Golden ( Órga (Scottish gaelic -archaic)) Ratio: Factoring the Fibonacci Myths

The podcast examines the Fibonacci sequence, a numerical pattern where each value is the sum of the two preceding it. This sequence eventually stabilizes into the golden ratio, a mathematical proportion frequently associated with natural beauty and artistic perfection. The source aims to distinguish between historical myths and mathematical reality, specifically addressing false claims that this ratio dictates ideal human facial features. While debunking the idea that the ratio is a universal law for aesthetics, the material highlights the genuine wonder found in the intersection of geometry and the natural world. This overview encourages a more grounded understanding of how these famous numbers actually function within science and design.

Transcript

Unknown Speaker  0:00  
All right, welcome back to the deep dive. Good to be here. I have to say, this is a topic I have been really itching to get into. Oh, this one, yeah? Because it feels like it's, it's everywhere, yeah? But it's also, you know, nowhere. At the same time, it's this weird paradox. It's an incredibly messy file, a beautiful one, for sure, but messy, and honestly, it's one of the few topics where you can find yourself arguing about, like the structure of a seashell and the dimensions of a galaxy, almost in the same breath. Exactly we are talking about the interception of nature and numbers, specifically, one number, the Divine Proportion, the golden ratio. And before we get into, you know, the heavy math or the ancient pyramids or any of that, I want to start with something small, something that actually triggered this whole deep dive for me. Okay, I was scrolling through Reddit in this community called plotter art. Are you familiar with this niche? I am actually, yeah, it's a fascinating corner of the internet for anyone who doesn't know. It's basically where people write code to drive these physical pens plotters, and they draw these incredibly complex, precise geometric shapes. It's where computer code meets ink on paper. It's beautiful, exactly. So I saw this post. It was by a user named magic cart, and it wasn't a building or some abstract grid. It was a visualization of sunflower seeds. Ah, I think I know the one you're talking about. Just this mesmerizing, spiraling drawing of florets radiating out from a center point. It looked completely organic like you felt like you could touch it and feel the fuzzy texture of a sunflower. But it's not a sketch, not at all. The caption said it was generated using something called Vogel's 1979 formula, which is a purely mathematical algorithm. It's a plot. It's logic. And it just stopped me in my tracks, because you look at the image, and you have this immediate emotional reaction, oh, that's nature, that's life. But then you look at the source, and it's cold, hard logic. It's code, and the core of that code the one variable that made it look so so alive was this very specific angle, approximately 137.5

Unknown Speaker  2:08  
degrees, the golden angle, the golden angle. And that reaction you just described, that feeling of Wait, is beauty just a math equation. Yes, that's the tension, isn't it? That is the central tension of this entire topic. We see something beautiful. We find a number inside it, and we immediately want to believe that we found the, you know, the source code of the universe. It feels magical. It really does. It feels like we've peeked behind the curtain of reality. It does. And that feeling has given this number a better PR team than any celebrity in history, right? I mean, you hear about it in The Da Vinci Code. You hear that it's in the Parthenon, the Great Pyramid. Oh, everywhere the Parthenon, the pyramids, the Mona Lisa, the shape of hurricanes, exactly even the spiraling arms of galaxies. It's always presented as this universal law of beauty. But, and this is why we do these deep dives, we have a stack of sources here that are, let's just say, a little less romantic, little more skeptic, a lot more skeptical. We've got mathematical critiques. We have biological onalcs papers from the American Mathematical Society, and a lot of them seem to suggest that we might just be seeing things kind of pareidolia, you know, seeing faces in clouds, or, in this case, seeing a very specific number in places where it might not actually be exactly so that is our mission today. We are going to take this number phi, and we are going to, well, drag it through the mud of scrutiny, right? We need to separate the actual mathematical marvels, and I know there are some real ones in here from what one of our sources, very bluntly called pretentious nonsense. I love that phrase, and I think to do that fairly, we have to strip away all the mysticism first. We have to just look at the tool itself. We can't judge the philosophy until we actually understand the map. Okay, so let's start at Ground Zero. What actually is the golden ratio? I see the symbol right? The Greek letter Phi. I know. If I punch it into a calculator, I get 1.618 and a bunch of other numbers. Where does that come from? Did we just invent it because we liked the way it looked? No, not at all. We discovered it. Yeah. And it arises from a really, really simple geometric problem. Let's do a little thought experiment. Okay, imagine you have a stick, just a straight line segment. I'm with you holding a stick, and you want to break that stick into two pieces. One long piece, we can call it a and one short piece, B. Okay, snap done two pieces. Now you could break it anywhere, right? You could break it exactly in half, so A equals B. You could break off a tiny, little sliver at the end. But the golden ratio comes from a very specific, a very elegant requirement, which is you have to break it so that the relationship, the ratio between the entire stick, the original length, and the long piece, is exactly the same as the relationship between the long piece and the short piece. Okay, let me. Let me visualize that. So the whole thing divided by the long part is the same number as the long part divided by the short part precisely the.

Unknown Speaker  5:00  
Whole is to the large as the largest to the small. It's a harmony of scale. That's it. The little part relates to the big part. Is the way the big part relates to the whole thing. It's almost a fractal concept, really kind of self similarity, right? And if you write that down as an algebra problem, so A plus B, A equals AB, and you solve for that ratio, you end up with a quadratic equation, yeah, x squared minus x minus one equals zero, and the solution to that equation is five. That's one. It's one plus the square root of five, all of that divided by two. And when you work out the decimal expansion, you get 1.61803398

Unknown Speaker  5:34  
and, well, it goes on forever without repeating. It's an irrational number, it is. But that definition, you know, the stick breaking thing. It's a bit abstract for most people. The way most of us encounter this number is through something else entirely, the Fibonacci sequence. The Fibonacci sequence, yeah, from math class, from movies. It's 0112358,

Unknown Speaker  5:53  
right? And the rule is so simple, you start with zero and one, and then every new number is just the sum of the two that came right before it. So zero plus one is one, then one plus one is two. One plus two is three. Two plus three is five. Five plus eight is 13. Eight plus 13 is 21 it grows pretty fast. It really does. It does but here's the magic trick. If you take that list of numbers and you divide any number by the one right before it, watch what happens. Okay, let's try it. Five divided by three is 1.666

Unknown Speaker  6:22  
repeating. Okay, go a little higher up the chain. All right, 13 divided by eight, that's 1.625

Unknown Speaker  6:27  
getting closer. Go higher

Unknown Speaker  6:29  
55 divided by 34 Let me see that should give you about 1.6176

Unknown Speaker  6:34  
and 89 divided by 55 is 1.6181

Unknown Speaker  6:38  
you see it right as you march up the Fibonacci sequence that ratio. It swings back and forth, little above, little below, but it gets tighter and tighter and tighter it converges on phi. So the Fibonacci sequence is almost like like a numerical attempt to get to the golden ratio, but using only whole numbers. That is a fantastic way to put it. Yes, and it's not just Fibonacci, which is what's so cool. There are other similar sequences, like Lucas numbers, for instance, they start with two and one instead of zero and one, but they follow the same rule of adding the previous two terms, and they also converge to phi. They also converge to phi. It's like phi is this? This gravitational center, any sequence that grows by that specific additive rule eventually gets pulled into orbit around 1.618 Okay. That makes sense. It's like a mathematical constant that these sequences are aiming for. But I want to bring up something from our source material that sounded a bit, I don't know, a bit strange. One of the articles, I think it was from the American Mathematical Society refers to fee as the most irrational number. It does sound like an insult, doesn't it like you're being irrational. Fee just calm down, right? But in math, irrational just means it can't be written as a simple fraction, like one half or three quarters pi is irrational. The square root of two is irrational. So how can phi be the most irrational? Is there, like a hierarchy of irrationality? There actually is, and this is probably the single most important technical concept for us to really grasp if we want to understand the biology stuff that's coming later. Okay, so think about approximations. We all know Pi is 3.14159

Unknown Speaker  8:07  
and so on. But in school, what do we often use instead 22 over 722, over seven. And what is that as a decimal? It's like 3.1428

Unknown Speaker  8:18  
something. It's really close. It's very close. It's a simple fraction, small numbers that gets you almost exactly to the right answer.

Unknown Speaker  8:26  
Pi is, in a sense, easy to approximate with a fraction. Okay, phi is difficult. It is the hardest number to approximate with a simple fraction. Why? What makes it so different? It comes down to something called a continued fraction. I won't make you do all the algebra, but just picture this, you're trying to write a number as one plus one over something and that something is one plus one over something else. It's this nested ladder of fractions that goes down forever. Okay, I'm picturing a fraction inside a fraction inside a fraction, like those Russian nesting dolls, but made of division signs. That's a perfect analogy. Exactly. Now, the numbers you put in the denominators, the bottom parts of all those little fractions, they determine how fast that whole ladder converges to your target number. If you can put big numbers down there, like a 10 or a 20, that fraction gets really small, really fast, and you snap to the answer very quickly. Okay, so big numbers in the denominator mean a fast shortcut to the answer, right? But for fee, for the golden ratio, that continued fraction structure is composed entirely of the number one, just ones all the way down, and one is the smallest possible positive integer, the smallest possible so, because all its denominators are as small as they can possibly be, the fraction shrinks as slowly as humanly possible. So it converges lazily, reluctantly. Converges reluctantly. This leads to a theorem mentioned in our notes, called hurwitz's theorem, which basically puts a mathematical seal of approval on this. It proves that you cannot find a rational fraction that approximates fee well without using absolutely massive numbers in that fraction.

Unknown Speaker  10:00  
Variation. It just stays slippery for longer than any other number. So if I'm trying to pin it down with a simple fraction, it just keeps sliding away from me. It refuses to be boxed in. Keep that in your pocket. That exact idea, that slipperiness, that refusal to settle down into a simple repeating pattern, that is going to be the absolute key to why nature actually uses it. Okay, it's not about magic or beauty. It's about avoiding a beat. Avoiding a beat. Okay, foreshadowing. I like it. But before I get to plants avoiding beats, we have to deal with the humans who are desperately trying to find them, because we have projected this number onto basically everything we've ever built. It seems we have an obsession with finding it. Yes, so let's talk about that big one, the universal law of beauty. The core idea is that if I show you a rectangle with sides in the ratio of one to 1.618 your brain just lights up. You involuntarily think that is the most perfect esthetically pleasing rectangle. Is that true? It is one of the most persistent and frankly unfounded myths in both psychology and art history, ouch, and we can largely blame one person for it, a 19th century German psychologist named Adolph zeissing. Okay, so he's the hype man. He is the ultimate hype man for phi zeising was a researcher in the mid 1800s who was just philosophically convinced that there had to be a single universal law connecting nature and art. Okay, so he went out and he measured everything, I mean, everything leaves, animal skeletons, insect bodies, ancient Greek statues about it. He found it because he was looking for it. He wrote these enormous, sprawling treatises claiming that this one ratio permeated the entire universe, organic and inorganic, micro and macro.

Unknown Speaker  11:39  
He basically started the rumor that the golden ratio is the literal blueprint of reality. But when we actually test this, I mean, surely modern science has checked if we actually prefer these rectangles over others. Oh, they have many, many times, starting all the way back with gusta Fechner in 1876 who was actually trying to scientifically verify zeissings claims. The results are underwhelming, underwhelming. How, like, people just don't pick the golden rectangle. Data is all over the map. I mean, if you show 1000 people a whole bunch of different rectangles, some tall and skinny, some almost square, some golden, you just don't get a massive statistical spike at 1.618 so what do people pick? Well, it varies. But interestingly, a lot of people tend to prefer a rectangle with a ratio of about 1.414

Unknown Speaker  12:23  
that is very specific. What is 1.414 it's the square root of two. It's the aspect ratio of a standard piece of a four paper. Ah, so we might just like it because it looks like a sheet of printer paper. We're just used to it. It's most likely a familiarity bias. The point is, there is absolutely no evidence of some hard wired biological switch in our brains that flips when we see fi we like all sorts of different shapes depending on context, culture and you know what we're used to seeing. Okay, fine, so our brains aren't hard wired for it. But what about the great masterpieces, the Parthenon in Athens I've seen the diagrams. I know you have. You take a photo of the Parthenon, you draw a golden rectangle around the front facade and it just it lines up perfectly with the roof and the columns. Yes. This brings us to what I like to call the fat marker problem. Fat marker problem, okay, explain. If you print out a picture of a complex building like the Parthenon, and you use a thick enough magic marker. You can make a line hit any two points you want it to hit. You can fudge it. You can absolutely fudge it. The Parthenon is a great example. You say it lines up,

Unknown Speaker  13:29  
but lines up with what exactly the edges top and bottom, but which edges the Parthenon is actually famous in architecture for not having straight lines, right? It's all slightly curved. The ancient Greeks were masters of optics. They knew that long straight lines appear to sag or curve to the human eye. So to counteract that, the floor of the Parthenon actually curves upward in the middle. The columns all lean inward slightly. The columns themselves even swell in the middle, a feature called endotasis. So it's a wobbly rectangle at best. It's an incredibly subtle, complex and beautiful geometry. So if you're zeising, or you're a modern tourist with a ruler and a camera, where do you start measuring? Do you measure from the bottom step, the top step, the base of the actual column, the top of the cornice? If you have five or six different possible start and end points. Statistically, one of the combinations is going to give you something close to 1.6 Exactly. And even more importantly, we have the records. We know a lot about Greek architecture. We have writings from people like tennis and calacrity, the designers they never once wrote about phi. What did they write about? They wrote about simple elegance. Whole number ratios, four to nine, two to one, things like that. They were engineers, not mystics. They wanted things to be structurally sound and look visually harmonious, using ratios they could actually work with. So the Parthenon is a no. What about Da Vinci, the Vitruvian Man, the guy standing the Circle in the Square, that has to be golden ratio, right? It's like.

Unknown Speaker  15:00  
The poster child for human proportion, Leonardo is another victim of our desire to find these patterns. Now it's true, he was good friends with a mathematician named Luca Pacioli. And Pacioli wrote a book literally titled de divina proportione, the Divine Proportion. He did that seems like a smoking gun, doesn't it? It really does. It does until you actually read the book, Pacioli praises the golden ratio for its mathematical properties, its elegance, but he doesn't explicitly tell artists, hey, you must use this ratio to compose your paintings. So it was more of a math book than an art instruction manual, exactly. And when you look at Da Vinci's own extensive notes, and I mean the man wrote down everything, he talks constantly about ratios, but he talks about thirds, halves, quarters, simple whole number divisions. The Vitruvian Man itself is based on the writings of Vitruvius, a Roman architect who used, you guessed it, whole number fractions. So we're retroactively applying this secret mathematical code to Da Vinci that he himself never actually used. We want him to be a wizard who knew the secrets of the universe, but he was just a genius observer of the world around him. However, I have to stop us here, because we have to give credit where it's due. There is one major figure in art and architecture who did deliberately use it, okay, Le Corbusier, the famous 20th century modernist architect, ah, the concrete and glass guy. That's him. He developed an entire system of architectural proportion called the modular and he explicitly intentionally built it from the ground up, using the golden ratio and the Fibonacci sequence. So he believed in it. He believed it created a harmonic rhythm that was related to the human scale. So if you go and measure one of his buildings, like the Williston, you will find fine everywhere, but you find it because he intentionally put it there, correct. It's an artistic choice he made, not some mystical law. He discovered, I see. Okay, so art history is,

Unknown Speaker  16:55  
well, it's largely a bust when it comes to these ancient claims, but now we have to go to the big one, the one that feels the most significant, the Great Pyramid of Giza, the Giza Plateau. Yeah, this is where the math gets really interesting and really contentious. And we have a source here that's specifically critiquing the theories of Graham Hancock. Now, for any listeners who don't know, Hancock is a huge figure in the alternative history world, and his argument is that the Great Pyramid isn't just a tomb, it's a geodetic monument, that it somehow encodes the precise dimensions of the Earth, right? And the claim is incredibly specific. He says that if you take the height of the Great Pyramid and you multiply it by the number 43,200,

Unknown Speaker  17:36  
you get the Earth's polar radius. And if you take the perimeter of the base of the pyramid, and multiply it by that same number, 43,200,

Unknown Speaker  17:45  
you get the Earth's equatorial circumference. And that number 43,200,

Unknown Speaker  17:50  
it's not random, is it? The claim is it's related to time. The argument is that it's related to something called the precession of the equinoxes. That's the slow wobble of the earth on its axis like a spinning top. It takes 72 years for that wobble to move one degree across the sky, and 43,200

Unknown Speaker  18:08  
is a multiple of 72 Okay, so let me get this straight. The theory is the builders of the pyramid knew the exact size of the Earth. They knew the complex physics of the Earth's wobble, and they built a perfect scale model of the planet using a time number as the scale factor. That is the claim. And you know, if it were true, it would completely rewrite human history. It would imply that ancient Egyptians, or whoever built it, had knowledge that we've only really acquired in the satellite age. There's always a but we have a rebuttal here from a mathematician, Thomas W schwerter, and he breaks this entire claim down with two simple geometric axioms. Yeah. Schroeder's critique is really elegant because it just cuts right through all the noise. Let's look at his axiom one. Okay. He says, basically,

Unknown Speaker  18:50  
if you build any pyramid based on simple circle geometry, you will accidentally embed the number pi into its proportions. Slow down. How do you build a pyramid which is square based on circle geometry. Okay, imagine a circle on the ground. It has a radius, the line from the center to the edge, and it has a circumference, the distance all the way around it. Right? Now, imagine you're an Ancient Builder, and you decide on a simple rule for your pyramid. You say, I'm going to make the height of my pyramid equal to the radius of that circle, and I'm going to make the perimeter of the pyramids base equal to the circumference of that circle. Okay, so height equals radius and base perimeter equals circumference. I can see that if you do that, the relationship between your pyramids base and its height will automatically mathematically involve the number two times pi. You don't even have to know what pi is. You just have to use a wrong drum or a piece of rope tied to a stake to measure out your lines. And because the earth is a sphere, because the Earth is approximately a sphere, its own dimensions also involve pi. So if you build a PI based pyramid, it will naturally be proportional to any sphere, the earth, the moon, the.

Unknown Speaker  20:00  
Basketball. So it's like building a model car. If I build a model car at a perfect one to 10 scale, the ratio of the wheels to the roof is the same as the real car. It doesn't mean I understand how a combustion engine works exactly. It just means you copied the shape. If the Egyptians used a simple, practical method, like a rolling wheel to measure the base, they would have encoded pi by accident. Okay, that's a pretty strong argument, but then Schroeder just drops the hammer with axiom two. This is the contradiction part. This is the logical contradiction. Remember Hancock's claim the pyramids height matches the polar radius and its base matches the equatorial circumference perfectly using the same scale factor, yeah. Why is that a problem? Because the Earth isn't a perfect sphere, right? It's an oblate spheroid. It's a little squash Exactly. It bulges out at the equator. It has a bit of a spare tire, as you said, right? So on the actual Earth, the ratio of the equatorial circumference to the polar radius is not exactly two times pi. It's slightly off because of that squashing, yeah, but a standard geometric pyramid built with the method we just described is exactly two times positive. Ah, I see. So you cannot possibly model a squashed sphere with a perfect pyramid and have both measurements line up. You can't the geometry forbids it. You can choose to match the height perfectly, or you can choose to match the base perimeter perfectly, but you cannot match both simultaneously with a single scale factor. It's a mathematical impossibility. So Hancock's math has to be fudged somewhere. Schroeder points out that there's a lot of cherry picking going on. You have to pick a specific height for the pyramid, and remember the top is missing, so we have to guess the original height, and you have to pick a specific value for the Earth's radius, which also varies, right? And when you actually run the numbers using Hancock's own preferred scale of 43,200,

Unknown Speaker  21:47  
the result you get for the Earth's radius is about 6332

Unknown Speaker  21:51  
kilometers. And is that close. It's close ish, but it actually falls outside the accepted range of the Earth's possible polar radius. It's a near miss that's being sold as a direct hit. It's a bullseye only if you draw the target around the arrow after it lands. That's the danger of numerology. Yes, you look for a number hard enough, and you have enough variables to play with, height, width, slope, angle, and you allow yourself a little wiggle room of a few percent, you will find your number every single time. It's the Cloud Dragon. Again, it's the Cloud Dragon. All right, let's leave the deserts of Egypt and go to the ocean. Because if there is one single image that has sold the golden ratio to the world, it is the nautilus shell, the poster child. Absolutely you see it on greeting cards, on science textbooks, on motivational posters, that perfect expanding spiral. The claim is that with every quarter turn, the spiral gets wider by exactly fee by 1.618 I hate to be the one to break everyone's hearts, but this is another one that's, well, it's a bust, or at least a partial bust. Oh no, don't tell me. The Nautilus is a fake. It's not a fake, but it's not what the textbooks claim. It is. Mathematicians, people like Keith Devlin and Clement Falbo have gone and measured hundreds of these shells in places like the Museum of Natural History. Yeah, a true golden spiral, the one you draw using the squares and quarter circles. It flares out pretty widely. Gets big fast. The Nautilus is tighter. It wraps around itself more snugly, so it's not growing as quickly. Exactly, exactly. If you measure its growth rate, its expansion factor, it's usually around 1.33 not 1.618 1.33 versus 1.618 I mean, that's a significant visible difference. It is. It is absolutely a logarithmic spiral. Nature loves logarithmic spirals because they allow an animal to grow without changing its fundamental shape, right? But it is not a golden logarithmic spiral. Okay, but I want to play devil's advocate here for a second. We have a source from a guy named Gary Meisner, who runs a website that's pretty much dedicated to phi. And he argues there is some nuance we're missing. There is, and it's a fair point. Meisner's argument is that maybe we're holding the shell up to the wrong standard, the classic golden rectangle spiral grows by a factor of phi every 90 degrees every quarter turn, right. But what if the Nautilus has a different rhythm? What if it grows by phi every 180 degrees every half turn, or by phi squared every full 360 degree turn? So if we just change the intervals of how we measure it, Meisner did this, and he found that if you use those alternative phi based spirals. You can find actual Nautilus shells that fit the curve remarkably well, sometimes within a 2% variance. So it might not be the golden spiral from the textbook, but it could still be a phi related spiral. It's phi adjacent. And I really like this conclusion, because it respects the messiness of biology, right? These are shells being built by living mollusks, not plotted by a computer. They're going to vary. Some are fatter, some are thinner. We shouldn't expect laser like precision from a sea creature. So it's close enough that our intuition isn't completely crazy for seeing a connection, but it's far enough that the math textbook is technically wrong, exactly. It's a beautiful compromise between math and nature.

Unknown Speaker  25:00  
Yeah, okay, we've been pretty brutal so far in this deep dive. No Parthenon, no pyramids, a shaky nautilus shell. I'm starting to feel like this whole Golden Ratio thing is a bit of a scam. Is it actually useful for anything real, or is it just a ghost we chase? Oh, it is very, very real. And this, for me, is the best part of the whole story, because when you stop looking for fill in places of mystical beauty, and you start looking for it in places of blind efficiency, you find it everywhere. Efficiency, not beauty, efficiency. Let's go all the way back to the beginning, to your Reddit sunflower, the made magic cart drawing. Okay, think about the problem that a sunflower has to solve. It wants to pack as many seeds as it possibly can into that head. Yeah. It doesn't care about art or esthetics. It cares about survival,

Unknown Speaker  25:48  
about density, right? Any waste that space means fewer seeds, fewer offspring, and it grows from the center outwards, so it pushes out a new seed. It rotates a certain angle. It pushes out another seed. It rotates again like a little biological rotisserie Exactly. Now let's imagine some bad choices for that angle. What if it turned exactly 180 degrees, half a circle each time seed turn 180 seed turn 180 you just get a single straight line of seed, right? You'd have all the seeds in one row with huge empty spaces on either side, a total waste. Okay? What if you turn 90 degrees each time? You'd get a cross shape forearms, still tons of wasted space in between, exactly. And even if you pick a more complicated fraction, like, say, 23 of a turn, or 1537, of a turn, eventually, because it's a rational fraction, the pattern will repeat. You will eventually start placing new seeds in spokes that line up with older seeds, and that's bad. It's bad for packing seeds, and it's even worse for leaves. If you are a plant stem and you put a new leaf directly above an older leaf, what happens? You block its sun, you starve the lower leaf. So you need a rotation angle that guarantees you never line up perfectly with where you've been before. You need a number that prevents that resonance, that repeating pattern. You need a number that is the absolute hardest to turn into a simple fraction, the most irrational number. There it is. Boom. This is where that abstract math from the beginning pays off in a huge way, because phi is the most irrational number, it creates what's called the Golden angle, about 137.5

Unknown Speaker  27:20  
degrees, if a plant uses this angle to space out its new growths, it mathematically guarantees that no matter how many leaves it grows, 10, 100 a million, no two leaves will ever perfectly overlap and block each other. That is incredible. It's not a choice. The math forces the optimal geometry. It optimizes the plant's exposure to space and light, the breathtaking beauty that we see those interlocking spirals of seeds that's just a byproduct of the plant trying to solve a very mundane packing problem in the most efficient way possible. And is that why we see the Fibonacci numbers in the spirals themselves, the 34 and 55 Yes, that's the visual result of this process. If you count the spirals winding clockwise and counterclockwise on a sunflower or a pine cone, you will almost always find two neighboring numbers from the Fibonacci sequence, 34 and 55 or maybe 55 and 89 on a really big one, nature is essentially counting in Fibonacci, because that is the whole number pathway to fee that makes so much more sense than God. Just like this shape, it's this shape works better than any other shape, evolution is a pragmatic designer. It finds what works and it sticks with it. So that's biology, which is amazing, but we have one more source here that I have to admit it completely blew my mind, quantum physics, something called cobalt, niobate, yeah, this is really cutting edge stuff. A few years ago, a team of researchers were studying this very specific magnetic crystal, cobalt niobate, at temperatures incredibly close to absolute zero. We're talking super, super cold, as cold as it gets, and they were applying a magnetic field to it, to force the crystal to change its state, to enter what's called a quantum critical phase. A simple way to think of it is like vibrating a guitar string. The atoms in this crystal can vibrate at different specific energy levels, okay, like different notes Exactly. And they found that when the system was in this critical transition state, the lowest two energy states, the two main notes the atoms could play, resonated in a very specific ratio. Let me guess the ratio was precisely the golden mean, 1.618

Unknown Speaker  29:20  
wait in the subatomic vibration of a frozen crystal in the fundamental structure of the matter itself. This suggests that phi isn't just a trick for plants. It might be a fundamental solution to a physics problem of balance and stability. Just like the plant wants to avoid overlapping leaves, the quantum system might settle into this ratio to perfectly balance its energy states during a critical transition. So it is a universal law, just not the one that zeising thought it was. It's not a law of things look pretty. It's a law of things stabilizing efficiently, or, even better, a law of things avoiding synchronization. In fact, engineers use this exact principle in.

Unknown Speaker  30:00  
Real World. If you're building a huge suspension bridge, one of your biggest fears is resonance. Right? The Tacoma Narrows Bridge disaster the video where the bridge starts wobbling violently in the wind and just tears itself apart. That's it. The wind was pushing the bridge at the exact same frequency that the bridge itself naturally wanted to vibrate. Resonance is destructive. So if you want to design a complex system that is highly resistant to being pushed into a destructive resonance, you want its internal frequencies to not mix well, you want the most irrational relationship possible. So engineers will sometimes deliberately design components to have frequencies in the ratio of one to five. They will specifically to dampen vibrations and prevent that catastrophic feedback loop. Because phi refuses to sync up. It just refuses to get in step. It is the ultimate non conformist number. It's the punk rock number of the universe. I love that, yes, it refuses to dance to the beat. That's incredible. So we've debunked a lot of the myths, but in their place, we've found something that is arguably so much cooler, I think so. But I want to wrap up by coming back to the psychology of it. Why do we still want to believe it's in the pyramids and, you know, in the proportions of our own bodies? Because we are pattern matching machines. It's what our brains evolve to do. If you're an early human on the Savannah and you see a strange disruption in the pattern of the tall grass. You think Tiger. If you're wrong, you're a little spooked for a minute. But if you fail to see the pattern and you're wrong, you're dead. We are biologically biased to find connections even when they're not really there. So we are hard wired to see the Cloud Dragon we are. And when you look at a really complex, noisy system like the stock market, you see the exact same thing play out, right? Traders use something called Fibonacci retracements. They do. They have this belief that if a stock price falls, it'll often find support and bounce back up right at the 61.8% level of its previous move, which is the inverse of phi it is, and guess what? Sometimes it does bounce there. So it's real. It's real, but probably not because of some magnetic law of finance. It's most likely because 1000s and 1000s of traders believe it will bounce there, so they all set their automatic buy orders right at that price. It's a self fulfilling prophecy. It's a perfect example of a self fulfilling prophecy. We create the order that we are looking for in the chaos. So if we synthesize all of this, if we boil it all down, the golden ratio is not a magic key left by ancient aliens. It's not the secret code of the Illuminati used to design the Parthenon. No, it is a specific, fascinating and very elegant mathematical solution to a recurring problem of efficiency. It appears in nature where things need to be packed tightly, or where systems need to be kept from shaking themselves apart, and it represents this lovely paradox, as our notes say, the most interesting thing about the golden ratio is that it is the number that is the least capable of actually being a simple ratio, and that profound irrationality is its superpower. It keeps the universe dynamic. It prevents the gears from locking up. So here's a final provocative thought for you to chew on. We see phi clearly in the sunflower. That's biological optimization. We see hints of it, real evidence in the quantum realm with cobalt neobit, that's fundamental physics. And some astronomers even argue that the logarithmic spirals of galaxies have characteristics related to phi. That's gravity working on a cosmic scale. Just hints, mind you on the galactic scale, but compelling ones. But if this most irrational number appears at the quantum level, the biological level, and potentially the galactic level. Does that suggest that the fundamental laws of how to organize stuff efficiently are the same across every single scale of the Universe? Are we looking at the fingerprint of optimization itself? That is the question, isn't it? And honestly, looking at a sunflower and seeing it as a master class in mathematical survival, to me, that's infinitely more beautiful and more awe inspiring than any myth about Ancient Greece. I couldn't agree more. So next time you see a flower or a pine cone, count the spirals you might just see the universe doing its math homework. Thanks for diving deep with us today. My pleasure. See you next time.

Transcribed by https://otter.ai