-Misconceptions-MAA.pdf

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