SYSK Selects: Fractals - Whoa

I'm Josh Clark, hanging on by my fingernails with me as always as Charles W. Chuck Bryant, doing much the same as we are about to start speaking on stuff you should know about. Fractals. Yea, more math theoretical, Matt even Yeah, a new branch of geometry. It's non Euclidean since you brought it up, Okay, very new. Euclidean geometry was like three B C and fractals are so there's a little bit of a gap there. There is a little bit of a gap and uh, there's a lot

of animosity among the Euclideans towards Fractillians. They need to loosen up and look at some of those far out pictures. I know, you know it's funny. Did you watch um, did you watch that one doc on? Yeah? Okay, did you see the other? The Arthur C. Clark one. It was made in like maybe eight s eighty seven and it had nothing but like um delicate sound of thunder rip off music going on the whole time. It was

really really trippy. Well, I posted a picture I don't know if you saw today on the stuff you should know all of the of the Mandel brought set. Its beautiful it is and it's very cool. And I didn't even say what it was. I just posted it, and like I'd say, about half the people were like, very cool, man, this is rad I love the Mantal brought set like fractill talk about fractals. And then the other half were like, well you guys tripping out like what you did a

grateful dead day. That's actually math, believe it or not. But it does look very it's very tied eye in nature and that's why the hippies like it. Plus also, I mean, if you've ever seen a fractal play out on a computer screen. Yeah. Um, so we are talking about fractals. I don't I don't necessarily want to give a disclaimer. Chuck and I are not theoretical mathematicians. We're not even like normal mathematicians. I balanced my checkbook my hand just to keep that little part of my brain going.

So I don't like forget how to add and subtract later on in life. I make myself do that, and I don't let myself jump ahead. I show my work. Yeah. Um, and that's about the extent of math in my life normally. See, I was the kid in math that when they said you're not allowed to use calculators, I would go, like, there are calculators in life, so why can't we use them. Yeah, Like they made calculators so we didn't have to do maths, right.

But at the same time, I find that shoddy because it's like, you're not you're not You're just circumventing learning something, and it's like the calculators there to support you after you know what you're doing. I disagree. Well, I think this is a pretty prime example of like going around to get to the end. So when when I was researching this, I was like, Okay, well, they don't really know what they're doing with this stuff yet, so we can just totally be like, well it's it's there, anything

you wanted to be and nothing at all. And then like I started looking a little more deeply into I'm like, oh, no, they do kind of know what they're doing. We really

don't know what we're talking about. So I feel like I have, just from researching this, a little bit, um something of a grasp of what fractals are, a little bit For those of you who who don't know what we're talking about, like, take a second to um look up just typing fractal and search images on your favorite search engine and you'll be like, oh, yes, of course it's a fractal um And that's what we're going to talk about, because fractal fractals are a new field, like

we said, in geometry, and they do have use and they have usefulness that I think people haven't even considered yet. But the the stuff that they have figured out how to use it for is pretty amazing stuff. Can I say what a fractal is at least so people know they should clear it all up? It is a geometric shape that is self similar through infinite iterations in a recursive pattern and through infinite detail exactly. So there you have it. Boom, Do we need to even continue? No?

But um, and that sounds like really that put me off, Like this article was pretty well done by a guy named Craig Haggett. I don't know who that it is, freelancer. I guess um, it's a pretty well done article. But that a sentence like that can put a person off pretty easy. And he even put it, you know, he made a joke about it, like, oh, you know that,

you get it, you know whatever. But um, when you think about it, if you take that apart, one of the hallmarks of fractal fractals, um is that they are a very complex result from a very simple system. And there's like basically three hallmarks two fractals that you just pointed out right. There is um self similarity, which is if you if you cut a chunk, like a microscopic piece of a fractal off and compare it to the whole fractal, it's going to be virtually the same. Yeah,

like or a fern. And the cool thing about fractals is is to me the coolest thing is that fractals. The point they made in the Nova documentary is that all of our math up until they discovered fractals and described practicals was based on things that we basically created and built. Like all geometry, right, Euclidean geometry, you have length, width, and height, which should view the three dimensions, right, yes, for like pyramids and buildings and combs and all those things.

And you it's extremely useful and we've done quite a bit with this. But what Euclidean geometry, as far as the fractal geometrists or geometers um insist, failed at is when they said, okay, look at that mountain. That's a cone. It's an imperfect cone, it's a rough cone, but it's a cone shape, right, So yeah, Euclidean geometry holds sway.

What the fractal geometers say is, yeah, you could say that it's a cone, but if you tried to measure and describe it as such, you're not going to come up with a very descriptive, a very um detailed description of that mountain. So what's the point What fractal geometry does is it says we're going to describe that mountain in every little craig and peak possible. And so what you have is the fractal dimension, which exists in conjunction

with length, widthin height. And what the fractal dimension describes is the complex city of the object that exists within those three dimensions as well. That's right. So finishing my point, the cool thing about fractals is that everything that we had done previously in geometry were because of things we've built. Practicals help describe things that were have been here since the beginning of time in nature, and one of the truest examples of that is the fern. Right with self similarity.

You take a little snippet off of a fern, although you shouldn't do that. Let's just look at it. Uh, it's gonna look the same as the larger part of the fern, and then the whole fern itself very self similar but not necessarily exact. No, it can be. There is a form of self similarity that is exact and precise, but in nature that's rare, if not just completely not found. Right, that's right. So you've got self similarity, which is the smaller part is virtually the same or looks the same,

or structured the same as the whole UM. And this process of self similarity UM going larger smaller in scale

is called recursiveness, right, And recursiveness is UM. Like you know those paintings where it's like a guy I think Stephen Colbert, the one that he gave to the Smithsonian has recursiveness in it, where it's a man in a painting standing in front of like a mantle, and above the mantle is the painting that you're looking at, and then it goes on and on and on and on and on anything that's infinitely repeating, right, same with if you're in a dressing room and there's a mirror on

either side of the wall, you just keep going on infinitely. It's recursiveness and with fractals the recursiveness of self similarity. Right. So there's two two traits. UM is produced through this thing called iteration, that's right. And that's where you say, here's the whole I'm gonna put it into this formula, and the formula has has the formula. The output of the formula produces the input for the next round of that same formula. It's a loop exactly, so it's self

sustaining and it can go on infinitely recursion. Right. That's right. So what we've just come up with is a fractal is anything that has a self similar structure and it's recursive through iteration. That's right. Okay, So um A, really I came upon this kind of easy, one easier explanation of a fractal from Ben wal Mandel brought site. He died, by the way in two. He seemed like a pretty

good guy. He was definitely thinking different. Um. And the way that Mandel brought described a really easy way to think of a fractal is um. There's this thing called the Serpinsky gasket, and you take a triangle and you can combine them into a bunch of little triangles and

spaces triangular spaces that form a larger triangle. Right, So that that one initial solid triangle is called the initiator, that's the original shape, and then all those other triangles combine that form that larger triangle or a self similar version of that larger triangle to the original triangle. That's called generator. Right. So the formula for creating a fractal would be to go into that generator, the version that has all the little smaller triangles that make up a

larger whole triangle. And say, all the ones that look like the initiator, the original just solid black triangle, take that out and swap it with the generator version, and all of a sudden you have one that's exponentially more detailed. There's more to it, And that's a fractal. That's all there is to it. You know what else is a fractal?

What the coastline? Yeah, that was a big one. Lewis Fry Richardson was an English mathematician early twenty century, and he very brilliantly said, you know what, if you take a yardstick and you measured the coastline of England, you're gonna get a number. If you take a one ft ruler and measure the coastline, you're gonna get a different number. If you take a one inch ruler and measure the coastline,

you're gonna get a different number. And it's basically infinite in that the smaller you go with your your unit of measure or your tool is the larger number you're gonna get. Because the coastline is so infinitely varied in its little nooks and crannies, right exactly It's a very

cool way of thinking about it. There's a second part of that to Chuck, is that so depending on the you, what you're using, the measure, the tool you're using, the measure, the number, the perimeter of that coastline could go on infinitely, but it still contains the same finite amount of space

within its paradox. That is a big time paradox because things aren't supposed to be infinite and finite at the same time, right right, Um and uh Lewis Fry Richardson he basically established in that coming up with that paradox, this kind of revolution and thought that fractal geometry is based on that. You can have the infinite mixed with the finite. You can get it from pretty simple formulas

that create very increasingly complex systems, right Um. And Fry wasn't the He wasn't He was the first guy to really kind of put forth this idea of thought, but

he wasn't the first one to notice this paradox. Yeah, and before people even knew they were fractals, there were there were artists like da Vinci that saw this pattern and tree branches that was um I know in the Nova documentary and the article, they point out the uh Katsu Chica Hokusai Japanese artists created the Great Wave off Kanagawa, and uh those are fractals. It's a it's ocean waves breaking and at the top of the crest of the waves are a little self similar or waves breaking off

into smaller and smaller self similar versions. And that's a natural fractal, or in this case, it's a depiction of one. So they were you know, early African and Nabajo artists were doing this and they didn't realize that they were fractals and that there were fractals all around us. No, they just saw crystals in a snowflake or another good one, yeah, exactly. Um, they were just they saw that there was what they were looking at was a repeating pattern that was self

similar and recursive. Right, yeah, that's it. That's a fractal, right. Yeah. And and Ben Wha Mandel brought was the first one to say, you know what, we can we can use math equations to actually apply to this. And he was a big star for a while, and then they sort of turned on him and said, you know what, this is all cool and trippy looking, but it's useless. Right, and he said, oh yeah, screw you guys, watch this and he wrote another book which started to uh give

some practical applications which are pretty exciting. UM. So the whole thing, the whole principle that is based on UM is that you can take a formula and plug in a very simple UM, well, a relatively simple formula like mantle Brod's formula. Will take that one. For example, his is um ZED goes to ZED squared plus c. Right, that's what it's called. If you're in England, zed, we say z Z. Well, anyway, Zed goes to which is

and the goes to is the key right here. This is what makes it fractal goes to means that um,

it's an error. It's an equal sign. It looks like an equal sign with a part of an arrow pointing towards ZED, the other point pointing towards the rest of the formula, which means that the the there's that feedback loop where it's like, okay, once you have the number that this punches out, you have, you feed it back can and you'll get another number and knows, just keep going and going and going, and every time, remember you're swapping out the original the initiator for the the detailed

version the generator, and it's just getting exponentially more complex with just that one iteration of that very simple formula. UM and Mandel brought set Uh. This is the one that's like it's probably the most famous one. That's the one that the Deadheads like because it's like this crazy juxtaposition between like black and like different colors and everything. And with his formula, two things happen with the number that you put in. It either goes towards zero or

it shoots off to the infinite. And what they did for this for the the Mandel brought set fractals was they assigned a color to a number based on how quickly it goes off to towards infinity. Right, so let's say that you have like four, If you plug four into this and in ten generations, it'll it'll become an infinite number. UM. Then say that that would be grouped into a blue color like ten generations blue, eight generations

is red, ninety generations is orange. See what I'm saying. UM. And then the other direction, like say if you put in four point two or something like that, it'll go towards zero and any number that eventually will go towards

zero is represented as black. So what you have then, is this really intricate depending on where you're zooming in or out on the fractal, this intricate change of colors, and what you're really just seeing our numbers that are plots on a plane, and that's your fractal, and then the black parts are numbers that will eventually be be zero. Right, And most of the mental mental brought set is black. Yeah, but if you zoom in, like that's the whole point. You zoom in on one of those little uh what

do we even call those little spikes? Uh? I guess you could call it a plot. A plot, and it's gonna look like what you just saw. And the Nova documentary is very cool when they zoom in on these, it's sort of mind blowing. Yeah, it is very I strongly recommend watching that because they explain it way better than us. Well, it helps to see it for sure, Oh yeah, big time. So um or draw it as I have done. It's a pretty nice little fract Yeah.

So we've talked about fractals, We talked about the Mandel brought set, we talked about where they started to come from um and the the idea. Remember Lewis fried Richardson, he was talking about measuring the coastline and going off into the infinite, but still containing a finite amount um. A guy came after him named Helga von Coke. He came up with a Coke snowflake, which is pretty cool.

If you take a straight line, or you take a triangle, and then on each side of the triangle in the middle you bust out the middle into another triangular hump. You do that over and over and over again. It goes off into infinity. Although it contains a finite amount

of space. The perimeter goes off to the infinite. A guy named Georg Cantor came up with the cancer set, which is you just take a straight line and you take the middle out of it, and then for each of those two lines that produces, you do the same thing and it just keeps going on and on and rather than going to nothingness like you're like, well, if you take a six inch line, eventually you're gonna bust it down and nothingness again. That doesn't happen. They found

that it goes off to the infinite. So they realized Ben Wall mantel Brought was plugging all these into computers, because that's what it took people realize this, Like George Cantor um Man I hope that's how you say his first name. He was he was working in the eighteen eighties, Um Gaston Julio came up with the Julia sets for

producing a repeating pattern using feedback loop. All these guys were like nineteenth century early twentieth century mathematicians and it was strictly theoretical until the late seventies when guys like Mantel Brought who worked at IBM, started feeding these things into these new fangled computers and seeing the results like this fractals like the mantel Brought set that he saw right, So Um, almost immediately there was a practical use for fractals that came in the form of c g I. Yeah.

They interviewed that one guy in the documentary um who worked on the first c g I shot in motion picture history, which was Star Trek to the Wrath of con and Uh. He was tasked with making a c g I uh land surface like mountain range and pretty mind blowing with it. Yeah, and he did. I mean, now you look back and it kind of looks silly,

but at the time it was completely revolutionary. And once he learned about fractals in the geometry and the math of fractals, it was pretty easy for him, and he made it seem like he was like, oh, well, this is the key, this is how you do it right. So well, and it is kind of easy, especially if you know what you're doing with computer programming and math, because what you're basically doing to create a fractal generator is teaching your computer to to do something within a

certain formula. That's your fractal formula, right, And so what Lauren Carpenter, the guy who created the the Star Trek to landscape for the first c G all c g I shot ever, what he basically did was created a computer program that said, hey, computer, I'm gonna give you a bunch of triangles. Because I think that was the

earliest stuff he was working with. Um, I'm gonna give you a bunch of triangles, and I want you to take those triangles and generate a new fractal set from it, right, And then I want you to do it again and again and again, and then every third time I want you to start turning them forty degrees, so it's going to change the pattern slightly, and then all of a

sudden you have these infinite variations. The reason why when you go back and look at that shot that it still looks kind of you know today, is because the computer he was working at didn't have the computing power to do that many times. Now we have higher computer computing power, and so what we're doing is telling our computers to keep going and going and going, swapping out that initiator, that one single black triangle everywhere it can find it in this pattern, this pattern of triangles in

the fractal with a brand new fractal. So it's just creating more and more and more and more fractals, which creates a finer and finer and finer resolution, which makes something look all the more real. Yeah, like the part in the doc about the Star Wars, I was making

the lava splashing. It's amazing, Yes, it was because they showed the first one they did it looks kind of plain, and then once you fed it through this infinite feedback loop, it just like shattered and and and uh, fractured, not fractaled, although I want to say fractaled off and just look more detailed, more detailed, more detailed, until it looked like lava splashing. Right, it's pretty amazing. Well, that's where the

word fractal comes from. Is um Mandel brought coined in to say, to indicate how the things fracture off and they form irregular patterns. Um, you can create a fractal that that is regularly repeating, but it doesn't look as natural. And with like say, if you're creating lava, you've got to have that one rule that like every third generation

kicks forty degrees or whatever the rule is. That just kind of throws a little bit of dissimilarity and too because if something is too self similar, it's not going to look right. It's not gonna look natural, it's not gonna look real, which kind of leads you to think, chuck. Then that there is a an application for studying natural phenomenon using fractals, right, while there are I guess all kinds, Um,

well this isn't so much natural. But the documentary interviewed Nathan Cohen who was a ham radio operator and his landlord said, dude, you can't have that huge antenna hanging out of your apartment so he started bending wires a straight wire into essentially a fractal and found that on the very first go it got better reception, um, merely by the fact that it was bent in that way and it was self similar. So he eventually used that

two I hope make a lot of money. I got the impressing that he did, okay, um by applying that technology to cell phones. Um, and the way they describe it as all the different things a cellphone can do, if you were to have a different antenna for each one of those functions, it would be like carrying around a little porcupine. So what cell phones now are based on is a fractal design called Manger sponge. Minger sponge, Yeah, I think, man, and uh it's basically a box fractal.

And if you crack up in your little cell phone, you're gonna see it wired that way. Yeah, You're going to be looking at a fractal. It's a square, right, and then within it are a bunch of little squares in a recursive, self similar pattern. And you, friend, are looking at a fractal. It's all around us. Yeah, Um, it's also all around us in nature. There's uh in that same uh documentary that NOVA program. There was a team from I think University of Arizona. There's a team

of academics. Yeah, that was pretty cool. Who um, we're trying to figure out if you predict the amount of carbon capturing capacity an entire rainforest has just by measuring UM and figuring out the self similar system that a single tree in that rainforest UM has. That makes sense? Well, it does, but it's kind of a leap. It's like, okay, so as one tree does it follow the same system

that the whole rainforest does? And they apparently found that yes, in fact it does, right, The same branching UH system found in that tree is similar to the the growth of the trees in the rainforest as a whole. Pretty cool. Yes, UM. Tumors in the human body. UH. One of the keys to getting rid of of cancer is or any kind

of tumors. Spotting these tumors early on. But with our ultrasound technology you can only get so small and so detailed that you can't see some of these natural fractals that you know, your blood vessels are fractals essentially, just like the branches of a tree are UM. So they are now using geometry too. Now if I'm not sure if I got this right, but I think it shows up. It shows the flow of the blood because ultrasound can pick that up through these fractals when they can't even

pick up the vessels themselves. Is that right, early earlier tumor spotting, which right, well, for all intents of purposes, they're looking at the vessels by finding the blood because they see where it's flowing. But yeah, depending on the pattern that it follows. If it follows like a like a tree branching shape, it's healthy, right, yeah. And then the tumors, all the veins are all bent and crooking, going in all crazy directions. The read out of a heartbeat, yeah,

it's not consistent. It's a fractical yeah. So they use fractal analysis now to study your heart rate and use that to better understand how arrhythmia happens through math. So

there's the especially with natural systems. That's kind of like the biggest contribution that UM Fractal geometry is produced so far, I think, aside from c G I is what medical uh well, just the that whole understanding that was first really kind of um voiced by Lewis Fry Richardson with the coastline that there's, um, there are natural systems out there that we can't really that we're not quite paying attention to, we don't really know how to deal with that.

We're trying to apply something like Euclidean geometry to something that you can't really use that for um. That that's what fractal geometry is really contributed so far as basically say, hey, there's a lot of natural systems out here that are self similar and recursive, and now that we kind of see in the fractal world, we see them everywhere and we have a better understanding of them. And one of the best examples of that, I thought was figuring out

how larger animals use less energy than smaller animals. They use energy more efficiently, and um, this is a kind of a biological paradox for a really long time, and these guys figured it out using I guess kind of the um same kind of insight that fractal geometry has. That if you take genes and genes are the mathematical formula or the equivalent of a mathematical formula, and you uh feed in uh, these genetic processes, what it's going

to put out. Is this self similar recursive pattern to where the bigger the organism is, the more this thing goes and goes and goes, the less energy it's going to use because there's more of it and it doesn't require very much energy to produce past a certain point. So if you have a very small animal, it's using a lot of energy to do these things to carry

this out. But there's that economy of scale because you're still using a relatively simple formula your genetic code, right UM, to carry out a very complex, seemingly complex UM system, which is your organs or you as an organism. So in the end, an elephant uses less energy than a mouse, yes, because they're both using the same formula, the same input. And then eventually you reach a point where it just gets easier and easier and easier to to use something

simple to create a complex system. I love it. I do too. Uh. I got one more thing. You heard this guy, Jason Paget, Huh, this is pretty crazy. UM. This guy like nine years ago, I think UM was mugged and to come Washington got hit in the back of the head really hard, knocked him out, and he acquired UM a form of synesthesia in which he sees fractals from being hit in the head. And um, basically it's an acquired savant savantism, which is pretty rare to

acquire this later on. Um, and this guy hated math, and his family used to make fun of him, he said, because he was the worst at pictionary. Uh, I couldn't draw a thing, couldn't draw a lick. Now, this guy can draw reportedly mathematically correct fractals by hand, and he's the only person on earth that can do this. And you should see these things. They're like, you know, a huge you know, two by two fractal that looks like

it was plotted by like a supercomputer. And this guy does these by hand now out of nowhere because he got hit on the head. That's pretty amazing. Yeah, it's crazy. He got him in the fractal center. Huh he did. That's strange that we would have like that ability latent in us, you know. Yeah. Well, they studied his brain, of course, UM, and they found that the two areas that lit up in the left hemisphere were the areas

that control exact math and mental imagery. So they have it well, and he's you know, he's fine with it, although he says that he's a bit obsessive about it because he's it's one of those deals where everywhere he looks now he sees fractals. Oh yeah, Well, I got the impression that people who are who are fractal geometers have the same thing. Yeah, you know, they're like, click at that cloud. I I can figure out how to describe it completely. Yeah with math. Yeah, it's crazy, um.

And then it's everywhere canopies of the trees. Like. I got that impression as well that once you start seeing fractals in natural systems, like then everything becomes um fractals and a lot simpler to understand. I realized today that I have always doodled in fractals. Oh yeah, yeah, because I can't really draw, so whenever I doodle, it's like

it's all aways been um little fractal shapes. Like I would draw some kind of geometric shape, then split off from that and make it smaller, and in the end they're sort of like fractals. Oh your fractal tree that you showed me, it's pretty awesome. So you got anything else? Uh No, I would strongly urge you to read this article a few more times. And then maybe go off and read some more about fractals, because we definitely have not covered all of it. I watched that Nova documentary. Yeah,

that's good stuff. What is it? Chasing the hidden dimentioned? Is that what it's called? And you call it chasing the dragon? Well, there's the dragon curve fractal. It's pretty boss, That's right, it is boss. Um. So you want to type fractals in the search bar how stuff works dot com to start, and that will bring up this very very good article. And I said search bar, which means it's time for a listener mail. Josh, I'm gonna call this, uh,

don't eat your peanuts around me, jerk. Yeah. Remember when the Air Traffic Control remark that never heard the announcement that, uh, no one can eat peanuts on the plane. I've flown a lot in my life and I've never heard that before. So Ian Hammer writes in on the Air Traffic Control episode, you were talking about peanuts being completely absent on some flights, And as a person that is really allergic to peanuts,

I can shed some light. My allergy is bad enough to wear the smell of peanuts, which is really just the presence of peanut molecules in the air will cause me to get itchy and swollen. Uh. In the case that I am in contact with a peanut have the superpower of becoming a balloon, and I'll swell up to the point where I will be dead in a matter of minutes. I can delay the anaphylactic shock for ten minutes, give or take with an injection of epinephrin, and this

will only work twice twice in his life. I think, so, um, if I do have a reaction, I have twenty minutes plus the fifteen minutes I have before normal anaphylactic shock would kill me. There really is in a way to save me in that instance, unless I can be administered the proper treatment that you can get only at a hospital. Because you can imagine when a plane is at thirty feet there's not much can be done to get me to a hospital within that thirty five minute time frame.

So flying can be a pretty scary thing when someone near you. Besides that they really want a peanut buttercup. People do this sometimes and it's a real pain to have to deal with. I just wanted to give you guys an overview of peanut allergy sufferers when it comes to flying. Keep up the incredible work. Look forward to seeing a TV pilot Ian Hammer. So incredible, is right? If we were insensitive to that, then all apologies. He didn't indicate that, but I know we weren't. I just

remember being surprised. Yeah, I was surprised, but I knew allergies could get bad. But man, that I think on the plane, I was like, what I've known about this since I saw an episode of Freaks and Geeks wherein one of the characters almost died because like some bully at school like gave him some peanuts. Oh yeah, was that it was the Martin Star character, the analog to Paul from Wonder Years, Okay, which was, Um, I can't remember his name. I book for some weeks. Yeah, it's good,

good show. Um, well, let's see allergies. How about a practice story if you know something about fractals that we don't, or can correct us or explain it better than we did, which I'm not sure that that's much of a long shot. Um, we want to hear about it. You can tweet to us at s Y s K Podcast. You can visit us on Facebook at facebook dot com. Slash Stuff you Should Know, or you can send us an email to

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