The Fibonacci Sequence As Seen in Flowers

What looks to us like a beautiful purple coneflower (echinacae purpurae)
We’ve all noticed beautiful patterns in nature: the clouds in the sky, rocks in a river, or waves in the ocean. Yet while some patterns seem random – the different colored flowers in a field, say – others are much less so – like the way the seeds are arranged in certain flower heads. In fact, it’s here that nature and mathematics intersect in a way that makes us wonder if everything in nature isn’t organized in a strictly mathematical way.
Here we’re going to be talking about the Fibonacci sequence in nature, with a little floral illustration to help guide us along. Follow us on a journey that will see us examine some ordinary flowers such that they suddenly appear extraordinary.

This ladybug doesn’t seem to mind the spiky qualities of the coneflower head it’s on. Or maybe it’s simply mesmerized by the flower’s symmetry!
The Fibonacci sequence (pronounced fibonarchee) got its name from a mathematician of the Middle Ages, Leonardo of Pisa (c. 1170 – c. 1250). The alias “Fibonacci” is most likely the shortened form of the Latin “filius Bonacci,” meaning “son of Bonaccio,” for his father’s name was Guglielmo Bonaccio.

Flower head with what looks like shades of the Fibonacci sequence
Though Leonardo was originally from Pisa, his father Guglielmo worked as kind of a customs official in what is today the Algerian town of Béjaïa. Through his schooling in North Africa, interaction with merchants, and travels around the Mediterranean, young Leonardo got acquainted with arithmetic and the HinduArabic number system. He soon found that the Arabic symbols 0 through 9 were far superior to – and userfriendlier than – the thencommonly used Roman numerals (I, V, X, and so on).

A common daisy with a green center? The amateur botanist’s view
So convinced was Fibonacci by the number system he had encountered that he helped to introduce it in Europe, even writing a book about it that was published in 1202. Liber Abaci (or “The Book of Calculating”) was a groundbreaking work that convinced quite a few of his European contemporaries and fellow mathematicians to switch to the ‘new’ decimal system.

White chrysanthemum with teethlike petals
So how does all this history relate to the pictures of pretty flowers shown here? Patience! Soon, all will be revealed.
In his book Liber Abaci – chapter 12, to be precise – Fibonacci describes the number sequence with which he would become so strongly associated. He used an example taken from nature – namely, pairs of rabbits procreating in a field to produce single sets of offspring that are also able to reproduce, after one month. Thus, starting with 1 pair of rabbits, in the next month you get 2 pairs (1+1), followed by 3 in the third month (2+1; remember, it’s still only the first pair that’s able to beget bunnies at this stage), then 5 (3+2), and so on.

Beautiful orange strawflower
In fact, Fibonacci most likely didn’t invent the sequence: he was very familiar with the arithmetic discourse of the time and probably took the ‘rabbit problem’ from somebody else. He did, however, popularize the problem and, more importantly, its accompanying series of numbers through his work.

Golden strawflower, the petals and center of which seem to suggest the Fibonacci sequence
This series, which is based on each new number being the sum of the two previous numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89), would eventually become well known in Europe. However, it wasn’t until the 19th century that the sequence would get its current name, coined by the French mathematician Edouard Lucas in honor of the erstwhile Fibonacci.

Ah, the sunflower! Loved by many for its bright appearance, its head also has florets whose patterns are popularly seen as an example of the Fibonnaci sequence in nature.
Looking at these sunflower florets, we can see a kind of spiraling pattern emerging. It’s amazing to learn that their display – or at least, the arrangement of the outermost florets – corresponds to the Fibonacci sequence. One researcher, H. Vogel, even proposed a mathematical model for the flower or seed patterns in sunflower heads, back in notquitesodistant 1979.

Chamomile photo with Fibonacci spirals highlighted
In the example of this yellow chamomile (actually another species in the sunflower family) we can see that the flowers’ disk florets are arranged in spirals following Fibonacci’s famous series. There are 21 darker blue spirals and 13 turquoise spirals. Ring a bell? Yes, 13 and 21 are consecutive numbers in the Fibonacci sequence.

Beauty up close: Seeds in the middle of a sunflower arranged in a wonderful spiral
Returning to the sunflower, we can see that all the seeds are packed as tightly as possible, with each at an angle to its neighbor to allow as many as possible to fit into the flower head. Also, the number of spirals (as seen before in the example of the yellow chamomile) typically corresponds to two numbers that follow each other in the Fibonacci sequence. Fascinating, isn’t it? But it gets even better.

Coneflower with ruby red center
Looking at this beautiful coneflower shot, we can’t help noticing the symmetry again – and this is where we need to talk about ratios. As we’ve learned, the pattern of the florets in flower heads like the sunflower’s isn’t random. And what’s more, spirals found in nature like these can be produced mathematically using Fibonacci ratios. Different plants exhibit different ratios: taking the example of leaves alternating up a plant stem, for example, one rotation of the spiral might touch two leaves, meaning the ratio could be 1/2.

Purple flower with yellow seeds
Here’s another example of how tightly Nature tries to pack her goods – squeezing as many seeds as possible onto a flower to guarantee maximum distribution. The purple and yellow color combination is simply divine, too!

This spoon chrysanthemum (talk about an apt name!) might seem to show signs of the Fibonacci spiral in its center.
But we digress. From here we’re going to move on to a key concept very closely related to the Fibonacci sequence: the golden ratio. You see, each number of the Fibonacci sequence divided by the previous number (for example, 2/1, 3/2, 5/3, 8/5, 13/8, etc.) will result in a “quotient” that, as the numbers increase, gets closer and closer to a “golden ratio” of approximately 1.6180339887. The proportions relating to this golden number have long been seen as being aesthetically pleasing, with the golden ratio applied to anything from the arts, music and architecture to the human body and nature. It’s no coincidence that angles approaching this golden mean are often evidenced in the growth of plants.

African beauty…
This beautiful and unusual flower is an Osteospermum whirligig or African daisy whirligig. To help safeguard its survival, the seeds in the center are packed as tightly as possible – economically following if not the Fibonacci sequence then at least some sort of spiral pattern, spirals being, the “lowestenergy configurations” from a physics point of view.

One can never look at too many purple coneflowers!
So, what do you think? Fascinating stuff, eh? What came first: nature as designer or science providing structure? It seems a bit like the chicken and the egg question. If only mathematicians had looked closely at nature earlier then it may not have taken so long before the Fibonacci sequence was discovered.
To see more stunning examples of the Fibonacci sequence in nature and learn more about the mathematics and the man behind it, follow the link. Meanwhile, we’re left scratching our heads after something of a math overload!