07 September 2007

How to build mountains

Mark Chu-Carroll of Good Math, Bad Math writes about how images of simulated mountains are constructed using a fractal process.

I'm kind of surprised to see how simple it is; basically you take a triangle and "pull up" a random point in the interior to get an irregular pyramid, and repeat this procedure on each of the faces.

One of the commenters there, mj, writes: "In a way it's not surprising that complex structure comes out of very simple fractal rules. The structures in reality, the real mountains, are also formed by relatively simple processes. A bit of wind and rain and erosion..." That's a good point. What's interesting is that there's no obvious connection between the rules used to generate the fractal and the real rules, but they generate the same sort of structure on a global scale. On a large enough scale, the low-level structure is simply hidden. Something similar happens with, say, random walks; a cloud of random walkers allowed to dissipate will eventually approach a Gaussian distribution regardless of the underlying lattice. An observer who could only observe on a large scale couldn't tell what the underlying lattice is. There are much deeper ideas of this sort; for example, we don't know whether the universe is actually continuous or discrete.

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