For this historic 444th episode of Astronomy Cast, we talk about fractals. Those amazing mathematical visualizations of recursive algorithms. What are they, how do you get them? Why are they important?
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Fraser: Astronomy Cast Episode 444: Fractals
Welcome to Astronomy Cast, our weekly facts-based journey through the cosmos, where we help you understand not only what we know but how we know what we know.
My name is Fraser Cain. I’m the publisher of Universe Today. With me, as always, is Dr. Pamela Gay, the director of Technology and Citizen Science at the Astronomical Society of the Pacific and the director of CosmoQuest.
Hey, Pamela. How are you doin’?
Pamela: I’m doing well. How are you doing, Fraser?
Fraser: Wrestling with technology.
So, for those of you listening, I am using a new microphone. Please, those audiophiles out there, let me know how it sounds from your end and I will – glad to hear your feedback. That’s the first thing; I’ve got three things to say.
So, the second thing – and this is, I think, pretty exciting – is that we’ve added a whole new show, which is called “The Full Feed”. And what this is, for those of you who don’t know, when we record Astronomy Cast once a week, on Fridays at 1:30 live –
Pamela: Pacific Time zone.
Fraser: – Pacific Time – and the show goes for, actually, an hour. And you, listening to the podcast right now, you’re only getting half of it. There’s a whole section where – before – where we wrestle with technology. There’s a whole part afterwards where we answer the questions live, with the audience, and get into some really interesting sounds and I really enjoy it. And I think, just between you and me, I think that live part of the show – the part afterwards, where it’s all just – you know, we’re just interacting with the fans and just talking about whatever’s happening in space exploration and things like that – it’s really entertaining. So, you should switch, if that’s what you want – if that’s the experience you want.
I can understand you can’t show up live for the regular show but if you want to be a part of the – you know, if you want to hear it while you’re doing a run or whatever, you can find it – Let me just find the exact link to make sure I get this right.
So, go to astronomycast.com/feed/fullraw and go ahead. And that is every part of the show. And, ironically, this episode – we just wrestled with my microphone for the first ten minutes. So, other times, it’s gonna sound much better but, for now, you know – this episode, specifically – not our strongest opening. But I hope the final part is going to be way better.
Alright. You ready, Pamela?
Pamela: I certainly hope I am.
So, for this historic 444th episode of Astronomy Cast, we talk about fractals: those amazing mathematical visualizations of recursive algorithms – and those tasty broccolis. So, what are they? How do you get them? Why are they important?
Alright, Pamela. Now, you threw this one in. You’re a bit of a fractal fan, aren’t you?
Pamela: Well, it’s more a matter of it was unbeknownst to – I’m guessing – most people out there listening; the natural way to conclude our “Death and Destruction” series, because it turns out that a lot of really cool death-and-destruction-related things have a fractal nature. So, for instance, fault lines are fractal in nature; coastlines that have suffered from landslides and lots of other destructive things – fractal in nature.
And, hey – it seemed like a good way to celebrate it being 4s all the way down, and to maybe start to transition to talking about some things that aren’t quite death and destruction.
So, I guess – What identifies something as a fractal? You know? I mean, like, we’ve all seen these kind of Mandelbrot pictures of, like, blobbies and when you zoom in, they get very – they’re the same; they’re kind of recursive. But, like, what is the gist of this?
Pamela: So, you know, what’s fun about trying to look up: What is the best way that I can describe fractals without inadvertently offending a mathematician? Because here’s where I have to admit, I would much rather write software to do maths for me than actually do maths.
Fraser: And last time, we did the show on infinity and got no end of grief from the mathematicians. So, we’re gonna go light on the math.
Pamela: Well – and what was awesome, in trying to look up how to best describe this accurately, was I kept finding articles that say: “Mathematicians can’t agree on a mathematical definition of what a fractal is.”
Pamela: And – yeah. So –
Fraser: Well, then, we’ll just live in the middle there.
Pamela: So, it boils down to a few key ideas:
So, first of all, fractals are self-similar at scale after scale after scale. This means that, as you look at them at more and more detail, what you’re seeing doesn’t actually change.
So, like human beings – we are not fractals. If you zoom in on a human being, you’re going to go from seeing human body; to seeing body part; to seeing skin; hopefully, to seeing individual cells; to maybe seeing, like, a cell; to looking in the – It is not fractal; it is a different thing at every zoom.
But when you start looking at things that are fractal in nature – snowflakes being one of the ones that we all try and make as small children – when you look at them at more and more detail, every time you zoom in, it’s like, “Wait, the view didn’t change.” No, it didn’t; it’s fractal.
Fraser: ‘Kay. And so, how do you get a fractal?
Pamela: It’s simply a matter of something that is the same at pretty much all scales – and is complicated. So, like, lots of things aren’t sufficiently complicated to be called a fractal. So, the way you get them is you have a detailed, complex system that is irregular in ways that is repetitive, and scales.
So, one of the simplest ones to try and understand is what’s called a “Koch snowflake”. This is where you take a regular, happy, equilateral triangle and then, on each of those three sides, you divide it into bits and add an equilateral bit. So, you now have on each side of the triangle – it now becomes a triangle bit, another triangle bit, another – and they’re all the same. And then you take each of their edges and you split it into thirds again. And you keep adding these equilateral triangley bits and, as it gets added and added, the edges get more and more complex but when you zoom in on these edges, they look absolutely identical at all scales.
Fraser: What is the point?
Pamela: Well, so, what is fascinating is it started kind of as a mathematician crossed philosopher thinking hard. And so, when you start looking at the history of fractals, it goes back to the 17th Century, where you had people thinking about the little voice in your head – the homunculus – and how that voice in your head had a voice in its head, had a voice in its head; and its little “men in the head” all the way down.
And so, it started with this idea of recursion. And then it moved from being a philosophical-mathematical concept, to people starting to play with – as calculus began to be a thing – are there lines that refuse to be what’s called “differentiated”, which is a way in calculus that you can approach a line and start to figure out: What is the area beneath the line; what is the inflection point? There are all sorts of cool mathematical things that you can do if you can differentiate something, but not all lines can be differentiated.
And fractals are just like, “Nope, not gonna do it.” Because, since there’s not really anywhere that you can, like, line across a fractal – because they’re constantly bending and twisting and turning – and so, they’re undifferentiable. And so, people are like, “Let’s define all the lines that have this characteristic.”
And from there, people started to think about, “Well, now that we’re finding all of these lines that are complex, recursive, not-differentiable, are there any things about them that are consistent?” And it turns out that the way that fractals scale, over size, they end up with different fractal numbers.
So, a way to think about this is, normally, you have something that has a length. If you want to look at its area, that scales by the square – factor of 2. If you want to figure out what the volume of that thing – taking it into three dimensions – is, then you have to take that factor of 2 and raise it to the 3rd power. So, the volume is 2 times 2 times 2. So, you have this nice, linear way of – with an integer – going from line to surface to volume.
Fractals are just like, “Nope, not gonna behave.” Fractals – as you look at increasing dimensionality – they don’t go as an integer; they go as a fraction – fractal/fraction – it’s what they do. And different things have a specific scaling factor.
And this has cropped up in things like – you can look at Jackson Pollock paintings and all of his paintings have the same fractal number. And they’ve identified forgeries because the forgeries had a different fractal number – because Jackson Pollock paintings are basically splotches of paint at all different scales. And when you start looking at how does it change as a function of scale, that change – that’s its fractal number.
Fraser: And, I mean, in sort of the perfect fractal, there is no difference. It’s almost like – I don’t know. It feels like it’s like the scale of the universe or something like that, right? Like, at a certain point with the universe, as you scale out, you start to get to that cosmic web of galaxies. And as you go further out and further out, then you’re just seeing larger and larger – But that’s actually not true. There actually is a –
Pamela: No. Universe isn’t actually fractal.
Fraser: Yeah, the universe isn’t fractal.
Pamela: That’s one of those things –
Pamela: People kept thinking, “Universe must be fractal.” No, it’s not.
Fraser: Right, right – and that it actually does have a maximum scale; that you move out to a certain point, and you’ve got these cosmic webs – these columns and walls of galaxies and voids in between them, like – I don’t know. Like some kind of tasty chocolate bar. But then, you go further out and there’s no larger version of that pattern; it’s done.
So – But with a true fractal, you kind of – when you look at it, you really don’t know how far you’ve zoomed in.
Pamela: And what is kind of awesome is how this can be applied to understanding how different systems have the potential to build up inside themselves, which is one of those really hard things to explain so I’m going to back out a little bit.
If you look at a coastline, you might casually say that the California coastline is like a thousand miles long. And that’s simply: You draw a line up at the border between California and Oregon; keep it going down; draw a line that ends when you get to the point where California meets Mexico. Cool – nice straight line. Well, coast isn’t actually a straight line; so, let’s approximate it with a nice, simple curve with one inflection point. That’s now longer.
But, you know you have bays – you have, like, Long Beach, San Francisco; all of these different places where you can take a ship in and get into safe harbor. So, you start adding inflection points. But then you start realizing, “Wait, there’s deltas; there’s –” and there’s really no end to how long the coastline is because, as you measure at a greater and greater fine-scaled measurement, you keep notching your tape measure up more and more and, at a certain point, California has an infinitely long coastline.
And that’s part of the fractal nature of coastlines. And fractals, in general, don’t have a length – which is kind of cool to think about – because every time you zoom in further, you find smaller and smaller line segments; smaller and smaller curve segments.
Fraser: That just blew my mind! Okay, hold on a second.
So, if you take like some kind of little loopty-loop shape – you know, like a roller coaster or whatever – and you follow the length of the roller coaster, you’re gonna get some kind of length. And if you zoom in, then you’re gonna start to see little bumpies on the rails and that’s gonna actually increase the length of the roller coaster. But at a certain point –
Pamela: Roller coasters are not fractals – just for anyone in the audience being confused.
Fraser: No, no, no – but this is my point, right?
Fraser: Which is that, you know, you could zoom in on a roller coaster and you could get a little bit of a longer pathway because you’re actually –
Fraser: You know, you’re kind of following the bumps along the actual rail that the roller coaster runs on. And so, it’s going to be longer, right? But then, at a certain point, there’s going to be some maximum – where you’re literally hitting the length of the atoms – and at that point, there’s no possible longer version of the roller coaster.
But in the case of a fractal, if you’re following along that line and you calculate the length of that line at whatever rough resolution you’re looking at, and then you zoom in a bit more – you’re going to see a longer line. And you’re going to zoom in a bit more and you’re going to see a longer line; and more and more and more. And that goes on forever.
But, obviously, these things don’t exist in the real world in that way.
Pamela: But –
Fraser: It’s that they’re a mathematical representation, right?
Pamela: But things like coastlines, it’s really kind of amazing to think – When you’re saying how long a nation’s boundary is; how long a coastline is, you really have to say: Averaged over five-kilometer-long segments to start to make sense of this.
And if you’ve ever flown over the deserty bits of the American Southwest and the Canyonlands, you’ll notice that the places where the water runs down out of the mountains – rivers have this amazing branching structure, where they just seem to branch and branch and branch as you get further and further from the main body of the river. And that branching structure is actually also fractal in nature. Where, as you go down to just the single sand dune with all of the droplets of water running down it, you keep getting a finer grade fractal structure.
Now, here, you are going to start to get limited by the size of a sand grain and the size of droplet of water, but the nature of this is fractal in nature.
Fraser: And that if you looked at the river system of, say, the entire United States; and then you looked at the river system for the final kilometer of a river going into the ocean; and you looked at a meter of a little piece of a little rivulet that’s coming off the side of that river, you are going to end up with almost precisely the same structure, it’s just a matter of scale.
Pamela: And it’s in trying to figure out how to mathematically describe these things that fractals become so powerful – because the fact that there is this non-integer scaling factor that allows you to go from one zoom level to the next to the next; that non-integer number is actually unique to different situations, where you’ll see a different fractal number, depending on what the landscape of the river it’s passing through is. We find different fractal numbers for different fault lines. But across the fault line, that fractal number – as long as the landscape stays consistent – that fractal number stays consistent.
Fraser: And we really needed computers, right? I mean, this is a mathematical field of study that really required computers –
Fraser: – to explore fully, because a computer doesn’t mind calculating the same math 4-gillion times, again and again and again, right?
Pamela: And with computers, fractals actually became extremely important for animators and people who are trying to generate landscapes because you need to look at: How does something look the same or different – at a variety of different scales.
So, in your video game – when you’re generating the coastline; when you’re generating hair, for instance; because at one scale, you see individual strands; at another you see chunks; at another, you see the whole dang ponytail. And at a certain level, we fully realize this because of how we draw cartoons, how we paint paintings. But, when it starts to be: How do we visualize things, the fractal nature starts to come into play.
Fraser: Very cool.
Now, do you have a favorite fractal?
Pamela: I love the idea of coastlines. And part of the reason I love it is I grew up in New England, where it was always people making fun of New Hampshire for its lack of coastline. It was like, “New Hampshire, what’s wrong with you? Do you not like the ocean?” And then you get into the: Well, New Hampshire’s filled with lakes. So, does it actually have more waterfronts when you start taking into account lakes? But then, you not only have to do that, but you also start arguing over, “Well, do you take into consideration all of the little coves and all of that?” And it’s just – It’s a fun argument that brings fractals kind of to life.
Fraser: I have a similar situation, which is that – as many of you know – I grew up on a small island, off the west coast of Canada, called Hornby Island. Go ahead, Google it. And the island itself is maybe 5 kilometers across – maybe smaller – and so, we could go anywhere.
And one summer, my friends and I decided that we were going to walk around the entire coast – just walk on the beach – and fully circumnavigate the island. And we knew that it was actually – We’d heard some other people had done it – a lot of work. And so, we got up. I think we started at 3:00 in the morning and walked until about 10:00 at night and didn’t quite reach our starting point, even after that entire day. And yet, you know, we knew all of these beaches; we’d been on all of them many times before, throughout our lives, but that curving back and forth – that following the exact contour of the beach line – was a much bigger, more complicated, job than we thought it was going to be.
Pamela: And this starts to come into play when you think about: Why do apartment buildings have such bizarre footprints? Well, it’s because everybody wants windows, and they want as many windows as possible. And so, you start having that fractal nature to the outside of your building. You start with a cross and then you turn it into four tees and then, maybe you start adding things sticking out – It’s a way of maximizing surface area by volume.
Fraser: Super, super interesting.
So, I mean, are there – I mean, you’ve talked about coastlines; you’ve talked about rivers. Are there other fields of mathematics; other fields – I mean, is there any connection to cosmology? I mean, we talked about how there literally is a – specifically, of the large-scale structure, it’s the opposite. For a while there, it was something that they even maybe anticipated; that there would be some kind of larger-scale version.
Fraser: You know, where are some other applications of fractal mathematics?
Pamela: So, the best not-on-Earth example I could give you is Saturn’s rings. And what’s amazing with the rings is, when you look at them with your friendly, handy-dandy, little, tiny Galileo scope – which is like the cheapest, best, most awesome telescope for viewing Saturn’s rings. Spend 50 bucks, view the rings. So, when you look at them with that, you see a couple of gaps but you can tell it’s a series of rings.
Well, if you look at it with just binoculars, it’s like “a” ring. So, you go from “a” ring with binoculars, to a few rings with Galileo scope, to – well, go find yourself a 30-inch telescope; now, you’re looking at more rings. Put Cassini right next to the rings and it’s rings upon rings upon rings. And the more we look at the structure, the more we see this constant breaking up into grooves and bands – and so, beyond the Earth, the rings of Saturn.
But here on the Earth, we find fractals in so many different places that we don’t really recognize as fractals until you start trying to, well, figure out how long things are. So, a lightning bolt – which branches out into all of these different bits – if you zoom in more and more on that lightning, you see more and more bits, and more and more bits. And lightning bolts are fractal in nature.
One that I want to learn more about is apparently the texture of mountain goat horns – with the grooves on the horns – has a similar nature to Saturn’s rings, where it’s just grooves upon grooves upon grooves.
Algae often has a structure. I’m going to destroy this poor piece of broccoli’s name but Romanesco, I think, is how it’s sort of said – Romanesco broccoli.
Fraser: Have you ever seen this? We have this in the grocery store here.
Pamela: I actually – I bought some seeds for it this year.
Fraser: So did I! Yeah, yeah. I’ve got some in the fridge – some seeds – that we may grow, although I find broccoli is just too much of a pain. I just buy it in the store.
Pamela: You know, I’m gonna try because my 72-seed starter thing had an extra two rows that I could start things in, so –
Fraser: Well, you let us know if that broccoli ever makes it to the kitchen table.
Pamela: I will, I will.
Fraser: My guess is that it will end up somewhere in the mouth of a –
Pamela: Ground hog?
Fraser: – of a caterpillar somewhere.
Pamela: Oh, well.
Pamela: Yeah, this is what net is for – this is what netting is for.
Fraser: No. Well, we don’t need to go down that rabbit hole in this episode but, you know, I’ve started to build a garden inside the house. It’s surprisingly pest-free.
Well, that’s really cool. So, I mean, is this – I mean, there was this big rush of fractals back – I’m trying to think. Like, in the early ‘80s, mid-‘80s, I remember there was, like, some great scientific magazines; there was, like, computer programs that you could run. Is it a continued field of study? Or is it sort of, you know – once the computers came on, was it sort of – Did people dig through it and now, you know, there’s less researchers working on fractals these days?
Pamela: Well, I think it’s less sexy than it originally was because it hits a relatively new thing. The first use of the word “fractal” was only in 1975. So, we – you and me – were not that old, but we’re older than the word fractal.
It was one of the first things that software engineers – and heck, little kids learning Logo. You could learn how to write recursive software and create fractals with your happy, little Logo turtle.
Fraser: The turtle! Yes! I remember that. Oh, that’s awesome. I was about to say that too – it’s the turtle, yeah.
Pamela: It’s turtles, all the way down.
Fraser: It is.
Pamela: It’s fractals, all the way down.
Fraser: Turtles, all the way down.
Pamela: And so, there is this whole: It’s a new idea; it’s super-exciting and we have technology to play with the new idea.
Fraser: Are some places that you like – out on the web, on the internets – where people can look at and play with fractals? Have you been – Has, in sort of your preparation for this episode, are there some resources that you’ve encountered?
Pamela: If you Google simple things like “fractal animations”, “fractal artwork”, there are amazing graphical artists out there. Just do a Google image search and it will kind of take your breath away.
And, I have to admit, looking at all of this – more than going, “Ooh, I must share this” – my reaction was, “Ooh, I must program this.”
So, this is the kind of thing where I’m thinking Processing.org, the software there – when I find a spare few hours, I’m going to be coding myself up a fractal generator.
Fraser: So – never.
Alright. Well, thanks Pamela. I look forward to your fractals.
Pamela: My pleasure.
Male Speaker: Thank you for listening to Astronomy Cast, a non-profit resource provided by Astrosphere New Media Association, Fraser Cain and Dr. Pamela Gay. You can find show notes and transcripts for every episode at astronomycast.com. You can email us at firstname.lastname@example.org. Tweet us @astronomycast. Like us on Facebook or circle us on Google Plus.
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Duration: 28 minutes