…or a not too bright 5-year old. I watched Nova the other night, where they did an atypically poor job of explaining the uses of fractal geometry. I’m a very visual person, and if math or physics can’t be explained in terms of its practical application, I have difficulty understanding it. Algebra, trig, calculus: no problems. But this one threw me. Why would I want to explain in mathematical terms how a fern is shaped? The non-randomness of tree growth was pretty cool, as it shows a sense of order to the universe, rather than it all being just random chaos. What is the point of making one of those fractal figures by bisecting triangles and making more triangles?
It’s partly because they look pretty, and partly because they illustrate interesting things. For example, you can have a two-dimensional shape, with a finite area, but without a finite line defining the boundary of the shape. In addition, some natural objects (such as coastlines) seem to have fractal-like properties even though they aren’t defined as fractals, but just arise naturally.
Fractals are interesting for a variety of reasons, but two that are easy to explain are:
Very, very complex patterns can be generated from trivial equations. The classic Mandelbrot pattern is a good example: Mandelbrot set - Wikipedia
This has promise for image compression algorithms.
Fractals appear similar at any scale. If you zoom into the Madelbrot set, at any scale, the edge appears similar. Some things in nature appear this way too, like coastlines (mentioned above). Fractals can be used to generate realistic-looking nature images.
It’s pretty helpful if you want to draw a lot of ferns without too much effort. If you have an animated movie with a lot of plant life, you start to appreciate that sort of thing.
Fractal geometry is also useful in the study of certain dynamical systems – given a long enough time, a dynamical system will evolve towards a certain end stage, called the attractor. This can be a point (like a pendulum after it’s stopped swinging), or something more complicated – a line, a plane, or even a fractal set. Attractors of the latter kind are called strange attractors; systems with chaotic dynamics have strange attractors, thus, fractal geometry is important in studying chaotic systems, like the weather; indeed, one of the first and most famous strange attractors, the so-called Lorenz attractor, was introduced to study convection in the upper atmosphere.
A slightly more ‘real world’ example are fractal antennas, which exhibit frequency invariance, i.e. the same electromagnetic properties across a range of frequencies.
Way back when in the dark ages, when I was in grad school, there was this total idiot who always sat up front in class and asked the stupidest questions. Real groaners. He was universally hated.
Centuries later, PBS thinks I want to see that numbskull again.
Okay, so it looks like somebody (or lots of somebodies) spent an inordinate amount of research time so we can have better special effects in movies and can draw pretty pictures. Please tell me that no tax dollars were harmed in the production of this discipline.
No, that would be a lie. Most pure research in universities is supported by tax dollars, one way or another. But there is a lot of money in some of the results of this research, including drawing “pretty pictures”, since compression algorithms lie behind having manageable file sizes for graphic images and moving pictures on computers. No compression algorithms would mean no movies on DVDs and no YouTube.
Yes, tax dollars went into it. No, the applications aren’t obvious to you. That doesn’t mean it wasn’t worth funding. Most of the technologies you rely on daily are based on research that at the time didn’t appear to be useful.
Actually, I kind of liked that Nova episode–but then again I’m an engineer, and they mentioned a few things that piqued my interest. I learned that an apparently finite line can be made infinite with more iterations of the particular function. 'Splained a lot of stuff, and I thought I learned something.
Tripler
I just want to know what makes Mendelbrot look like Eric Cartman on his side.
I like Nova because they don’t talk down to their audience. But I was having problems figuring out what this one was about. I also thought that the finite line being infinite was an amusing bit of philosophy.
Giles: Ah-HA! DVDs and Youtube. Now THAT I can understand.
Nah. Let’s put this another way: physics models behaviour of real-world objects using mathematics. A straight line can represent something moving at a steady speed. A curve may be something accelerating, like a ball falling down. A sine can describe an oscillation, a pendulum for instance. Those are all fairly simple things.
Reality, basically, never actually works that way. Reality is very complicated, and often very dependent on initial conditions, which makes exact calculations impossible. So, what we need is simple equations to describe complex objects. Fractals are complex objects described by simple equations, so we can use them to attempt to model certain kinds of complex systems, like the weather, or even the stock market.
And if you think that’s somehow esoteric, you’ve already got a potentially chaotic system once you’ve got three particles interacting with each other.
More or less what Chefguy said. Mathematics becomes **useful **when it accurately models something interesting. Then we can learn about it via models. Fractals are good at modelling lots of real-world phenomona that “normal” mathematics is not.
This one’s actually not too hard to explain. You know that snowflake shape they kept showing? That encloses a finite area. You start with a triangle, and make little triangles in the middle of the sides, than you make even smaller triangles in the middle of all the resulting sides. You keep repeating this on to infinity, and you get a shape that, no matter how far you zoom in, is always jagged and never smooth. It is, therefore, infinite in length, because you keep meandering among little jaggies at infinitesimal scale, and never get where you’re going. But, the area within the shape is still finite. It’s a closed loop, and doesn’t extend off into infinity in any dimension along the plane.
The coastline of Great Britain is a good example. If you use an infinitely small ruler, you can increase your resolution to the point where the length of the coastline is (practically) infinite. You spend so much time measuring around the subatomic particles that make up each sand grain that you never get anywhere. But the area within that infinitely long curve is clearly finite. Because the residents of Great Britain don’t have an infinite amount of space to spread out in.
The first part, that it encloses a finite area it correct. Just put a big square around the whole thing. Clearly finite. But being infinitely jagged does not establish infinite length. Take a square a make a staircase diagonally across from the lower right corner to the upper left. Make the zigzag as fine as you want. (You can make it a “process” like the Kock snowflake by replacing each step by two steps if you want.) But the length is always twice the length of the square. (This is done is a well-known “proof” that the square root of 2 = 2.)
To prove it is infinite length, you have to calculate the limit of the sum of the sides. Well, okay, actually it’s a product. At each iteration the length is 4/3 longer.
Note that fractal compression was pretty much a no-go from the start. Barnsley and Sloan (of a famous fractal book) didn’t really figure out things well, patented stuff when the issue of free compression methods was at its peak, etc. (I knew Barnsley at the time so this wasn’t a surprise.) The extreme slowness of the method and the results many times looking too “computer generated” have essentially killed it. At some point, other people might start looking into it and a whole other method that is more practical might come out.
The value of Barnesley’s work was the idea of a fractal as a fixed point of a deterministic transformation - the idea was to start with the target object, figure out its transformation, and then one could basically reproduce the original object to any desired resolution by repeating the transformation.
One can certainly argue about the quality of the approximation methods, but the idea is rather potent.
I watched that episode and it seemed to me that possible practical applications of fractal geometry were clearly spelled out. Modelling weather systems, detecting abnormal heartbeats by their fractal signatures, finding cancer early by screening for abnormal fractal patterns in the blood vessels around cancerous tumours, estimating the total size and effects of a rainforest on carbon dioxide levels, and using fractal patterns in your everyday cellphone antennas. Clearly, the potential of fractals is almost unlimited, and we are just beginning to understand their applications to the real world.
Exactly. The list goes on and on and on. If we could explain it to a not very bright 5 year old, as requested in the OP, then we would have it in the elementary school curriculum. Fractal geometry is the maths, and then you have all the implications in chaos theory and in so many other fields - it is just so enormous a topic, no-one would ever attempt to explain it to a five year old.
I have, however, attempted to explain it to many gifted school kids over the last 15 years. The reaction is astounding. There is nothing I have taught which has caused such a huge impact on the young minds I dealt with, but I never did it in a quick course. Always with time for the students to plot hundreds of fractals and just play with them. Time to explore the amazing Sierpinski Gasket (the triangle thing) and see the link to Pascal’s Triangle and all sorts of other seemingly unlinked aspects of maths, time to play with a simple game and find they were replicating the stock market crash of 1987, time to question the very nature of art, time to discover that you can have really simple equations, iterate them (apply them repeatedly) and not be able to predict the outcome - they are chaotic - that cannot be absorbed in a short time. It can be quoted, but not understood just how profound a statement it is. The world can never be predictable, and you can discover that, kid, with a really really simple equation and a calculator.
Then there are the incredible people stories, like that of Mitchell Feigenbaum and of Gaston Julia.
I did my masters degree on evaluating fractal geometry and chaos theory in the curriculum, and developed courses which I have taught in schools - and online to kids all over the world.
This is not a topic you can fully appreciate from a TV show. You can only glimpse the wonders. I would only teach fractals to gifted kids (gifted is not measured in IQ anymore - task commitment is a part of the picture) because to appreciate fractals means being willing to put in the time and effort to think about the implications and let it sink in. The maths of, say, the Mandelbrot Set is mindbogglingly simple while the output (not the flat picture of it) is mindbogglingly complex and beautiful. You have to zoom and zoom and deepzoom to appreciate it. There are thousands of aspects of fractals which each, taken singly and allowing time to sink in, will just inspire you and totally change the way you see the world. I still find new inspiration and understanding each time I delve into the topic.
But if you want an idiot’s guide to fractals, I am afraid it can’t be written. Fractals are far more profound than that.