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Asked by sanban to Chris, Dave, David, Fiona, Jack on 25 Jun 2013.
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Jack Miller answered on 25 Jun 2013:
Hi Sanban,
Fractals are shapes that look the same at any scale. Often they’re generated iteratively, through an equation or algorithm that’s just applied recursively. They crop up in many areas of science, often modelling nature. There are also deeper connections between non-integer ‘fractal’ dimensions and theoretical physics, which I’m sure David would love to talk more about.
Fractals are generated in nature when processes occur with (usually) simple, self-repeating rules. For example, you can generate a shape similar to this broken brick of perspex (
) by drawing a straight line on a plane, finding its midpoint, and subdividing it in two, drawing two more lines and forming a ‘Y’ shape overall. Then, take those two short lines, find their midpoints, subdivide them in two. Repeat the above steps infinitely many times, and you get a fractal. Fractals often have counter-intuitive properties. For example, the Koch snowflake — formed by taking an equilateral triangle, dividing each of its three sides in three, and making the middle section the base of a new equilateral triangle (and repeating those steps) — has a finite area, but an infinite perimeter. They’re really cool, really pretty, and crop up a lot in nature (due to things obeying relatively simple recursive rules like those I’ve described above, especially in biology).
Hope that helps!
— Jack
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David Freeborn answered on 25 Jun 2013:
Hi Sanban,
As Jack says, Fractals are structures that look the same on every scale. Each time you zoom in on a fractal, you get the same sort of structure.
Like this on the Mandelbrot set (probably not worth watching for the whole 15 mins…)You can get either perfect Fractals like the Koch Snowflake (
), or imperfect ones. For example, we can model the coastlines of islands as imperfect fractals, and also mountain range and ice sheet structures (
) up to a point, but if we zoomed in far enough (to the level of e.g. molecules), the fractal structure would break down.We use fractals a lot in science: in almost all complex systems, predicting the weather, economics etc. One cool feature is that you can have an infinitely long perimeter for objects of a finite area- because the perimeter has an infinitely complex structure. That will even happen with simple fractals like the Koch snowflake.
One cool application is in computer graphics. To generate good computer graphics, you need layer upon layer of structure. Graphics designers have started using fractal-type algorithms to generate realistic objects like mountains and trees. This has led to big improvements in the qualities of graphics. As you zoom in, more layers of structure are generated.
We occasionally do use fractals in Quantum Mechanics. One problem with Quantum Mechanics at the moment is in calculating path lengths. They have a fractal structure, because however far we “zoom” in, there is a small probability of random quantum events disrupting the path. Right now, all our efforts lead to infinitely long quantum paths, which isn’t ok. I suspect a new underlying theory will sort out this problem.
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