What sorts of problems can fractals solve?

After doing a bit of research on fractals, I was wondering what sort of real-life applications do fractal have and in what way would they be used to help solve a problem. I already know people use fractals to map coastlines and borders, but is there any other 'smaller' problems that could be solved by using the principle of fractals and what would be some examples of these and how do they work?

Also, is there a way to generate a problem of a certain situation that could be solved by a fractal or its principles?

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For many such problems from many branches of science, see Mandelbrot's book The Fractal Geometry of Nature. – GEdgar Oct 5 '12 at 14:02
For some people fractals solve the problem of finding a job. – Asaf Karagila Oct 5 '12 at 16:57

Fractals are used in many computer games to render realistic graphics for mountains, landscapes & 3D terrains, especially for flight simulations, computer games, digital artworks & animations. Rather than storing a huge amount of detailed height data in the computers memory, fractal-based algorithms generate the data 'on-the-fly' to render realistic landscapes in real-time. Algorithms for generating fractal-based landscapes take advantage of the 'self-similarity' which exist in the natural world to render visually rich and complex images with attention to fine detail at all scales to appear smooth and not suffer from 'pixelation' even when zooming in close.

These algorithms use methods such as recursive subdivision and fractional Brownian motion to generate a 3D landscape which is then smoothed using a variety of techniques such as image filtering and polynomial interpolation (splines & Bezier curves) to generate photorealistic images of rolling hills & other natural scenes.

Pioneers in this field include Ken Musgrave, digital artist and CEO of Pandromeda (http://www.pandromeda.com/) and Loren Carpenter who went on to form Pixar Studios.

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I was pretty impressed by the digital sundial.

The theory was developed by Kenneth Falconer whose book Fractal Geometry: Mathematical Foundations and Applications may be of interest.

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Based on the idea that solutions of the wave equation tend to resonate poorly on domains with irregular boundary, Bernard Sapoval and Marcel Filoche invented the fractal sound barrier, for use along highways, etc. My understanding is that it is extremely effective, though not yet common because of its higher cost.

There is a story that an early installation encountered problems when local authorities decided they didn't like the color of the sound barrier, and had it painted! The paint filled in the material's fine pores, destroying its fractal structure and leaving it no better than an ordinary wall.

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Not clear of what you mean by 'smaller' problems but anyway I suppose you should take a glance of the journals 'Fractals' and 'Chaos, Solitons & Fractals'.

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The second journal you speak of doesn't have a very good reputation... – J. M. Oct 5 '12 at 13:20
@J.M.: The journal has existed anyway since fairly long time. Were there plagiaristic papers or what do you mean? – Mok-Kong Shen Oct 5 '12 at 15:36
Did you click on the link I gave? – J. M. Oct 5 '12 at 15:50
[I overlooked the link you gave.] I am shocked by the story told in the wiki article. On the other hand IMHO one shouldn't consider that past event to be a barrier against reading that journal. While the modern productivity of the scientists rapidly goes up, the moral of publications deplorably failed to be well controlled in quite a few notable recent cases. – Mok-Kong Shen Oct 6 '12 at 18:44
Well, it's hard to trust a faucet that used to produce dirty water... – J. M. Oct 6 '12 at 18:45

The Weierstrass function is technically a fractal since it exhibits self-similarity and is a great example of a function that is everywhere continuous and nowhere differentiable. The idea being that no matter how closely you look at the curve, everywhere is a "corner" and so non-differentiable.

Also, l-systems are used to model plant life and biological systems using very simple recursive rules.

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