What can be a example similar to mathematical Koch flake that could be found in nature... where for a given enclosed volume its surface area is indefinitely large?

  • $\begingroup$ Indefinitely large, I don't follow what means, unless you mean an inability to exactly measure it, if you mean infinitely large, nothing in nature, except maybe the universe, is infinitely "large" $\endgroup$ – user108787 May 26 '16 at 14:24
  • $\begingroup$ Actually meant disproportionately large in relation to mass, like maybe in occlusion salts, some catalyst surfaces .. do not exactly know exactly. $\endgroup$ – Narasimham May 26 '16 at 14:42
  • $\begingroup$ Sorry about that, glad you got an answer though. Regards $\endgroup$ – user108787 May 26 '16 at 15:14

Mathematical fractals do not exist in nature. There is however a huge amount of phenomena that behave 'fractally' within a finite range of scales. Take the typical example of the coast of Britain that can be arbitrarily long depending on how finely you are willing to measure it.

The link goes to a map where only a portion of the coast is visible. If you use it to measure the length of visible coast you will get a finite number (about $50$Km). However, if you zoom in, more details will become visible and your measure will give you a bigger number. If Britain was a mathematical fractal you could continue this forever and measure an infinite coast. In reality however you can only zoom in up to a certain point. When you reach the size of a beach the definition of 'coast' must be updated to take tides into account. When you zoom into sand grains, you need to know the tide schedule very precisely and I have no idea of what happens on the scale of atoms. Anyway, you should stop at some point.

In the same way, if you zoom out, the coast gets more convoluted and the length of the coast grows faster than the size of your window. As before, this only goes on forever for mathematical fractals. In reality you end up seeing all of Britain at once and 'fractality' stops.

Very roughly, and using google maps I find that the coast of Britain is a fractal if I look at it on scales ranging from $100$m to $100$Km. At larger scales, Europe becomes visible and at smaller scales the map is not detailed any more.

If you accept the physicist's definition of a fractal: 'a normal fractal but within a finite range of scales', then there are fractals everywhere in nature. Check this out, for example.

Actually, if you look into critical phenomena you can find scale invariant physics over huge range of scales (potentially from the size of a few atoms to the size of your sample). In practice the thing that restricts your scaling range is the distance of your system to the critical point.

  • $\begingroup$ Thanks for your detailed answer. ..yes the boundary is set when you stop looking into it further. $\endgroup$ – Narasimham May 26 '16 at 15:13

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