On Sunflower’s Golden Spirals

Photo credit: Esdras Calderan/wikipedia (CC BY 2.0) https://commons.wikimedia.org/wiki/File:Espiral_de_semillas_de_Girasol.jpg
I’d like to talk about patterns. In particular, the pattern that you might see if you look at the sunflower above.*
(Photo credit: Esdras Calderan/wikipedia (CC BY 2.0) https://commons.wikimedia.org/wiki/File:Espiral_de_semillas_de_Girasol.jpg )

Before you even think about it, I’m sure you can see spirals. Maybe you see them whirling clockwise, maybe counter-clockwise. Let your eyes refocus and another set of spirals will appear. They almost seem to pop out in your vision, but hang on; which is it? Are they going left or are they going right?

We have an incredible talent for picking out patterns out of noise; you can recognize a friend in a crowd or a familiar song over construction noises without thinking about it. Our sense for patterns is so sharp that we see faces in the moon or in potato chips, or shapes in the clouds. This is probably for the better; thinking you saw some food, some hidden danger, or even a friend where there is none is a lot safer than missing the one that actually is there, so I’d definitely take some silly crossed signals in exchange for this power of ours.

These are harmless examples, but there is a dark side. Gamblers see patterns in their wins and losses and make catastrophic bets. Con-artists exploit us, claiming to tell the future or read minds. Confirmation bias is a dangerous habit that has pervaded our political discourse, where we pick out evidence and patterns in data that suit our preferred answer. We don’t do this with ill-intent; it’s something our patterned-tuned brains do beyond our control. We can only fight it if we watch ourselves, think twice, and double check the news we forward it to our friends.

We also see patterns on another level; we find curious connections throughout the world, linking ideas that don’t seem related. Sometimes it looks like magic, others like design. Sometimes, it’s our minds searching for something that’s not there. As a scientist, this can be frustrating for me. I see articles about psychic powers and fake science, dangerous alternative medicine, and this prevailing tendency to make science mystical and unknowable. I think many people would be surprised as to how much they can understand with a little patience. We don’t need to scrutinize every detail in our experience, but I don’t like it when people assume that that is beyond them. Sometimes, with some care, the microscope lets us peel back the veil of nature and find the truth behind a pattern.


Photo credit: Max Ronnersjö/wikipedia (CC-BY-SA-3.0) https://commons.wikimedia.org/wiki/File:Aeonium_tabuliforme.jpg
The Fibonacci sequence and the Golden Ratio are patterns that pop up all the time in nature and in media. The Fibonacci sequence follows a simple rule; I start with the first two numbers, 1 and 1. If I add these numbers I get 2. If I add the 2nd and 3rd numbers (1 and 2) I get three. Add the 3rd and 4th I get 5, etc. The sequence looks like , etc. It sounds like the kind of thing a bored mathematician would do for fun, but it has a peculiar habit of showing up all over nature. Plants seem especially fond of it; you can see it in the arrangement of leaves on a stem, the scales of pineapples, and as it happens, the florets of a sunflower. If you go back to that first picture of a sunflower and counted the spirals in the seeds, you’d notice something interesting. I can pick out spirals at a bunch of different angles and directions, but the number is always a Fibonacci number.


Photo credit: Esdras Calderan/wikipedia (CC BY 2.0) Fibonacci Spirals added by Spencer Thomas

This is a peculiar quirk of the way these florets grow. The plant spirals out as it produces them, following a rule – each seed is some angle from the last. This angle happens to be a full  divided by , where  (the Greek letter ‘phi’) is the Golden Ratio, about equal to 1.618.

Like the Fibonacci sequence, the golden ratio appears everywhere in nature. People have known about this number for a very long time; the ancient Greek sculptor Phidias (400s BCE) worked it into much of his art. A quick google search will tell you how people have associated it with the ratios of beautiful faces, sections in pieces of music, etc. The ratio itself also has some neat properties, for example  (in fact  is sometimes likened to ’s little brother).

So what does  have to do with Fibonacci number? The two are intimately related. If I divide the 1st and 2nd Fibonacci numbers (1 and 1), I get 1. The 2nd and 3rd (1 and 2) give me 2, the 3rd and 4th give me 1.5, then 1.666…, then 1.6, etc. If I keep picking later and later Fibonacci numbers, I get closer and closer to . That’s that mystery solved, but why does a sunflower care? Sunflowers probably don’t know math, but they’re also not stupid. They’re carefully optimized by evolution to make the most out of what they’ve got; their mission is to fit as many seeds as possible onto their face. As a material scientist, I could tell you the very best way to do that looks like this:



Diagram provided by Spencer Thomas

It looks a lot like a honeycomb and that is no mistake. This is how bees achieve the same goal, but the sunflower kinda wrote itself into a corner. The spiraling mechanism that sunflowers use to grow can’t make a honeycomb; it’s terrible at making packed arrangements, always leaving some empty space. Instead of completely altering how the sunflower grows to solve this problem, evolution tuned it to do the very best with what it has, and  with its Fibonacci spirals happens to be the optimal turning angle.

It was shown by J.N. Ridley**  that this is the best possible way to pack seeds on a sunflower’s disc and this video is a beautiful demonstration of the idea. What it comes down to is that  is almost 21/34, and it’s almost 34/55, and almost almost 55/81, but these are all really bad estimates. By comparison, 22/7 is a pretty good estimate for Pi. You need really large numbers to get a ratio that’s close to, so a turning angle of  is a sunflower’s best hope at making the messiest spirals it can.

Give yourself some credit; that sunflower is doing everything it can to hide its spirals, but you can still see them clear as day!

Spencer Thomas recently received his PhD in Materials Science and Engineering from the University of Pennsylvania. He is now doing his Postdoc at North Carolina State University in Raleigh. He also happens to be Katie’s brother. Spencer studies metals at the atomic level; the way atoms are arranged in a material can change its properties; one can take ordinary metals make them stronger, more flexible, corrosion resistant, even radiation resistant.

Spencer believes that no matter who you are, good communication can put scientific concepts within reach. The modern world demands scientific literacy and it is the responsibility of scientists to make that possible.