Why Is the Fibonacci Sequence So Common 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. 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 reflect various patterns found in nature, from the swirl of a pine cone'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.


In 1202, Italian mathematician Leonardo Pisano (also known as Leonardo Fibonacci, meaning "son of Bonacci") wondered how many rabbits could come from a single set of parents. More specifically, he posed 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 bear young 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 to a new pair, leaving two pairs total. When month three rolls around, the original pair of rabbits produces 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 for a total of five pairs of rabbits.

The first Fibonacci numbers go as follows: 0, 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, each integer is the sum of the preceding two numbers. This set of infinite sums is known as the Fibonacci series or the Fibonacci sequence. The ratio between the numbers in the Fibonacci sequence (1.6180339887498948482...) is frequently called the golden ratio or golden number. The ratios of successive Fibonacci numbers approach the golden ratio as the numbers approach infinity.

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.


The Golden Ratio in Nature

Romanesco cauliflower
Take a good look at this Roman cauliflower. Its spiral follows the Fibonacci series. 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 astonishing variety of objects, 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:


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 pine cones, 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) 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 Fibonacci spiral, also known as a golden spiral, is a visual expression of the golden ratio. In the illustration above, areas of the shell's growth are mapped out in an interesting pattern of squares that use only Fibonacci numbers. If the two smallest squares have a width and height of 1, then the box below has a measurement of 2. The other boxes represent squared numbers in the Fibonacci series
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 maximized seed arrangement.


Misconceptions About the Fibonacci Sequence

While experts agree that the Fibonacci sequence is common in nature, 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].

Mathematician George Markowsky pointed out that both the Parthenon and the Great Pyramid have parts that don't fit inside a golden rectangle or conform to the golden ratio, something left out by people determined to prove that the golden ratio exists in everything. The term "the golden mean" was used in ancient times to denote something that avoided access in either direction, and some people have conflated the golden mean with the golden ratio, which is a more recent term that came into existence in the 19th century.


Originally Published: Jun 24, 2008

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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