The Golden Ratio: Is It Myth or Math?

Here’s a sanitized version of the provided YouTube transcript:

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A number so perfect that we find it everywhere—sacred geometry, a mathematical property hardwired into nature. Secrets of the golden ratio—what’s the answer? Is there actually one special number that underlies everything, from sunflowers to seashells, from pineapples and pinecones to the pyramids and the Parthenon? A number that links beauty in art, music, and the human body? One number that connects nature’s order to the rules of mathematics? Some people think so, but as Uncle Carl says, extraordinary claims require extraordinary evidence. So let’s take a closer look at what the golden ratio is really about.

Hey smart people, Joe here! Which of these rectangles is the most perfect? Give them a look. Which one feels the most balanced, the most beautiful? Did you pick this one? That’s a golden rectangle, and many believe that this shape is the most aesthetically pleasing quadrilateral. This one? Not so much. The golden rectangle’s ratio of its long side to its short side is exactly this—this is the golden ratio, abbreviated as phi (φ). These numbers after the decimal point go on forever without repeating. Like the better-known pi, phi is an irrational number.

It’s irrational because it can’t be written as the ratio of two integers. For example, the number five is rational because we can write it as the integer five over the integer one. The number 0.5 is also rational because we can write it as 3 over 4. Even 0.3333 (infinitely repeating) is rational because it can be written as 1 over 3. But what about the diagonal of a square whose sides are one unit long? The Pythagorean theorem tells us that the diagonal has a length of the square root of two, which is a number that can’t be written as a ratio of two nice and tidy integers—it’s irrational. Likewise, phi also can’t be written as a simple integer ratio.

An ancient Greek named Euclid was one of the first to notice this. Euclid was big into geometry; in fact, most of the geometry we learn in school is named after him. Around 300 BC, Euclid wrote a book called “Elements,” a collection of most of what was known about math at the time. Until the 20th century, it was the best-selling book ever, other than the Bible. He noticed a special way to divide a line where the ratio of the whole to the longer segment was the same as the ratio of the longer segment to the shorter one, and that ratio is phi. Euclid called it the extreme and mean ratio. The names phi and golden ratio didn’t show up until almost the 20th century.

The Greeks and mathematicians of that time didn’t think of numbers like we do—as strings of digits from zero to nine. To them, phi was this ratio, just like to them, pi wasn’t 3.14159, etc. Pi was just the ratio of a circle’s circumference to its diameter. This is literally the golden ratio, and you can do some interesting things with it. The ratio of the long sides of this triangle to its short side is, you guessed it, phi. This is a golden triangle, also called a sublime triangle, and the angles of that triangle are 72, 72, and 36 degrees.

If I divide one of the long sides according to the golden ratio and make a smaller triangle, it’s another golden triangle—same angles and all. The length of these sides to the base is one over phi. We call this shape the golden gnomon. If I take one golden triangle and attach two golden gnomons on the side, I get a regular pentagon.

Let’s overlap two golden gnomons and add a smaller golden triangle on the side to make a pentagram. Going back to our golden rectangle, if we put another golden rectangle here, another inside that, and so on, and draw a curve through all these shapes, we get a shape called the golden spiral. If that looks familiar, it’s probably because you’ve seen an image like this before on the internet.

There’s even more strangeness: if you multiply phi times itself, that’s the same as one plus phi. Take one over phi, and that’s the same as phi minus one. This is a weird number. So what? Phi is weird. There are infinite numbers, so some of them are going to be a little strange. What makes phi special is that it shows up in a bunch of unexpected places that are pretty far off from geometry class—or at least people claim to find phi in a lot of unexpected places.

This is the interesting thing about phi: as cool of a number as it is on its own, it’s achieved this almost mythological status. Many people say that because we find it in so many places, it can’t just be a coincidence; it must be a sign of some deeper secret about the universe.

Where does the real story of phi end and the myth begin? If there’s one person responsible for the mythological status of phi, it’s Leonardo of Pisa, aka Fibonacci. Around the year 1200, Fibonacci was responsible for bringing Hindu-Arabic numerals into common use across Europe. These are the numerals we use today—zero through nine. Merchants quickly realized that doing arithmetic with these was much easier than with Roman numerals, which were in use at the time.

To teach people how to use these new numbers, Fibonacci wrote a math textbook called “Liber Abaci,” which means “The Book of Calculation.” This book was full of math problems to teach people how to add, exchange currencies, and divide and multiply with these new numbers. Tucked inside chapter 12 was a problem about rabbits. Imagine you have a pair of rabbits in a field—one male and one female. Starting from the second month she’s alive, every female reproduces each month, making a new pair of rabbits—one male and one female.

How many rabbits will they produce after one year? You can pause and work it out if you want, but it ends up looking like this. You might notice something special about the number of pairs of rabbits each month: the number of pairs is equal to the sum of the previous two months. After 12 months, you’d have 144 pairs. This is the famous Fibonacci sequence. You can carry it on forever—just add the previous two numbers to get the next, and so on until the end of the universe or until you get bored.

The reason we’re talking about the Fibonacci sequence in a video about the golden ratio is that as you go on in the Fibonacci sequence, the ratio between numbers gets closer and closer to phi. In fact, any sequence of numbers that follows the Fibonacci rule—adding the two previous to get the next—trends to phi. The Lucas numbers follow the pattern, and the difference between the terms all trends to phi.

I know it’s weird, but Fibonacci never made that connection himself. A guy named Johannes Kepler did a few hundred years later—the same Kepler who figured out the math that explains how planets move. It’s after that, when the Fibonacci sequence and phi got together, that the myth really took off, and people started to claim these numbers were more than just numbers.

Despite phi’s seemingly mystical mathematical origins, the truly mind-boggling aspect of phi is its role as a fundamental building block in nature. Plants, animals, and even human beings all possess dimensional properties that adhere with eerie exactitude to the ratio of phi to one. Phi’s ubiquity in nature clearly exceeds coincidence.

But does phi, the golden ratio, the divine proportion—whatever grand name you want to give it—really show up everywhere in nature, or is it our pattern-sensing brains making us think that we see it everywhere?

Let’s look at some places people claim to see phi. The human body is said to have an ideal ratio of a person’s height to the distance from their navel to their feet as phi. Beauty standards in different cultures vary a lot, and people come in too many shapes and sizes for that to be a rule.

The Great Pyramid of Giza, the Parthenon, Notre Dame Cathedral in Paris, the Taj Mahal—a handful of ancient buildings that people claim were built with golden ratio dimensions. The thing is, for an object that’s fairly big, like a building or complex like a body, there are so many ways to measure it and so many measurements you can take that some are bound to be somewhere around a golden ratio apart from each other.

For example, this video is a 16 by 9 aspect ratio—that’s pretty close to 1.6, but it’s not phi. 16 by 10 would be even closer. Lots of places that people claim to see phi in nature are just plain wrong.

For instance, the ratio of one turn of a DNA helix to its width is often cited as being 34 angstroms high per turn and 21 angstroms wide—both Fibonacci numbers. Unfortunately, that’s incorrect.

Here’s the key thing: in any example that you find, phi isn’t approximately 1.6, give or take—it’s exactly this. If people go measuring things looking for the golden ratio, they often measure them in ways that ensure they find the golden ratio. Our brains love patterns, and once we learn a pattern, like the usual arrangement of a mouth, a nose, and two eyes to make a face, we see that pattern everywhere.

This brings us to the nautilus shell. A nautilus is a mollusk that swims around with a spiral shell. It’s become the official mascot of the golden ratio in nature. The claim is that if you trace the spiral of this shell, each ring is a golden ratio away from the next smallest ring. However, people have measured many nautilus shells, and they aren’t golden spirals. The ratios vary quite a bit, like snail shells or sheep horns.

It’s an example of what’s called a logarithmic spiral. Each turn of the spiral grows by the same proportion because the nautilus grows at the same rate, but that proportion isn’t phi. That’s too bad because logarithmic spirals are cool, but no one pays attention to them because of this obsession with phi.

My point is, close is not enough. If phi is a fundamental building block in nature, we should be able to show that it’s more than a coincidence—that there’s some reason behind it being there.

Not every claim about seeing phi in nature is fake. Phi does show up in nature in a really interesting way. If you’ve ever looked closely at a pineapple, a pinecone, a sunflower, or an artichoke, you’ll notice that all of these plant parts show a special kind of spiral.

For example, on a pineapple, there are spirals going one direction and spirals going the other direction. If we count these spirals, we find 13 spirals in one direction and 8 spirals in the other direction. Those numbers sound familiar because they’re both Fibonacci numbers.

Maybe that’s a coincidence, so let’s count something else. Pine cones also have spirals. We find 8 spirals going one direction and 13 spirals going the other direction—again, Fibonacci numbers.

We can also find Fibonacci numbers in sunflowers, roses, cauliflowers, and even in the branches of a monkey puzzle tree.

Why is there this connection? Imagine you’re a plant. You eat light, so the more sun you can catch with your leaves, the better. As a stem or branch grows, where do you put your leaves? If you place one leaf down, and then go a fraction of a turn, you can maximize sunlight exposure.

It turns out that if we use any fraction of a circle with a whole number on the bottom, our leaves will eventually overlap. However, if we use an irrational number instead, it turns out that phi might be the most irrational number there is.

Using phi as a guide, we can fill in our leaves without overlapping, leading to a Fibonacci number of spirals. This pattern is not only useful for catching sunlight but also for catching rain and funneling water down to the roots or for packing more seeds and flower petals into a small space.

This explains a lot about why plants do this, but the how is more complicated, and scientists are still working out many details. What we do know is that plants have had millions of years of evolution to find the best way to do all the cool plant stuff they want to do.

It’s not like every plant follows this rule; plenty of plants follow different rules and work just fine. In evolution, what matters is that something works well enough, not that it reaches some mathematical perfection.

But you can’t deny that this is kind of beautiful, and that’s probably a big part of why we’re attracted to this pattern more than other plant patterns. Beauty can take many forms. Some artists, like Salvador Dalí or the architect Le Corbusier, occasionally used the golden ratio purposely in their work, but there’s plenty of beautiful art that makes no use of the ratio, too.

Remember when I asked you to pick the most beautiful rectangle at the beginning? Many of you probably picked one other than the golden rectangle. Math follows very particular rules, while things like beauty and life are a bit messier because the world is a messy place, and that’s part of the beauty, isn’t it?

That’s okay because sometimes in the middle of that mess, if we look hard enough, we can find some order. After all, stay curious!

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