Geometer Explains One Concept in 5 Levels of Difficulty | WIRED

hi i'm keenan crane i'm a geometer and professor of computer science at carnegie mellon university and today i've been asked to explain fractals at five levels of increasing complexity so fractals are a sort of strange kind of shape that have detail at all different levels fractals appear all over nature they rise naturally in computer graphics because we want to make beautiful images of the natural world fractals are also really interesting because they show how extremely simple descriptions can give rise to really complex shapes what's your name myra have you heard of fractals before um no absolutely not so a fractal is something that we see every day but it's a little bit hard to describe it's it's a shape that if you look at it really far away or you look at it really up close it has kind of a similar appearance and actually that word fractal sounds a little bit like fraction right so actually fractals are in some way fractions but for shapes have you ever seen a movie called moana yes yeah moana lives on this like beautiful island right this island has lots of trees on it some artists had to make all those trees how do you think they do that they um try to find something similar to that on the google and they try to picture it in their head saying what would what would it look like if they it was animated so somehow they have to explain to the computer what a tree looks like so that the computer can draw the tree for them yes and that's kind of what we're gonna try out today in fact we're gonna have you build a fractal a tree is a really good example of a fractal because if you look at the whole tree and then you break off a big branch of the tree yes it almost looks like that branch that you broke off is itself another tree so our rule is that every branch splits into two smaller branches okay so your tree is totally beautiful but it took a long time to make right how would you feel if i said now you have to make a thousand of those trees i'd be like oh my that's a lot of work yeah it's a ton of work if you think about moana's island it has these tens and hundreds of thousands of trees on it and that's why we need computers to help us out because computers are really good at taking these really simple rules like put two branches on every other branch and doing it really really fast i want to know how to draw a fractal on a computer if you want to learn to draw fractals then you might need to learn a little bit about programming programming like coding like coding exactly that's what actually a lot of the artists for the movies are working with code rather than with a paintbrush so in your own words how would you describe a fractal to somebody i would describe a fractal by saying that when you see a object if you'll zoom in closely you'll see that it's it's broken down into pieces so as you zoom in more closely you'll keep seeing smaller and smaller pieces definitely that is totally what fractals are all about yes what are you studying in seventh grade these days i think we're doing geometry so what if i told you that there are shapes where you can't measure the length of the perimeter it's big but um all of the sides are kind of like so weird that they wouldn't be able to pinpoint anything specific to add together the shape is just super super complicated and so you can't actually measure the length yeah right yeah so that's already a really good idea of what a fractal is it has some really really interesting detail at all scales that makes it hard to talk about standard quantities like lengths and areas and volumes would a planet or an asteroid be a fractal yeah so if you look at kind of the wrinkly surface of an asteroid each one of those little wrinkles adds a little bit to the surface area and so it's really hard to say what is the area of an asteroid let's take a look at a little example of where fractals actually show up in nature what we're going to try to do is we're going to try to say what the length of the coastline of great britain is so we're going to start out with the blue ones which are spaced further apart we're going to ask you to connect the blue pins with a piece of string so we can get a measurement of the coastline so we have our first measurement of the length of the coastline and we're going to do it over again but this time we're going to use the white pins which are spaced closer together so we measured the coastline again and what do you think happens there's more in and out therefore probably takes more strength for this one i think i agree with you but i think to really validate our scientific experiment we should probably compare the length of the string a lot of extra slack in that ping string so what we noticed with great britain is we were never able to measure its length it just kept getting longer and longer this is called the coastline paradox where there's not really one definite number you can assign to the length of the coastline but it depends on how you measure it like if we kept going to really get this to be accurate we might have to like actually go down the beach and start measuring these tiny little details along the coastline but really there's so much detail we would probably never get a definite answer for how long the coastline is fractals also give us a really nice language we're talking about how smooth or rough some kind of shape is and actually a lot of people engineers and scientists use this idea of fractals in this language of fractals to compare different shapes not in terms of their size but in terms of their roughness so based on everything we talked about today how would you describe a fractal to somebody in your own words it's a shape that you can't describe as a shape yeah totally like it's a shape that you can't use sort of the ordinary language that we use to talk about shapes you really need some other ideas or concepts to talk about [Music] do you play any video games um i really like hearthstone minecraft what happens in minecraft if you get really really close to a shape well it just looks like a block yeah it looks really blocky and it's the same thing not only with shapes and games but also colors and textures and this is an especially big problem in virtual reality because you have no control over where people are going to go what they're going to look at how close they're going to get to various objects so fractals on the other hand are these things that have infinitely you know nice detail as you get closer and closer and closer and so this is something that can help us solve this problem in computer graphics of generating more detail one of the reasons why fractals are really nice for computer graphics is because the algorithms that we use to draw images also have this kind of recursive flavor what's recursion recursion is a function that uses itself or calls itself in its definition and basically with that you can figure out my new details such as searching for a value in a binary tree it's kind of like iterating through itself but it's often more sophisticated in that it's easier to write they're easy for a computer to execute right it just has to run this recursive procedure over and over and over again we can kind of get as much detail as we want or need when i think about fractals and kind of what they've done for computer graphics the example i think is subdivision surfaces did you did you encounter subdivision surfaces in your graphics class at um the name doesn't really ring a bell so subdivision surface is a way of describing a smooth shape on a computer rather than some complicated fractal shape so meshes and computer graphics are usually made of these flat polygons with kind of sharp edges and so the question is how can i get something nice and round and smooth out of these sharp looking polygons what i can do is i can start cutting one at a time the corners of this paper off it still looks very angular i still have these really sharp points right why are we doing this because i want to draw a smooth curve on the screen but all my computer can do is draw straight lines and that's actually too true of gpus it's kind of interesting gpus are basically really fast machines that can only draw one thing and that's a flat triangle and so if we keep doing this with our paper you kind of get pretty quickly the idea what's going to happen so when you go to see a pixar movie for instance every surface has been subdivided down so you have tiny little triangles called micro polygons that are even smaller than the size of a pixel how long does this process take well because people really need to use these uh subdivision services for everything people worked hard over the years to make this super super fast in fact subdivision services were invented basically at pixar there's this guy ed catmull and he was partially responsible for one of the most famous kinds of subdivision surfaces called catmo clark subdivision services and in fact recently won the turing award for these subdivision services what do you think are the current shortcomings in i guess applying fractals to computer science right now what is what is the cutting edge so we've talked a little bit about the positive features of fractals and procedural graphics which is you can write one simple recursive program and the computer creates a lot of detail for you so that's really nice right it's saving you a lot of work but the downside is you lose a lot of control so because the only thing you're describing is this short little program you don't have complete control over how this is going to end up and so adding more controllability to procedural graphics is something people have been thinking about for many years so how has our conversation changed your understanding of what fractals are all about i think it's really interesting to see the different ways fractals will be not only useful but necessary in being able to render these games and these different programs that are interesting in the metaverse or different media to be really beautiful [Music] hey thanks for joining us remotely yeah of course it's a pleasure i'm happy to be here do you have any sense of how you would give a really precise mathematical definition to you know what is a fractal it would probably have to be some kind of recursive definition like imaginary numbers i know the mandelbrot said we'll use those the mandelbrot set or the julia set you know the idea is always oh i'm going to apply some polynomial over and over again z squared plus c or something like that when i think about fractals i try to get away from these very specific examples and ask what is it fundamentally that makes a fractal of fractal and one thing that i think you can be precise about even if you can't really say exactly what a fractal is is you can talk about this idea of fractal dimension have you ever heard of that no i have not actually so if you look at this this piece of paper what would you say the dimension is well on the on the paper itself i would say it's two dimensional but the actual paper would be three dimensions because it has a very small thickness yeah great so so the real physical paper has some amount of thickness but when we model this mathematically we might ignore the thickness and say yeah this is really just kind of a two-dimensional sheet of paper and then you have your apple how many dimensions is the apple i would also say three and y three because it has the height and the width and uh there's a depth to it as well totally now as a little experiment take your piece of paper and just crumple it into a ball so what is it is the paper three-dimensional or is it two-dimensional it changed in dimensions depending on how it is shaped so it's not quite as solid as the apple but it's also not quite as two-dimensional as the original sheet of paper and so for this reason people associate this crumpled ball with a fractal dimension maybe something like 2.5 dimensions instead of two or three there are a lot of different definitions precise definitions of fractal dimension but i think the one that's easiest to understand is this thing called box counting dimension you have let's say an image and you want to decide what's the fractal dimension of this image so what you're going to do is you're going to count the number of boxes or you can imagine big pixels of this image that cover the boundary of this shape and you're going to see how does that counting compare with how the counts work for just an ordinary shape so so if i have a straight line and i start out with one big box that covers the whole line and now i shrink my boxes by a factor of two i just make boxes half as big how many boxes will i need to cover that line and if i cut that box in half again how many boxes do i need to cover the line a four but if you take a more interesting shape some kind of fractal shape like let's say the coastline of great britain and you start doing this box counting experiment something really interesting happens as you make these boxes smaller the number of boxes you need to cover the coastline grows faster than it would for just a straight line yeah i've i've heard about that where if you if you change the amount of measurements that for a coastline you can actually change like how much of the coast that there is but if you measure in mile lengths you're going to get a much different estimate than if you measure in one inch increments and so what this box counting is getting at is saying well i still can never decide what the length of the coastline is but what i can do is i can see how quick does the number of boxes grow relative to how it would grow just for an ordinary one-dimensional curve like a line or a circle is there any other interesting applications of fractals so procedural computer graphics which kind of came out of thinking about fractals is one answer to this question about how do you add more detail without for instance consuming tons of memory or requiring artists to paint ultra detailed textures so if if you can instead describe at least some aspect of what you're looking at in a procedural or recursive way then you can add as much detail as you need when you get close to objects do you mean like perlin noise yeah like perlin noise is a great example right perlin noise was one of the early ways of synthesizing additional texture at any level of detail that you needed to make things look natural and realistic i have a random question do you know how the research into fractals began you can look pretty far back in history to see kind of glimmers of this idea of fractals in science in the 19th century people were trying to look for examples of things in mathematics that were very unnatural so there was for instance this guy named georg cantor who was showing that you can have these sets with really strange properties or you can have functions with really strange properties this thing called the devil's staircase and so on and it wasn't until about a century later where someone named mandelbrot said oh actually this strange mathematics that was meant to show how unnatural things can happen is actually a perfect description of things that really do happen in nature and from there people really ran with it and said oh okay well if these fractal descriptions are good for nature we can also use that for creating really realistic and believable pictures in computer graphics [Music] i can say that when you reached out to me and i first learned about this program i immediately went to my computer and i implemented yet another version of like zooming into the mandelbrot set yeah just because i was so excited you know so that the thing is that it took me maybe 30 minutes a kid at home who's like 13 and 14 who just starts playing around with a simple computer program can make incredibly beautiful fractals yeah and i think that's one of the things that's exciting about it or you don't even need computers like i remember when i was a kid i played electric guitar and i had all these guitar pedals with different effects and oh what happens if you put the output back in the input and just run it through itself you're going to start hearing this kind of fractal sound right so today we've been telling people fractals are things that are somehow self-similar they have detail at all scales are there other ways you might introduce fractals to somebody or are there other things you might say this is what a fractal is i think you could get to the reasons why does a coastline look like a coastline at all different scales it's because the forces of nature tend to work similarly at all different scales and it's those forces that are doing very very simple things over and over and over again that are constantly creating the sense of detail i think about like the scale and variance of different physical equations like navier stokes is you know kind of scale invert you know this reynolds number that tells you how viscous things are but you can have the same kind of fluid behavior at all different scales and that's why you get turbulence at all different scales i actually actually remember the moment when i finally figured out how to make turbulence in computer graphics i was working for this company magi and disney was working on a movie i think it was weird science where they wanted to have a marble vaz and then about three in the morning i was at a diner nearby and i was pouring the cream into the coffee and i was watching it and i started watching it swirl around and i realized that what was happening was really simple the you had this line of cream and then it hits the cup and it folds and then the folds get caught up and then that folds again and it's a really simple process of folding within folding and i just went to the computer and did that yeah and things look like marble and look like flame and they look like clouds and they look like you just keep using those simple techniques yeah and i think it's really cool that this kind of fractal description of geometry or physics is also kind of baked into the nature of computation computation is recursive kind of by its nature and so it's kind of a match made in heaven that we happen to build these machines that also right you know behave the way that you just have to understand recursion exactly somebody once told me in order to understand recursion you just have to understand recursion there you go and then you get it all yeah but i think it's this point that that point with the coffee is important because the forces were operating at one scale they were operating at the scale of the coffee cup but over time they kept making details are smaller and smaller process at one scale plus time you get fractals i think that's also what's so beautiful about fractals is that you know if you think about sort of the history of geometry as well felix klein was looking at geometry by saying well geometry is all about invariance i have some group of transformations and i look at objects that are sort of invariant with respect to these transformations so if you just look at translations okay what kinds of shapes remain the same under translations well you get tiling you get wallpaper and if you start asking that same question what if i allow scaling in my transformations then boom you have fractals immediately come out of nowhere and everybody understands scaling scaling is a simple thing yeah now you work on kind of the future of virtual reality and augmented reality and extended reality but it's kind of interesting because i think if i think about exploring these infinite fractal landscapes in some way they still feel a little bit lonely or they feel a little bit poor of the kind of richness that we have in the real world this is where uh machine learning um starts coming in because you can start saying okay this is a very very rich virtual world but it's informed by you know my favorite mountains that i saw in italy once so you can start training these fractal worlds on things about the real world that have particular emotional resonance for us people outside of computer graphics and engineering and science and so forth have also thought a lot about and used fractals as a language for talking about nature for characterizing shapes and behaviors and so forth tissue or friction or all sorts of really important phenomena do you think that computer graphics has kind of dropped the ball in terms of saying you know this is not something that's exciting anymore to work on procedural uh descriptions and we're moving on from that well i mean if you look at any hollywood movie or you look at any of the game worlds that people are spending all their time in they're highly procedural they have to be and they have to make use of fractal techniques because it's basically a way to get enormous complexity without having to explicitly store the complexity and because they're able to use these relatively simple fractal techniques to make very complex looking natural things right lazy evaluation we love being lazy in computer graphics well it's also it's not even possible to explore to store an entire world in your computer absolutely you want to be able to sort of generate it on the fly i'm looking forward to things getting better we're not there yet so one of the things i guess that touches on is this question about controllability or the ease of with which everybody can create these worlds not just people not just mathematicians not just you know trained computer scientists one thing when i think about ken muskraves working about this program bryce this one i felt like i could really use what do you think we still need to do in terms of putting these kinds of tools in in people's hands making it easier for people to use proceduralism machine learning to build these kind of worlds i think in that case it came down to the fact that ken in particular had a mission to make accessible tools for people without sacrificing the power and the richness of making beautiful things i mean in a sense he was kind of the computer graphic bob ross so he made the land of happy little trees yeah yeah which i mean when you when you think about the techniques of somebody like a bob ross um they're fractal yeah i think this is what what's so beautiful also about mandelbrot's work is him saying you know it's not really about these exotic examples like even the mandelbrot set or julia sets or whatever yeah they're like really interesting mathematical curiosities but they get at this idea that fractals are kind of inevitable and bob ross probably never as far as i know never sat down and you know thought about recursive descriptions of trees or anything like that but it's just something that naturally comes to you as an artist well i mean you can go back to all the classic artists vinci's notebooks were just full of like this thing looks like that thing at completely different scales so he didn't have a fancy word for it but he completely understood it yeah it's really part of human nature or humans connection to nature yeah hopefully our discussion today has helped you see the world in a different way and also see how mathematics and art can come together to make beautiful images i hope it's inspired you to look at the world around you in a different way

Channel: WIRED

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Keywords: fractal, fractal math, fractal scientist, fractal shapes, fractals, fractals explained, fractals explanation, fractals wired, how fractals work, innovation, keenan crane, keenan crane wired, keenan crane wired scientist, math, math fractals, ott 5 levels, science & tech, science & technology, wired, wired fractals, wired keenan crane

Id: ofFLYfUEPVE

Length: 22min 33sec (1353 seconds)

Published: Mon May 30 2022