Math in Nature: The Hidden Numbers Behind Nature’s Beauty

In this episode of Wildly Curious (formerly For the Love of Nature), co-hosts Katy Reiss and Laura Fawks Lapole explore the fascinating ways that mathematics shapes the natural world. From the perfect spirals of nautilus shells to the fractal patterns found in trees and rivers, they dive into how simple mathematical principles underpin the complex beauty of nature. Whether you're a math enthusiast or someone who's always struggled with numbers, this episode will have you looking at nature in a whole new way.

Perfect for anyone curious about the intersection of math and nature, this episode sheds light on the hidden patterns that govern the world around us.

Transcript

Hello, and welcome to For the Love of Nature, a podcast where we tell you everything you need to know about nature and probably more than you want to know. I'm Laura.
And I'm Katy. And today, we are gonna be talking about, listen, this is gonna be shocking for all of our listeners.
We're gonna be talking about math, but it's math that's gonna be found in nature.
And how, like, even though there's a lot of beauty in the complexity of nature, it can actually be described using mathematics and whether or not this is a human construct or a fundamental law of the physical world. We're each gonna give you some different examples of what we found as, I was just gonna say interesting, but as interesting as one could expect an individual with a math learning disability would find in math.
I was fascinated. I just don't understand a lot of it.
I have no interest in math, partially because I suck at it so bad.
Yeah, but if I'm gonna be into math in any way, this is all very concrete, that's why I like geometry and not algebra. So this is, okay, if you-
Forget you imaginary numbers, nobody needs you.
Yeah, this is real stuff. I can see it, I could actually touch it. Yeah, I would much rather learn of math and nature.
Me too.
Well, we don't have nature news.
Well, Avatar is about to come out.
So it's like the news.
It is the news.
We've only been waiting for what, 11 years?
I think it's 13, which is ridiculous.
Ridiculous. The longest wait in between sequels. Not even, you know how a lot of people, I think, make a movie and then lay in the future, they're like, okay, maybe we'll make a second one now.
No, this one's been in the works for 13 years. We were promised a sequel 13 years ago.
13 years ago.
James Cameron.
Yeah, calling you out on our podcast. Yeah, no, I am so freaking pumped to go see it.
I need it to be as good as the first one.
God, I hope so. I so hope so.
I will murder. Burn the theater down if I've waited this long for it to just be a trash sequel.
Right? No, I don't think it will though, just because it is James Cameron, but still.
I definitely am gonna find an IMAX theater and go in all its glory.
I mean, I am very excited to see this. I wasn't gonna be really pissed off at, with the last Jurassic movie, if Dr. Grant and Sattler didn't make out, but I don't know, like I don't have any expectations for this one, other than it better be good.
Yeah, yeah, yeah, yeah, yeah. I have no idea what to expect. I mean, I've seen the trailer, but the last one, yeah, I have no idea what they're gonna do.
I just wanna see more of that beautiful world.
Yeah, right?
I don't even really care what they do.
That Laura and I would literally be some of the first two people to volunteer to be there.
Yeah, and the only way you'd get me into space, I think as we said before, promise me Pandora is out there, and I will go today, tomorrow.
Yeah, 100%. Anything else? Mars?
No, screw you.
Gotta be interesting.
No, I need trees, preferably giant trees that I can live in.
That are alive, legitimately alive.
That I can plug into.
Your hair, no less, plug into your hair.
It's just everything about it's great. I want to be in a V.
Well, did you want to go first, then?
Yeah, I think I'll set things up, because my first thing is fractals. And I think that-
You tried to say that so excitedly, but it's not gonna be.
Well, it's kind of funny. I actually thought a lot about one of my friends from undergrad, Sarah, who was a math major, and did her undergrad thesis on fractals and music, because she also loved music. And I didn't understand how you could find math and music.
Fractals, no less, which is I'm about to tell you, but so, Sarah, I've always been blown away by your intelligence. I'm not even close to that level, but I'm gonna try and explain what fractals are. And it is a fairly basic concept.
So fractals are complex patterns that are recurring and self-similar, meaning that you can see replicas of the whole at different scales and sizes. Basically, it's a pattern that looks the same or at least similar no matter how much you zoom in. Like you zoom in and you see it again.
You zoom in and you see it again. You zoom in, you see it again.
Now I'm really questioning how you see it in music, but continue.
Right, that's what I don't understand.
Right, right, we just got it.
Sarah, mathematician, I'll probably mispronounce his name, but I'm assuming it's Wenaw Mandelbrough, coined the term in 1975. So it's actually a relatively new term in the grand scheme of things. I mean, not that they haven't always existed.
Yeah, but still.
But not called a fractal. So to be considered a fractal, shapes don't have to be exactly the same when you zoom in. They just have to, and this is in quotes, display inherent and repeating similarities.
So, but that means-
So basically close enough.
Close enough. It has to, like, when you look at it, you're like, oh, it's pretty much the same thing. Zoomed in.
So that leads to like many, there are several different types of fractals, and I'm not going to go into that, because I want to talk more about them in nature. But mathematically speaking, the reason why a fractal can be thought of in math terms is that formulas can result in shapes. So there are things called fractal equations that results in these visual images through a process of iteration.
And iteration just means again, like repeating again and again and again. So there's an equation, you input a value, you do the equation, then that result is put into the next equation that's the same like equation. So you're just like getting a result, putting it in, getting a result, putting it in, getting a result, putting it in.
And it results in these crazy elaborate shapes. If you look up fractals, you'll probably see Mandelbrot's fractal, it's the most famous one, the guy who coined the term. And kind of the cool thing is that these insanely simple equations, like these fractal equations are really easy ones.
They're very simple, but they result in intensely complex shapes.
Well, and that's like what a lot of the math in nature is. And I talk about it a little bit, is even though nature can be so stinking complex, a lot of these things, like it seems very complex, but it's based upon very, very, very simple principles.
Simple rules, yeah, yeah, yeah. And another thing that I had no idea about, so there's fractals are the things, there's something called fractal geometry, and it deals with the fractal dimension, okay? Yeah, there's another dimension that I didn't even know about.
So there's 2D, that's flat stuff. Then there's 3D, so that's like a cube. Okay, so now picture this, Katy, you take a piece of paper, it's 2D, and then you trumple it up into piece, and you're like, here's a sphere.
It's not 3D though, because there's all these wrinkles that have, so there's space. It's not a hollow sphere, it's not a solid sphere, it's a crinkled up space filled semi-solid sphere. Those, that space of the crinkles, that's the fractal dimension.
What the heck? Yeah, just, I had no idea. So, yep.
So wait, why are they calling it? Okay, okay.
It's neither 2D nor 3D, it's halfway between, and that's the fractal dimension.
Interesting.
I don't know what, and fractal just means like it can be fractured. So I guess that's why they call it fractal dimension, whatever.
Interesting.
Anyway, a perfect fractal, I don't think exists in the natural world, but a perfect fractal would be infinite. Okay, it would never end. Okay, okay.
You could continuously get closer and never ever stop.
It would have to stop at some point though.
I mean, I guess it's like as weird as thinking about the universe being finite, you know. A perfect one would be infinite.
This is like an ant man situation.
I can't even, yeah. Before my brain explodes, let's continue on. So where can you find them in nature?
Cause that's really why we're here. And nature is freaking chock full of fractals. I'm gonna divide them into categories.
First up is plants. A fern, okay, is a fractal. Yeah, ferns are fractal.
So if you look at a fern, it's got that distinct feathery shape. Then you zoom in to the fern, and it's got the little leaflets on it. Those leaflets are in the same shape as the fern.
You zoom in on the leaflet, and the tiny little lobes coming off of it look like the fern. So there's actually a formula for this. It's called Barnsley's Fern Formula.
I love it.
That's your claim to fame, is your fern formula.
I would way have, it would have been way easier for me to learn math if you were like, we're gonna learn about ferns and how they are mathematical.
Then there's little Laura and little Katy, like we got this. Like we're finally on board to learn about math.
The fern formula is random numbers generated over and over again and put into the formula, which produces a unique fern shape. So it's not actually like his, he just figured out how to make a fern using math. Isn't that weird to think about?
Yeah, that's weird. But also that that's his claim to fame.
You know what I mean? I mean, I take it.
The fern guy, the fern math guy. Very neat.
The fern math, yeah, fern math guy.
So the branching of trees are fractals. So the trunk is the origin point. Each branch is like a smaller scale of the whole.
So when you look up at a tree, it's just branching, branching, branching, branching, branching, smaller, smaller, smaller, smaller. So that is a fractal, but that is a great example of one that is not an exact replica, okay? It just has to be similar enough to resemble the whole.
Gotcha, gotcha. And the reason, like, why do we see these fractals? I'm gonna try it.
There are some places where we know this is probably why nature does this, because there's gotta be a reason. And it's to optimize sun exposure. The more you branch, the more likely you are to be able to find the sun.
Okay, that makes sense.
Yeah, I'm like, oh, okay, well, yeah, that's easy to explain.
And I mean, if you think of, and I talk about this a little bit too, if you're going to expand, why not do the same thing over and over again?
Right, you know what I mean? Again, back to the simple, like you're doing the simplest thing to, with the least energy, usually in nature, right? Since plants have internal structures to supply water and nutrients, they also have branching systems throughout their roots and leaves.
So think of us, like in our veins and everything, they branch just like a tree branches. So the tree itself branches what is in the tree and plant branches, and the entire plant might have a fractal shape, such as broccoli, cauliflower, pineapples, or cacti. Those are considered fractal shapes.
The specific type of arrangement of these, like think about a cactus, like one of those little spirally cactuses, the leaves in that arrangement, that's called spiral phylotaxi. And it's the way the leaves are arranged in a fractal pattern. The reason why they think the plants do that is that it helps funnel water.
So they're creating a spiral to catch all that water, especially if you're a cactus. There's an even distribution to all leaves. Not a single one is being blocked out by another one because you are evenly distributed around.
And finally, that it relieves stress on the structure of the plant. So rather than having two opposing forces that are gonna bend one way or the other, it is absolutely even pressure around the whole plant. So they kind of figured it out.
They can get water, they can get sun, and they can get water, sun, and stress.
I really feel like humans evolved stupidly.
Plants are like, guys.
Yeah.
We're doing this so wrong.
So wrong.
And yeah, humans are just, nothing about our body is meant to work.
Could have been a pineapple.
It could have been a pineapple.
No, to be fair, that's true, that's true.
Okay, so not just plants. Crystals. Minerals impact the structure of the crystal, which can create fractals just like ice crystals.
So minerals are crystals, is crystals, that is all fractal patterns. Anatomy, the circulatory system and the respiratory system are both fractal patterns.
Why have I never realized that?
Yeah, the branching, the branching. Oh my gosh, there are some amazing images. If you look up the branching of a tree and then the branching of the human lungs, like the vascular system within a lung.
Incredible, it looks exactly the same. Our lungs are just trees inside of us.
Okay, maybe I take back that humans evolved stupidly.
Well, we try, I mean, yeah, we kind of did the same thing with some parts of us.
Our lungs were on point, everything else missed the memo.
Heart, arteries, veins, capillaries. So the anything that's got a circle cycle.
I was like, I don't know what you're trying to say. I wanna help you because you're struggling, but I don't know what you're gonna say.
Okay, geography actually has a lot of fractals in it.
Interesting.
So as rain, and really a lot of it comes down to water. So as rain falls, water runs downhill, collecting in larger and larger quantities, streams to rivers, to the ocean. And so those, again, branching patterns, it's the exact same image as a tree.
If you look at river systems, stream systems, it's a tree. So it's this branching. You're just seeing branching, branching, branching.
But another really cool, weird rule, Katy, I have no idea about, is that the curves in streams and rivers are always six times the width of the channel.
I'm thinking.
I mean, six, so the curve, like when it curves, you could measure that curve, which I'm assuming they mean the distance of the curve.
Okay, okay, okay, okay.
And it's six times the width of a normal, of it. Yeah. Just a weird flow rule.
Interesting.
Yeah, right, I kind of want to go on and measure a lot of different curves and streams. What are you guys doing?
What are you guys measuring?
The river stream.
The curvature of a stream.
What are you doing out here?
I'm fishing.
Well, did you know you're fishing in a curve that's six times the width of the channel?
Like a total crazy person.
Just slowly backs away.
Or throws a fish at you and runs.
Yeah, yeah, right.
The water carves the land into fractal shapes. So mountains are fractals.
Wait, wait, how is a mountain a fractal?
Okay, bear with me here, cause this was also like crazy. Same with coastlines, okay? And it's because if you look, like when you zoom in, it's just the same shape as the hole, okay?
So you're just seeing a smaller portion of the mountain.
Yeah, but it's something about, it's something about the detail. Like a mountains are eroded, and same with the coastlines.
Okay.
One of the examples they use, I didn't really get the mountain thing, but one of the examples they use with the coastline thing is that-
I was like, you're gonna have to convince me. I started on the six times the width of a river, but-
Right, right, right. And you're like, but a fractal of the- Okay, so-
Yeah, convince me.
I suppose it's the definition of fractal or something.
Okay.
But the smaller your unit of measurement, the more precise your perimeter. Okay, so the example they used is, let's say somebody asked you to measure the coastline, the East Coast. And you went out with a yardstick, okay?
And you're measuring-
That would suck.
With a yardstick, okay?
That would suck.
And then you'd get a length. And then they gave you a ruler and told you to do it. You would come back with a completely different number, okay, because it's more precise.
I see, I see.
Then they send you out with six inches, and you get a different number because it's even more precise. So it's a smaller of the whole. The more you zoom in, the more it's still, it's something about the definition of fractal.
But essentially, that it's an infinite zooming in, okay? Because you could go in with smaller and smaller and smaller and smaller units of measurement and still constantly get different numbers because of, I don't know, I don't know.
Listen, we're not math people, we're nature people.
Apparently earthquakes are fractals, like the patterns that the shaking follows, fractals. Weather, lots of fractals in weather. Lightning, duh, branching.
Yeah, and that's a hard part for me about fractals is it changes, I don't wanna say it, because it doesn't change the definition, it's more so like it changes how you're looking, because there's the branching, but then there's the zooming in. And I get that it's the same principle, but at the same time, it's not.
Well, it's easy for me to picture with a fern or a tree. But there are other some things that are not right. And the reason I never thought about why lightning creates fractals, why does it split so much?
And the reason is because air is not a good conductor of electricity. There's too many things it's running into. So it's running into a whole bunch of stuff.
Right, I never thought about it.
I never thought about that, okay.
But yeah, it's hitting things and splitting.
Why did I never think about that? Okay, continue.
I know, more you know. Then, because lightning is a fractal, thunder is considered a sound fractal.
But I guess it's the same thing with music. I don't know, but apparently it's-
Where's Sarah? Sarah.
It's because the way when the air collapses, it's collapsing in a fractal, but it's then hitting your ears with that sound in a fractal pattern, which apparently is how it's in music. So- Interesting.
I don't know.
Okay, okay, because whenever I was thinking- Okay, and again, we're gonna have to ask Sarah, but when I was thinking from the music perspective, I was still going off of the Fern example, where it's like-
Right, patterns.
I was like, what do you get? Like a four beat note, and then it's like, you know what I mean? It's like the same pattern when you're zooming in, but now, if you're saying how it hit your ear, I guess it would have to be the same.
Interesting. Don't know. Sorry, I'm leaving you guys.
I'm leaving our listeners with a lot of questions. Snowflakes, fractals. Not the branching pattern.
That's like lightning, but definitely. So the origin point is the center, and there's actually a very famous snowflake called the Coke snowflake, which is actually not a true snowflake, but looks like one.
Is it an individual snowflake? Like one specific snowflake.
Yes and no, it's not a real snowflake. It's a math snowflake.
Okay, okay, because I was like, there's a one.
And named it. Now I'm gonna go out and name the Laura snowflake. Is that one?
Oh, it's melted, it's gone.
Just very fleeting. You put it around on your finger.
No, but you might have seen something like this before. So it's like a fun math thing.
Okay, you cannot use those two words. Fun and math do not go together.
All right, I'm gonna say this to you, but really you just have to look it up to see it, because there's no way to explain a shape so easily. But what you do, and you guys, listeners at home, if you wanna do a interesting math exercise, you start with an equilateral triangle. Okay, remember that is all sides of the same.
All sides of the same triangle. I know.
Okay, math, Katy, we're here. This is the level of math I got.
Okay, then you're gonna divide each of the lines into three segments. Okay, so. Got it.
Then. So far so good. Then you draw an equilateral triangle that has the middle segment.
Okay, I'm gonna try and do it with my fingers. Okay, this is your line. You divide it into three segments.
Then you're gonna draw another. There it is, three segments. Then you're gonna pop up another triangle from that line.
Okay, and remove the line segment that's the base of the triangle, so it's just open.
And then, so the first time you do this, you end up with a six-pointed star. First, you got one triangle. Then you got a six-pointed triangle.
Then you've got a six-pointed triangle with points coming off of that. Then you've got a six-pointed triangle, so you keep going, you keep going. All right, so here's where I get to.
Math is crazy. Like, explaining math really is like explaining magic. Like, I-
It is, it is.
It's so crazy. So, although the perimeter is infinite, all right, we can increase, you know, when you're measuring perimeter, like we're talking about the coastline, every time I bump a new triangle out, I'm creating more perimeter.
Correct.
Okay, you're on the same board. I've got this for good.
Yes.
Area never increases from that original triangle.
Wait, what?
Don't ask me how.
I don't understand it, but perimeter is infinite and area is finite.
How?
I don't know.
We need more math people to help us.
Because I'm like, but it's bigger, but apparently it's not bigger. I don't know. I don't know.
This is a very interactive episode, listeners. We need lots of answers from you. We're just scratching the surface here of things that we do not understand.
The mysteries of the universe.
We're just here to teach you about things we don't even understand.
All right, I'll go to For the Love of Nature. We teach you everything that we don't know. Probably one that we still don't know.
Okay, that should be an episode where we talk about things that we don't know. Like, we'll just throw out things we don't know.
Like, really bizarre questions.
Rattling on questions with no answers. Okay, two more things. Still in weather.
Last one thing about weather is clouds.
Is what?
Clouds. Clouds are fractals.
Okay, you just really emphasize clouds.
Cloud. Turbulence, which that's such a fun word to say. Just say that with me.
I've never been drinking, I promise.
Laura's not even the one drinking.
Turbulence. It's a good word in your mouth. Turbulence impacts the way water particles.
It's a good word in your mouth.
Yeah, come on.
You've never said a word.
Everybody say it with me.
It's a great word to have in your mouth.
Turbulence. Turbulence. That's what I always, I laugh with the kids at work whenever I have to say pupa.
That's another great word to say. Pupa. It's that you, something about that.
Okay, anyway.
Turbulence impacts the way water particles interact with each other, and it is fractal in nature. Okay, last thing. Space.
Galaxies. We spend a lot of times on fractals.
Galaxies are the largest spiral fractal, obviously.
Ooh, I'm gonna talk about galaxies too.
Perfect. And that is the end of the fractals that Laura knows about. That's all I know.
And I'm done.
They just form, galaxies form giant spiral fractals.
Well, that's a perfect segue that you would talk about galaxies, cause then I'm gonna talk about... That despite my very immense math learning disability, I can do some math. Laura's triangle description, whatever.
I'm... Terrible description. You just have to look at the picture.
Of this triangle.
Well, it's not so much...
Of this snowflake.
It's not so much that, it's the whole perimeter versus area thing.
It seems like it's breaking the rules of math, but I guess it's not.
The rules of everything. I'm skeptical. All right, so the first math and nature that I'm gonna talk about is the golden spiral and Fibonacci spiral, which are in fact two different things.
However, they're basically the same thing.
First of all, I feel like a lot of people have heard the name Fibonacci because of secret code stuff. But the golden spiral.
The golden spiral, yeah.
Like it's the best thing in the universe.
Yes, well, because it appears so much and it's so simple. The golden spiral. The heavens open, which I'm sure like heaven's a fractal.
Yeah, probably. Probably.
All right, so even though they are two separate things, I'm going to be talking to about them as they're one, and together they're called logarithmic spirals. So there are more than just these two within nature that are, I guess, that there is like a math equation.
A logarithmic equation, I remember that word.
That it follows, it will constantly follow the same thing. So again, even though nature can be very complex, it oftentimes relies on very, very simple rules to simply exist, like very simple. And these spirals are definitely one of them.
So a true golden spiral is formed by a series of identically proportioned rectangles. Like you can box it off. So it's not like the shape of it, it's you can fit those spirals within, or the ratios within rectangles.
So let's say if you have a spiral, like a nautilus shell that I'll talk about, you can put this proportion, like this, the big section where the nautilus actually lives, in a rectangle. And then you can put a rectangle, and then a rectangle, and then a rectangle, and then a rectangle. So you can rectangle it off, and it's all perfectly proportioned, each part of it.
Does that make sense?
Have you just seen a picture of that?
Yes, yes, yes. If you just look up Fibonacci's spiral, you'll see the whole rectangle thing. So anyway, so you can go ahead and put those in there.
And so a golden, let's see here. Hold on a second. And then Laura, even that you said, like you think that some people will recognize Fibonacci because of the name or Fibonacci's sequence, and this is where a lot of people get the coding.
So Fibonacci's sequence is one, like for an example, it'd be one, one, two, three, five, eight, 13, and it would keep going. Each number in the sequence is the sum of the two numbers that precedes it.
Okay, gotcha, gotcha, gotcha. So one and then one because it's one plus zero.
Yep. And then one and then two.
One plus three.
Yeah, cause it's one, one, two, three. So then it's, okay, so one, one, two, three, five. So one and zero, one.
One and one is two. Two and one is three. Three and two is five.
Five and three is eight. And it just keeps going that way.
Gotcha. Yep. And so if you do that equation, it shows that shape.
Not only that, there is a math equation. There is an actual math equation that's like A minus B plus whatever.
Some kind of logarithmic thing. Yeah, yeah.
That makes it way more complex. But it's the same concept as Fibonacci sequence, which is why they call it Fibonacci spiral, because it's, again, like I said, it's like those rectangles that if you take the rectangle and then the rectangle and then the rectangle, and it all pour proportions out. So Fibonacci, he was, his actual name is Leonardo of Pisa, whose nickname was Fibonacci, which, who gets the nickname?
What does Fibonacci mean?
Well, I'm glad you asked, Laura, because.
His nickname is, I mean, you'd have to have a fun nickname that meant something.
Yeah, but it's not, well, it's not that exciting, but it's Phileas of Bonacci or son of Bonacci.
Oh.
That's all it is. So my nickname would be daughter of Phil. Yeah.
Whatever that Latin is.
Philia. Philia. Philia phyllis.
Philia phyllis?
I hope not.
Philopheus.
Philopheus. Everyone can now call me Philopheus.
Thank you very much.
All right.
So in the early 13th century, although the sequence can be traced back to 200 BC in Indian literature, the 13th century is whenever Fibonacci kind of took it. Well, I don't want to say took it. He basically took credit for something else that was done way prior to his existence.
And this sequence has produced a large amount of literature and has connections to many branches of mathematics. Fibonacci's sequence, all that stuff. So both of the spirals, though, at least I think-
Cause what makes the golden different?
I'm gonna get into that and-
Oh, okay, okay. Sorry, jump on the gun.
Two seconds, yeah. So one of the, I think that these are kind of the most iconic, like whenever Laura and I were talking about this episode, I knew immediately this is what I wanted to do, partially because I do suck so bad at math. And this is one thing that I did recognize.
So as Fibonacci spiral increases in size, it approaches the angle of the golden spiral because the ratio of each number in the Fibonacci series is the one before it converges on pi or the golden triangle number is 1.618. So everything around the golden spiral, even though they're freaking both spirals, again, both spirals, whatever, they're like, the golden spiral is based off of 1.618, and there's some stupid math equation that you need to get that. But then Fibonacci is that Fibonacci sequence, and it's the same ratio, but it's freaking, it's still-
Like it kind of comes out to look the same, but the golden one specifically is-
Yes, it's a freaking spiral, 1.618. Yeah, and that was done by a German mathematician, Martin Ohm, which I don't know if that's the same guy that we get the ohms from. At the unit measurement, but anyway.
Maybe.
I don't know. So further back in time, the spiral itself was described as divine because of its frequency that it was found in the natural world. In mathematics, the golden ratio is a special number that's approximately, like I said, equal to 1.618, and represented by the Greek letter phi, or just a circle with a line through it.
I don't know why Greeks have to make things way more complex than what they actually are. So the golden spiral is a pattern created based on the universal law that represents the ideal, quote unquote, in all forms of life and matter. And it genuinely can be found like freaking everywhere.
The ideal, I mean, I feel like that's very subjective.
That's also a very strong opinion. Right.
This is, listen, you are not the ideal. You have not reached the ideal ratio.
I mean, an idealist is ideal ratio.
Yeah, yeah, yeah. That's a very human centric of us.
We're not ideal. So in fact, it's all often cited as an example of the connection between the laws of mathematics and the structure of living things. So this is one, again, like if you're gonna talk about mathematics and nature, this is typically one you're always gonna come across.
So the golden spiral comes from, I already kind of talked about it. Now, where the Fibonacci's, it does have a rectangle, basically a golden spiral. I don't know.
I started reading more and more into it. Apparently, the rectangles that make up the golden spiral are like better rectangles.
Perfect. They're the ideal rectangles, Katy.
That's what they are. I just read about them.
Rectangle.
So anyway, so let's go ahead and jump into examples, because I'm so done talking about math already, and this is just the first one. Let's see. Yeah, so the first example I'm gonna talk about is of course the nautilus shell, because it literally is the spiral.
It is the spiral. So if you caught our squid episode, we briefly did mention nautilus because they are cephalopod. Nautiluses live in the South Pacific, hundreds of meters beneath the surface of the ocean.
And I started again with this one because it is exactly the Fibonacci spiral is. It starts as a small spiral and proportionately spirals outward in the exact ratio, which gives us the Fibonacci sequence. Which I also didn't know this too, but nautilus, they make their shell by mixing sugars, proteins, calcium, and other minerals.
And then they add the resulting crystallized material to the lip of the existing shell. I guess again, I just like never thought, how do you make it? Because they do keep growing and that's how they keep growing.
So I do think that's neat though, that that's how they do that because every time they're adding on to it, it still ends up being perfect.
In the same proportion.
What the heck?
We can't do that. Humans can't do that. I can't even draw a circle and have it look the same twice.
I can't draw a perfect circle.
Good Lord.
So anyway, but like we said earlier, a lot of these things that are found in nature are doing this because it is, I mean, apparently easy. I don't know how a nautilus does it. Like how the heck do they get it?
Are all snail shells Fibonacci spirals or they're just spirals?
I don't think so. I think they're just spirals because the proportion is very specific. Yeah, well, not only that is a very specific for like the Fibonacci, but also it is symmetrical, the right word I wanna use.
You can look at it and you're like, oh, boom, everything's the same. Everything's exactly the same. And I don't think it's snail shell is.
But another example of Fibonacci spiral is hurricanes. Okay, so by using computer, scientists have kind of been able to study the shape, well, we have been able to study the shape and size of hurricanes. And I was getting, God, we know how we always find these things that scientists weirdly have strong opinions on?
Yeah.
This is one of them. Some people are like, oh, it's Fibonacci spiral. Other people are like, no, it's not.
So it's apparently heavily debated.
I guess it's where you decide where the edges of those hurricane are.
That's exactly what it is. Is we're like, we can never fully decide it. Listen, and for my book, does it look like a spiral?
Sure, close enough. Like, especially because of how you can't, if you were to measure it, it is pretty darn close, close enough. And so of course you would look at it as the eye of the hurricane is the starting point.
And then you just keep going out in that Fibonacci sequence, and it should give you, I know that a lot of the pictures I was seeing was like Hurricane Katrina. They were using that one to show like how easily you can make a Fibonacci spiral over it.
Boy, I wonder if it was a golden spiral, because definitely then you'd be like, oh, divine, like that, I feel like people could really spin that some terrible ways. Right, this is some terrible ways. Like divine punishment.
Yeah, right, oh God.
Which it's not, it's not, there's no way.
It's not, but yeah, so that was interesting. Another example that is not fought over is a pine cone.
Oh, I think I ran into that myself because the pine cones are fractal.
Yep, it is. So if you pick up the pine cone and look at the bottom of it, you will quickly see the Fibonacci spiral. Now it's not like one single spiral, like a nautilus is, but it is a spiral within a spiral within a spiral.
So it's like on stock, stacked on top of each other. Yeah, so it's not like one big spiral.
Yeah.
It's they're stacked on top of each other. And like you have one that's like the perfect, the Fibonacci spiral, and then you have another one, and then you have another one. And so it's like a bunch of them perling together in the line.
And so then the last one I'm gonna talk about is the last example, it's out of this world.
Yeah, galaxies.
Galaxies, yeah. And this one includes our own, the Milky Way, and a couple of others you might recognize is the Andromeda Nebula and the M81 galaxy. Listen, I read so many different things about these three galaxies, and everybody's like, the one, the three most recognized, like the Milky Way.
Nobody's heard of the M81.
No, nobody's, but so many people are like, you must know it as the M81. No, nobody knows. Anyway, so this one, you can say that the golden spiral can be found in the shape of the arms of the galaxy.
Yeah, if you look closely and do the math. So again, debate, some people are saying, yes, it is the perfect golden ratio. Other ones are saying not so much because again, how can we actually measure it?
Define the edges.
Yeah, but like, come on people, just give the Milky Way a break.
It's not a KitKat, it's a Milky Way.
So for the Milky Way, several people theorize that if the measurement of orbital distances of planets started from Mercury, which is the first planet in our solar system, the average of the mean of planet orbital distances of each successive planet is taken in relation to the one before it, which is insane.
That is crazy that they follow a rule like that.
Yeah, the planets circling something is just like, you know what we should do? We should be perfect. Let's follow this perfect spiral.
I'm about it.
Yeah. So anyway, those are the three examples of Fibonacci's or the Golden Spiral. So it's everywhere.
If you start looking, I'm gonna have to, next time I go out on a hike, actually look for it. I mean, besides-
That's fame with Frafters, too. Right, you're just gonna start seeing them everywhere.
You're gonna see them everywhere. All right, Laura. Hit to your second one.
I kind of love the rule thing.
Okay.
Right? Not, because I also, I mean, nature's always breaking them, but I like that there are these underlying, hidden rule things.
They are, yeah.
So, right, you think I didn't know very much last time? Well, I really don't know very much about this one.
Hi.
But I found it intriguing, so I dove as deep as I could, and I think I did spend more than one hour looking at this. Universality, okay, it's a concept. And from what I could understand of it, universality is also known as disordered hyperuniformity, which is like completely opposite things.
I'm just like, wait, say again.
Disordered hyperuniformity. So absolute chaos was absolute uniformity.
I feel like that describes my life.
Yeah, disordered hyperuniformity. Yeah, I feel that. It's newly discovered in most instances.
So this is like, it's been starting to pop up here and there in the most random of places. And like now mathematicians are starting to take notice of this, what the result is, this disordered hyperuniformity is popping up all over the place, which is why they're calling it universality. So it's basically, if you want to sum it up, it's hidden order within chaos.
So yes, it's our lives. It's our lives. It's our lives, hidden order inside of chaos.
Uniformed disarray, the place between order and randomness. Okay. It determines the spacing between solutions.
All right, here's where it turns into math. It determines the spacing between solutions in a large matrix of random numbers. Okay.
It's when complex systems figure out the best way to work. So let's say a system is just crazy complex.
I hope my life figures out a way to work.
It's got to figure a way out, and it just does it. It just does it.
That explains so much in my life, though.
It actually, depending, I found another article where they're actually saying that it, weirdly enough, could possibly be considered a new state of matter.
This is where we're gonna have one of our crises again. We're gonna start to freak out here any moment now, folks.
Yeah, cause I, so it behaves like crystals and liquids. So that's where you get- That's where you get your ordered disorder.
You don't know what the heck. Very rigid, like a crystal. And loosey goosey, like a liquid.
This just describes like drinking Katy and non-drinking Katy.
That's all it really does.
The arrangement of it is highly uniform like a crystal. And it has the same physical properties in all directions, which is like a liquid, does. So order over large distances and disorder over small.
So I had to look up like a picture of this, and I'm gonna give you a very good concrete example of this.
Please do. Because basis, please do.
Let's say I have a circle and there's just disorder. There's something like I have thrown tic-tacs all over the floor, okay? And I throw a circle.
If it's a small circle, I'm gonna see chaos within that little circle. If I throw a hula hoop, I may see a pattern. It's like God, okay?
So you need to see those things.
He zooms in on one person, he's like, this person's just jacked everybody together. This makes a little more sense.
I'm starting to see a pattern here. Yeah, yeah, yeah, yeah, yeah. So that's what it is.
Mathematically, there is a formula. Like I said, it defines the spacing between eigenvalues. Don't know what that means.
Okay, it's gotta be- It's gotta be a name.
It is, it is a name.
Eigenvalues, one word.
Oh, it's one word.
One word, an eigenvalue in a random matrix. Okay, so where can you find it in nature? Because this is actually gonna explain it to us.
Okay, in nature, it can be found in, I'm gonna go through the ones I'm not gonna explain, then the last one I will. It can be found in-
Let me give you all these examples I'm not gonna explain. Go ahead.
Liquid helium, simple plasmas, the nucleus of uranium, human bones, sea ice, and that's nature, also in the internet, a bus system and climate change models. That's why it's called universality is because they're seeing this weird order from chaos in all these different places. The bus system was a really kind of like an older example and now like sea ice changing plasma.
Okay, well, what connected the dots for me is chicken eyes. Guys, I'm not, this isn't a joke. This is not a joke, people.
Okay, so, chicken. Chickens have, I'm already gonna get it wrong because of course I didn't write it down, but I'm pretty sure it's five different rods and cones. Okay, which is more than we do.
And all of these little like rods and cones are different actual sizes, like diameters, okay? When you look at a chicken eyeball, or like under a micro-
That I do, you know, that I do on the daily.
Or just stare deep into those chicken eyes. Now, you'd have to do it under a microscope so you can see the rods and cones. But if you look at it, it just looks like crazy chaos.
There's little dots all over the place, and those dots are different colors corresponding to the rod or the cone. But there is actually a pattern to this if you look at them as individual rods and cone clusters. Again, the order from disorder.
Somebody else described it as, let's say you have a box of marbles, and you shake it up, and then you open it. That is disordered hyperuniformity. Those marbles have found the best way to organize themselves from chaos.
Sorry, I'm laughing just because of your example.
You're like, you're looking down, you're like, and you shake it, and you open the lid, and then you look up at me.
Which is your example.
Apparently, it's the best way that the eyeball figured out how to fit all those rods and cones in. Yeah, yeah, yeah. So as I know we've talked about before too, I always think about our bodies as just being like little factories full of little people.
So it's just all these little guys trying to figure out how to best organize this eyeball.
A chicken, no less.
They're like, guys, we have to figure this out. And it looks like they're running around like crazy, but they've actually figured it out.
Like a chicken. They're just running around.
My desk at work, like, you know, things like that. Disordered hyper uniformity. That's all I got.
I tried, it's the chicken eyes the best way. If you want to know more about it, you're gonna have to read more. Definitely look up a picture of the chicken eye thing.
You realize if you just Google chicken eye, it's just gonna come up.
No, no, no. Look up universality chicken eye, and I'm sure it will show you the rods and cones.
That's what I Google on the daily, not just chicken eyes.
So yeah, it seems like it's a rule that they're just figuring out. But it's something about the distance between eigenvalue, whatever those are.
Well, thank God you went first, because I'm going to finish this up.
We're going to talk about hexagons.
Yay, shapes.
So Laura and I realized we were waying over our heads. We're going to wrap it up with shapes.
Shapes.
I got it.
At least hexagons aren't basic shapes, right? I mean, they pretty much are. Yeah, but hexagons are not when you learn in preschool, I don't think.
Right, I don't know.
I don't know either. I'll just find out soon.
Well, there's a lot of hexagons in nature.
We should be learning about them in preschool, I guess.
Yeah, right.
There's a lot of them.
There's a lot of them. And again, this one isn't quite as complex as what Laura and I have been talking about, but a hexagon, I'm not going to explain to folks what a hexagon is. A hexagon in the shape of itself is the shape that best fills a plane with the equal size units and leaves no wasted space.
Love it. So out of everything, this is the most organized.
The most organized of being like, I've absolutely got this.
This is not even a disordered hyper uniformity.
No.
It's just hyper uniformity.
Yes, just super, I've got this. All right, so hexagonal packing also minimizes the perimeter of a given area because of its 120 degree angles. So because it's like very specific angles that also factors in, not just, oh, we've got like the hexagon, but also that it's 120 degree angles.
That's what makes it like, listen, this is a fantastic shape. Forget your other shapes.
Why is it 120 degree?
Wait, 120 degrees?
Because it makes it completely.
Oh, like the angle.
Yeah, yeah, yeah, sorry, sorry, sorry. The corners, the corners.
Okay, gotcha, gotcha, gotcha.
Yeah, not like the hexagon. So with this structure, the pull of surface tension in each direction is the most mechanically stable.
Okay, that's what I was thinking. Because it's just like the fractal plant.
Yes, exactly, yeah.
But it's spreading of the stress.
Yep, yep, and that's what it is.
Or the load is sharing the load.
Yes, and so only three regular shapes with all sides and angles identical will work. Being ideal as far as filling in all the space, all that stuff. And that is an equilateral triangle, squares, and then hexagon.
So there's only three. Of these, hexagonal cells require the least total length of wall compared to triangles or squares of the same area, which is interesting.
Because are they just saying wall as in one side?
Yeah, yeah, to create.
Because you would think that.
All around.
I mean, is perimeter the same? I guess it could be.
Yeah. Well, it has to be if it's equilateral.
No, no, no, no, no. I mean, like, is the perimeter of a hexagon the same as an equilateral triangle, the same as a, but I guess it's just the size of the.
Yes, the size of what it is. But anyway, that's like the introduction because it's like hexagons. Like, it's just, it makes the most sense.
So the first example that I'm going to talk about are honeycombs.
And those guys know what they're doing. If any animal has their lives together, it's bees.
It's bees, yep, they know it. One weird thing, though, that I didn't know was that the honeycombs is when it's built, it's aligned to the Earth's magnetic field. Right?
That's so cool.
Bees, ma'am.
Bees, which we talked about before about how great they are.
They are.
I love when we get the hive in at work, the observation hive, we always leave an empty section, and they have to build their own. Usually, usually when you stick the frames in, it's already a honeycomb, but they have to build their own. And it is amazing to watch them do it because it is perfect hexagons.
Every single time.
And that they do change it, we just can't see it, but the size of the hexagon determines whether it's gonna be a male or female.
Yep, yep.
And like the queen can feel it with her butt.
With her butt of what it is.
That's a girl one. Right?
So let's see her honeycombs, in which bees store their amazing bee spit, essentially, because it's not vomit, because it never reaches the stomach. So technically, it's not vomit.
Yeah, good clarification. So just spit.
So as these champion spitters are making honeycombs, they really are done with precision engineering, produced by an array of prism-shaped cells, with, again, perfectly hexagonal cross-section. The wax walls are made with very precise thickness as well. The cells are gently tilted, again, like I said before, from horizontal to prevent the honey from running out, and the entire comb, as I said earlier, is aligned with Earth's magnetic field, which is just ridiculous bees.
Yeah, that's, yeah.
Come on. So there was a Greek philosopher, Papas of Alexandria, that thought that the bees must have a certain geometrical forethought. Like, they thought that bees were like, you know what I need to do?
I need to make a hexagon. Like, he thought that they must know.
How would we know if they don't?
Well, we're gonna prove it here in a second. And who else at the time, Greek philosopher, would have given them this wisdom, but God? So, you know, divine bees.
Another individual, William Kirby, in 1852, said that bees are heaven-instructed mathematicians, which I was never, that would never have described me in any of my lives. That's so cute, I love it. Then of course, here comes the party pooper of all former thought, Charles Darwin, Yeah, yeah.
Who wasn't so sure of this divine theory, of course. So he conducted experiments to establish whether bees are able to build perfect honeycombs using nothing but evolved and inherited instincts. So as his theory of evolution would apply.
So here we go. Darwin recognized that explaining the evolution of honeybeams, their comb building abilities was essential if his theory of natural selection was to be taken seriously. So apparently, I didn't know this.
I mean, I read like the, a lot of Charles Darwin stuff years ago, but I guess I never picked up on this. Because apparently, he did a lot of stuff with bees. I just didn't listen to him.
Yeah, I didn't know that.
So he started doing an experiment with bees at his home. And thankfully, he, as we all know, Charles Darwin documented like everything, thank God, because then we, it's easy to go back and to read what he was actually doing. And so Darwin found several things, but one prominent thing that he found was that bees acted accordingly to two antagonistic principles.
One, causing the bees to deposit and excavate the wax. The other, limiting the degree of excavation. In his views, bees set out to make circular shells, which became hexagonal due to their working under the constraints of two, and the next.
Okay, so Darwin found the two thing, that was one problem, that was the key thing. I mean, he found several different things that he was talking about. First of all, there are some bees that don't do hexagonal.
Like there's some Mexican bee that doesn't do hexagonal. They actually do circles, which seems to be a waste of space.
Well, I mean, I guess some animal, right, if you're gonna do, because so many native bees just build tubes for themselves.
And so that's when he found was that originally a lot of the bees set out to make circles, but then it ends up forming hexagons, just because as they're going and they're like laying these down, it's like the honey and everything is just like filling in the space. And it happens to go from like a circle. And as you keep piling them on, it just expands essentially into a hexagon.
Like they are making them into a hexagon, but at the same time, like that honey is filling the space. So basically the bees are making that shape because it's the easiest, because like the bees don't want to waste space. They don't want to waste time.
So they're just, that's the shape that works. So they start out making circles, but they kind of just like, as they're like packing it, I guess, and everything, it ends up being a hexagon. So it's just the easiest thing to do.
Again, seems complex, but super simple. Another example of hexagons in nature is a bubble raft. Now, I wish this was something you could ride on.
It's not, but this is actually something that you could do at home and see the results of for yourself. If you ever make a grouping of a bunch of bubbles all together, the bubbles actually do become hexagonal, or again, yeah, close up. Closest.
Close enough, which is my type of math, close enough. You'll never find a raft of square bubbles. I mean, you won't.
If four bubble walls do come together, they instantly rearrange into a three-wall junction with more or less equals of 120 degrees between them. And one website that I came across said to imagine a Mercedes-Benz symbol, so I ran out to all my Benzes in the garage. No, just kidding.
I'm a millennial, I'm poor. But unlike the bees with a bubble wrap, it's not like something is sitting there like trying to arrange the hexagonal product, you know what I mean? Like the bees are controlling that pattern, whereas like a bubble is just a bubble.
Like it's just gravity.
Yeah, it's just doing what it needs to do. And so they form it in the cells. But again, just like with the spiral, it seems that this happens because it is literally the simplest form, and nature has evolved in such a way that it is always going to take the path of least resistance and even more so without a creature controlling it.
Like if the bees aren't controlling it, like this is a natural shape, this is very easy, so let's just do it this way.
Well, right, and I mean, easy, I feel like it's a weird term to use for something that's not alive. But that there are, I mean, it all has to do with bonds of water.
Exactly, exactly. It has to do with the surface tension and everything. Again, this is the shape that makes the most sense to fill the space, to have, like we said, like every angle, every side of it, carrying like an equal load, playing its part.
Yeah.
So that's all I have. I have hexagons, much as I'm learning than anything else.
Yeah, the shape, but not the bee thing. I'm still like trying to wrap my mind around it. But what I was gonna say is, you know, at the beginning, we said that whether you believe, so through the research, and at the very beginning of this, when I was talking to Katy about what do we wanna talk about with this, is that there are two different viewpoints of mathematicians.
Some believe that there are universal laws that without us, there's still math, and we just discover math. And then other people are like, no, no. Math is a human construct that we use to explain phenomena.
Manamana. Boop, boop, boo-doo-doo.
What do you think?
I think it just happened. I still think it happens to be like, I don't wanna say the path of least resistance, but I think it's the path of least resistance in that humans have just stumbled upon figuring it out.
That's what I think. I think it's a human construct. I think that those things exist, but it's not math.
It's that it's just...
Because you think about anything with math, like you can assign anything to anything. I mean, it is interesting. I know with my golden ratios and everything that it is equally proportional and perfect every single time.
But I think if you're looking for it, you're gonna find it, sort of deal.
Yeah, definitely. I think I'm more along the lines of that humans have created math to explain things, not that we're discovering math. I don't know, though.
I don't know. I suck at math. I know so little about it that I really think it could convince me either way.
Yeah, I know. I'm dumb, so it doesn't even convince me either way.
It's really interesting. Oh, there is, if you're interested, the Mandelbrot, the fractal guy, he wrote The Fractal Geometry of Nature. Like that's, he discovered fractals in nature was how he kind of discovered fractals.
So if you are a person that listened to this and you're like, yes, math and nature, this is my thing.
I need more.
Then you need to check out the book, The Fractal Geometry of Nature.
Interesting. All right, guys, well, that wraps up talking about our episode today about math and other things that Laura and Katy can't understand. So we hope you at least learn something about math and let us know your viewpoint one way or another.
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Yeah, and join us next week because next week is our last episode of the season, which is crazy. It's crazy to think about, but I'm really looking for next week's as long as it all works out.
Right.
Now that you know more than you wanted to know, your curiosity should be peaked, and hopefully you don't feel overwhelmed and you care just a little bit more.
I'm overwhelmed.