The mathematics of nature
Patterns in the forest and the field: Finding the Fibonacci sequence in Kawartha-area flora
Much of this article comes from my latest book My Cousin & Me: And Other Animals, a natural history memoir of two skinny boys chasing life in the hinterlands of Central Ontario. Scattered over these pages are 360 rare wildlife photographs of wolves, bears, fishers, deer, eagles, moths, moose, mice, and more. Enthralled by the glorious life all around them, these boys came to realize that all this beauty is the result of evolution by natural selection — Charles Darwin’s great idea.
Every day my cousin and I spent in aimless ambling over our hinterland farm was certain to reveal some treasure. Flowers were our particular fascination; perhaps it was innate for our grandmother was a master cultivator of domestic varieties. When you don’t have technology you can at least have beauty, as our species has had since the great cave paintings at Lascaux and Altamira. Our grandmother knew a thing or two about beauty.
Unlike her, we concentrated on wild varieties of flowers. We were young Pythagoreans for we loved to count. Oh, not as lovesick adolescents might with “she loves me, she loves me not” but just a straightforward enumeration of the flower’s petals. Let’s begin at the beginning. We discovered that every healthy variety of flower always has a certain number of petals, even if the number were exceedingly large. A wider study reveals this regularity also applies to leaves, branches, and seed spirals. Consider the following eight examples of flowers picked from the field of boyhood dreams (see Photograph One):
In the shadowy fields all around us was a design by nature that had evolved over millions of years. And it is this pattern in the petals I wish to investigate (see Photograph One): 1,1,2,3,5,8, 13, 21, 34, 55, . . .
The numbers we had stumbled upon as boys were none other than the famous Fibonacci sequence named for a medieval mathematician. And the secret for finding the next term in his sequence, which as boys we had overlooked, is marvelously simple: add the preceding two terms to get the next.
For example, 3 +5 = 8, 21 + 34 = 55, and so on.
With composite flowers, meaning those with multiple florets on their heads, you can, by concentrating on the spirals, distinguish two Fibonacci numbers (see Photograph Two). The head of this sunflower has 34 spirals going clockwise and 55 going the other way. Hence, at the edge you have 34+55 or 89 special florets we call petals. The 55 spirals are more easily seen near the edge of the head, the 34 nearer the center. Count them! Depending on the size of the particular seed head, composite flowers have different numbers of spirals; nevertheless, they are always neighboring pairs of numbers from the Fibonacci sequence.
The pairs commonly found on sunflowers are 21/34, 34/55, 55/89, and, occasionally on a giant, 89/144. These spiral patterns can also be seen on the heads of many other composite flowers.
Besides sunflowers, daisies, and pineapples, nature’s best example of the Fibonacci pattern is the pine cone (see Photograph Three). These range in size from the width of your fingernail to the length of your forearm. In the figure reproduced here, we have a cone on the left with its spirals outlined on the right: 8 solid black or white counterclockwise and 13 banded black and white clockwise. Larger or smaller cones can have different pairs of numbers, but they’re consistently adjacent numbers from the Fibonacci sequence.
Some readers will point out that I haven’t shown examples of flowers with 1 or 2 petals or leaves, the initial terms in the Fibonacci sequence. So consider the jackin-the-pulpit with 1 leaf that was found by my cousin and me in rare places on our farm. Apparently, jack-in-the-pulpit is poisonous— today we would also label it transgender because most years it alternates from male to female.
Beside the jack-in-the-pulpit is the delicate perennial Dutchman’s-breeches with its 2 petals found commonly in rich deciduous forests everywhere. Except for the color, they reminded me of the bleeding hearts my grandmother grew.
The woods are wide and full of wonders, but we boys were mere counters, nibblers, and sniffers at her mysteries. Just two skinny kids roaming fields like foxes searching for whatever we could find. Here a quartz rock, there an emerald snake, and over there a woodcock’s nest.
I have recounted how my cousin and I had stumbled upon some terms in the Fibonacci sequence by counting flower petals. As noted, however, we didn’t discover the simple rule for finding the next term by adding the preceding two. You could say, perhaps, that we had no vision of this famous sequence. But you would be wrong. Be it ever so humble we did have a theory of sorts—an idea about what the next term might be before we actually found it. We noticed something curious concerning these numbers: an even number (E) is “always” followed by two odd numbers (O). We were generalizing beyond the evidence. See for yourself: 1,1, 2,3, 5, 8, 13, 21,34,55,89... O,O, E, O, O, E, O, O, E, O, O. . .
On the occasions when we applied this knowledge of odds and evens, it was confirmed. In effect, we were predicting the future based on the past, something science has always done. Our little theory was as much a creation as a sonnet by a fledgling poet.
How does nature mold such order out of chaos? This is the question of the ages. In the city, you can see order everywhere, from roads to buildings to street signs, but in the fields and forest signs of order are deep and elusive. Yet a little inspection of nature’s fabric shows subtle and beautiful patterns. So what is the purpose of all these Fibonacci spirals and their numbers, and how did they come to be?
Approximately 98 per cent of all species that have ever existed are now extinct. And all these extinct species have one thing in common: they did not leave copies of their DNA, be they seeds, spores, eggs, or babies. As for the plants, a Fibonacci spiral is the best method to PACK SEEDS CLOSELY. There you have it: the explanation for this Fibonacci frenzy. It’s all a matter of maximizing the seed head through close packing.
The plant, of course, knows nothing of Fibonacci spirals or numbers. Evolution has put it on a close-packing quest: produce more seeds, have more progeny, be fruitful and multiply, or perish. When only a tiny portion of all your offspring (DNA) survives, it pays to produce hundreds, even thousands or tens of thousands, of copies. Consider the salmon and the dandelion, the sunflower and the pine cone, the daisy and the milkweed.
The world of the living tolerates slight variations, while larger ones are punishable by death. All creatures great and small, dance upon the edge of life. In every ecological niche, life hovers around a cluster of particular behaviors and bodily adaptations. The mechanism that brought about these adaptations is the master narrative of the 20th century. As philosopher Daniel Dennett says, “If I were to give an award for the single best idea anyone has ever had, I’d give it to Darwin, ahead of Newton or Einstein.” He wasn’t referring to evolution, which had been discussed since the time of the Greeks. Even Darwin’s grandfather, Erasmus, wrote about evolution—an idea as firmly established as the Earth orbiting around the sun. No, Darwin’s dangerous idea was the discovery of the mechanism for evolution, the force that causes descent with modification — the force that brings order out of chaos. We call it natural selection.
Observations verify that approximately 90 percent of all plants worldwide display Fibonacci numbers. This is a confirmation of the consistency my cousin and I had found. But in nature, multiple forces are always at work, so outcomes tend to be a little messy. What about the starflower with its seven petals and the four-leaf clover? What about the remaining 10 percent? That is still an enormous quantity of plant species.
Science is a journey not a destination. My cousin and I were not wrong in our observations because of these exceptions (starflower and four-leaf clover) just incomplete. To finish this investigation, however, would take us beyond the scope of this article. For those who are interested, I have explored this larger theory in a book titled Immortal Ideas: Shared by Art, Science, and Nature. Readers should consult the sixth chapter, “Numbers: Natural” for all the details.
The Fibonacci sequence is the grain in the stone, the path in the forest, the long sought after pattern for the consistency we wouldbe Pythagoreans had blundered upon in the dreamtime of our youth. And so we began to know this place for the first time.