The swell of a wave right before it crashes, the complex whorls of flower petals, the distinctive lines of our fingerprints—Alan Newell studies all of them. A professor of mathematics at the University of Arizona in Tucson, Newell has revealed how simple mechanical and biochemical forces give rise to such a variety of natural shapes.
Newell also studies how certain kinds of spirals, called Fibonacci spirals, are widespread in nature, from the arrangement of leaves around a plant to the patterns on pinecones. Humans consciously or unconsciously mimic these forms in art and architecture. They show up in ancient stone carvings, Leonardo da Vinci’s paintings, and the proportions of the Greek Parthenon.
Newell spoke about Fibonacci spirals and plant patterns at the February 2011 meeting of the American Association for the Advancement of Science in Washington, D.C. Afterward, he chatted with SciCom's Sandeep Ravindran about how natural patterns form—and his striking hypothesis about how early humans might have perceived these designs and carved them into stone.
How did you start studying patterns in nature?
"When people ask me what I do, I tell them I’m a Renaissance scientist. If you tell them you’re a mathematician, they look at you like you’re a dentist or something."
After graduating from MIT with a graduate degree in applied mathematics, I worked on the turbulence of waves and light propagation and things like that. My first post-doctoral position was at UCLA, in a lab with a joint geophysics-mathematics connection. Pattern formation was just beginning to make some advances. I got involved in some of those early works on how to describe patterns mathematically.
Some of the problems you describe seem to stretch across many different fields. How do you manage that?
It wasn’t very hard. I never came from a background that emphasized field boundaries. In fact, if you think back to the old days, people were natural philosophers. Darwin, Hamilton, Maxwell, these guys were Renaissance scientists. They thought broadly. That’s very important in science, to see connections. And I enjoy this. When people ask me what I do, I tell them I’m a Renaissance scientist. If you tell them you’re a mathematician, they sort of look at you like you’re a dentist or something. So Renaissance scientist perks up people’s interests.
You’re showing that there’s a mathematical and physical underpinning to processes, not just in optics, but in all these biological forms. What has the reaction been from biologists?
I gave a talk in a pub the other day, and I said mathematics is like a good poem, which separates the superfluous from the essentials and then fuses the essentials into a kernel of truth. Smart people, no matter what their background is, appreciate good ideas. And some very good biologists I know like to be catholic in their views.
For a long time it’s true that the biological and zoological sciences worked descriptively: “Here’s a leaf, here’s a tree, and a tree is shaped this way because God made it so.” But I think biologists actually made a bit of a mistake. They started off being descriptive, then they went to everything [comes from] the gene, the genotype. That’s nonsense. If everything was determined by genes, the fingerprints of identical twins would be the same. But they’re not.
Why is that?
It’s like nature versus nurture. The genotype leads to a set of possible outcomes, and then somehow the influences and biases, the nurture part, come in and decide the particular configuration to take. Identical twins probably have fingerprints that are very alike in their overall structure, with a series of epidermal ridges on the tip of their fingers. But where the ridges end, and whether there’s a whorl or a loop in the middle of a finger, is random.
There’s been a back and forth between whether plant patterns form by mechanistic rules or by looking at what’s good for the plant. Where is the balance at the moment?
It’s still very open. The observation that most plants have these special patterns on them goes back to [Johannes] Kepler. Kepler was a very religious man, and he saw the hand of God in these observations. People in those days [in the 17th century] associated the appearance of order with some grand design.
In pioneering work in the early 1990s, Yves Couder and his colleague had this wonderful experiment using oil droplets in a magnetic field. Lo and behold, they saw lovely Fibonacci spirals. So they recognized that maybe the kind of patterns we see in plants was sort of universal behavior, and they took the point of view that it could be governed by discrete algorithms. That’s kind of like God having a little computer in the flower and saying, “Put the next one here.”
The truth may lie somewhere in between, but I think the mechanistic viewpoint is the right one. I think actual biochemical, physical, and mechanical processes determine what happens, and it so happens that it turns out to be good for the plant.
In your lecture, you talked a bit about how you can weave together mechanistic and teleological [designed pattern] explanations. I thought that was very interesting.
Well it’s interesting to me too, because why should there be a connection? It’s a question like, “Why is mathematics? Why does this seem like such a natural language to describe the Universe?” Optimal packing—how you fit things together in the best way—is a very challenging mathematical problem. So it’s quite amazing that the mechanistic viewpoint leads to pattern formation in which the places you put these flowers is the same as if God was designing them. The surprising number of deep connections from mathematics and physics continues to astound me.
What about the Fibonacci sequence? It shows up all over the natural world.
The Fibonacci sequence is defined as being a sequence in which the next number is the sum of the last two. So pick any two numbers, [such as] 1 and 7. You add 1 and 7, you get 8. Then you add 7 and 8 and you get 15. 1, 7, 8, 15, etc.—that’s a Fibonacci sequence. The regular Fibonacci sequence is the one where you start with 1, 1. So you have 1, 1, 2, 3, 5, 8, 13, etc. For some reason most [natural] systems start with 1, 1, so the Fibonacci sequence you see most often in nature is the regular one.
In theory you could get any Fibonacci sequence in nature?
You can. In certain pine cones, 84% of the time you’ll see the regular Fibonacci sequence. 12% of the time you see the double Fibonacci sequence, which is gotten just by doubling the regular one, 2, 2, 4, 6, 10, etc. And 4% of the time you get what’s called the Lucas sequence—1, 3, 4, 7, 11, 18, and so on. It really depends on how that flower or how that cone starts off. But the properties are all the same.
One of the interesting features about the regular Fibonacci sequence is that if you were putting down flowers around a circle, the angle between successive flowers is a golden angle, 137 degrees. And in all the Fibonacci sequences, the ratio of successive terms rapidly approaches the most irrational number, the golden number. [This ratio, also known as "phi," rounds off to 1.618.]
Is the golden number the same as the divine proportion?
Yes, exactly. People seem to like that divine proportion. It turns up in architecture. They like to design buildings with this ratio of the length of a room to its height. Why it turns up in that, what’s the physiology and psychology, I don’t understand.
Are we aesthetically drawn toward some of these natural patterns?
It’s an idea that fascinates me. I suspect there is some truth to it, but I’m not an expert enough to say. I do have a crazy theory to tell you about, though, that is rather novel, on megalithic art. If you look at all the etchings on stones, usually connected with burial places in Ireland and Britain and Europe, there’s tremendous similarity. There are chevrons and spirals, and the crests between neighboring spirals are on a common wavelength, a preferred scale. Now, is this a pattern-forming system? Well, obviously not. The rock doesn’t buckle into these patterns. Somebody drew them on.
Why did they draw them on? These guys, back in those days, like we do on Saturday nights, had a few beers. And they’d get together and chat. But they didn’t probably drink beer, even though it was around then. What they did mostly was chew weed. And many weeds make you feel good because they’re slightly hallucinogenic. Now, if you take a hallucinogenic drug like LSD, it lowers the inhibition on the visual cortex, so natural patterns start rising. I think that spirals just happen naturally on the visual cortex, and then after about 15 seconds, you start dreaming weird dreams, personal dreams like your granny chasing you with an axe. But for the first 15 seconds, and this is done experimentally, people are asked what they think they see. They start drawing common themes. They draw spirals, chevrons—exactly the kinds of things you see in these megalithic art curbstones.
So, I think megalithic art is a reflection of the first patterns you see on the visual cortex in the human brain when inhibitions are removed.
If those correspond to patterns in nature, that would explain a lot.
Yes. But that’s not science. You have to do the statistics of all the patterns you see in megalithic art, how they correlate with what’s seen in the visual cortex, and then perhaps you can draw some inferences. I’ve been too lazy, but I’m trying to get some anthropologist friends interested. When you get to my age you can have lots of crazy theories. I have a theory of quarks and leptons too, but that’s for another day.
Do you have a favorite pattern?
No. They’re all good guys.
Your website mentions that you have a manuscript in preparation about the universal nature of Fibonacci patterns.
Yes, there’s one out, but I’m not happy with it. It’s sort of scratching at the surface. I feel like there’s a lot more of the woods, but we’re still bumping into the trees. We don’t see the big picture at all yet.
Do you envision that at some point we’ll be able to figure out the universal patterns of everything?
No. I’m not one of these believers in the theory of everything. I leave that to the high-energy physicists. I do think we’ll gradually learn more and more, and we’ll see things as part of a bigger and bigger picture.
Sandeep Ravindran, a 2011 graduate of the Science Communication Program at UC Santa Cruz, earned his bachelor's degree in biology from Cornell University and his Ph.D. in microbiology and immunology from Stanford University. He has worked as a reporting intern at the Salinas Californian, the Stanford University News Service, and the San Jose Mercury News. He is now a science writer for the Proceedings of the National Academy of Sciences in Washington, D.C.
© 2011 Sandeep Ravindran