By Spencer Thomas, PhD
I’d like to talk about patterns. In particular, the pattern that you might see if you look at this sunflower.*
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.
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.
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:
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.
* As an aside -- I learned something else writing this article. The "flower" of a sunflower isn't actually a flower. Every one of those individual seed pod-looking things ("disc florets" clustered in the center, "ray florets" around the outside) is an individual flower. It's not terribly relevant here, but has made it a little tricky to talk about concisely! The whole head is called a capitulum.
It seems ray florets don't possess both male and female reproductive organs, but disc florets do, which means the disc florets can self-pollinate so the sunflower has some florets dedicated solely toward sexual reproduction (which is often considered healthier), while the disc florets can do both as needed.
Apparently, when people started figuring out how all this worked, it was considered a very scandalous line of inquiry! It's actually kind of interesting. Lots of flowers are capable of self-pollinating, but most of them only do it as a last ditch resort because diversity is good.
** Ridley, J. N. (1982). Packing efficiency in sunflower heads. Mathematical Biosciences, 58(1), 129-139.
By Spencer Thomas
Destin Sandlin, creator of the YouTube channel, "Smarter Every Day," is a rocketry engineer at the Redstone Arsenal. Arguably, his breakout video demonstrated the remarkable ability of chickens to keep their heads stable independent of their bodies. You can google “Inverted Pendulum” for an idea of how important a problem this is for engineers. His channel was a golden opportunity to get his children involved performing experiments and learning science by experience, while also supplementing their college funds. He also leveraged his success to help build an orphanage in Peru. His videos range from slow-motion videography of Prince Rupert’s Drops and explosions, to the mechanics of insect flight, to entomology adventures in South America.
In particular, I’d like to mention the recent Episode #182: Dominoes -- HARDCORE Mode. There’s amazing subtlety in something as simple and whimsical as a chain of falling dominoes and Destin captures it beautifully. Destin’s experience as an engineer sees a pattern, the signature of the corrections that rockets make mid-flight. Dominoes and rockets may seem unrelated, but no chain of dominoes is perfect and rockets have to fight through turbulence. We hope that our dominoes all fall in a row, and we seriously hope that our rockets don’t spiral out of control. Maybe dominoes can tell us something about stability, and studying dominoes might help us make better rockets.
This is where Destin makes an appeal for Basic Research. We don’t get some obvious economic benefit from understanding how dominoes fall, but we learn a lot in the process. There are many curiosities in the world and dozens of mysteries even in the most mundane aspects of daily life. For some, curiosity is enough. To others, these questions sound frivolous. However, we don’t always know what rewards we will reap when we empower thoughtful individuals to follow their noses and give them the freedom to explore.
History has shown that the rewards can be numerous, and sometimes fundamental. The transistor, the cornerstone of modern technology, would make no sense were it not for our understanding of quantum mechanics. The foundation of quantum mechanics was laid by scientists who were puzzled by the colors that objects glow when they get really hot (from red to white to blue). The eureka moment that solved that riddle changed everything about how we see the world at the smallest scales, and produced one of the most important technological revolutions of all time.
This is the foundation of science; dues that must be paid if we intend to advance. While the development of the transistor paid great dividends, it wasn’t remotely obvious that the specific color a hot poker glows would ever be understood as anything more than a novelty. Another example, we don’t study fruit flies because we want to develop medicine for fruit flies. We study fruit flies because their genetics are simple and easy to probe. As we improve our understanding of genetics in the abstract, we improve our capacity to provide treatments for people.
It is concerning that the US government share of basic research funding has fallen below 50% for the first time in the post-World War II era. While some of this is due to an increase in corporate investment, particularly on the part of the pharmaceutical industry, a significant part of it is because we as a nation are increasingly declining to contribute budget dollars. While the increase in private investment since 2012 is helpful, the findings of private and corporate investment are not as openly shared as public endeavors, including even basic data as to whether the research being conducted is actually basic or applied.
Basic research is not really conducive to business, at least not at the early stages that can have the greatest impact. The outcomes are too uncertain, desirable tangents are too frequent, and the timelines are too long. The risky and meandering path of basic research is often not good for business. It’s not just unrealistic to expect captains of industry to conduct this kind of research; it’s not really a fair expectation because people depend on them to ensure the bottom line and provide safe investments.
Occasionally, a singular individual arises like Elon Musk, who seems to regard profit as a means to innovation rather than the other way around. However, the ability, desire, and charisma required to make that work and bring people along is rare. People like him are important, but we cannot rely on them alone to address the issues that we face. Not only is his combination of qualities rare, we must also acknowledge that he's still very limited in what he alone will champion. The kinds of things he is trying to develop are still technologies with immediate practical application and relatively short-term monetary benefit.
This is not the pursuit for those who want to take advantage of the opportunities in the marketplace. This is the pursuit that creates those opportunities. We have to decide, as a society, if we wish to pursue them and how much we will invest. Philanthropists are important, but the truly altruistic are rare and, quite frankly, can't do everything we need alone. We still need public funding that supports the kinds of basic research that are only really feasible in universities and national labs.
MIT released a report in 2015 highlighting 15 research opportunities that could boost the US Economy. It also noted that while other nations are boasting great discoveries, our commitment has fallen from 10% of the national budget in 1968 to less than 4% in 2015. A 2014 article illustrated some of the extraordinary yields our past commitment enabled, from GPS, to the discovery of cancer cells and other medical breakthroughs, to LiquiGlide, which was named by TIME magazine as among the best inventions of 2012. From the article:
For more than 60 years, MIT and other American research universities have led the world in discovery and innovation—with benefits to the entire country—due to federal funding. This vital support, however, is now on the decline. In 1960, for example, 55 percent of MIT’s campus revenue came from federal research dollars. By 2013, it fell to 22 percent. Chisholm says the decline is disrupting the research process.
'Researchers are focusing on projects with a high probability of results, because these projects have a better chance of getting funded. What’s happening is faculty are doing safe things because they know they’ll work. They take fewer risks, but then the probability of discovering something really new and exciting goes down."
The challenges we face are great, and we will not meet them by hoping that great men will resolve them on the way to seeking their own fortunes. There are some endeavors that require us to come together and make investments into the pursuit of knowledge for the common good. Basic scientific research is one of them. From dominoes to rockets, from a quirk of light to the computer, we don't always know what we will find when we veer off the beaten path. It may seem like we’re merely taking the scenic route. However, we rarely find something truly new when we stick to the main road; the innovations that touch billions of lives lie in yet-undiscovered country.
Spencer Thomas, is a PhD candidate in Materials Science and Engineering at the University of Pennsylvania. He also happens to be Katie's brother. He studies metals at the atomic level; the way atoms are arranged in a material can change its properties. There are ways to take an ordinary metal and make it 10-100x stronger, but they return to normal over time by a process called grain growth. In his recent publication in Nature Communications, he develops a rudimentary theory, backed by simulations, for understanding the fundamental mechanisms of grain growth and what they mean for attempts to stabilize these materials. While we and many involved in the study of very small things are excited about that, here we look forward to sharing with you other things that stimulate his very sharp mind.