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We live in a 3 dimensional world. So, line and plane as 1 and 2 dimensional objects do not exist in reality although using these concepts are useful for modeling some problems such as motion in one direction. I have more difficulty about fractals. Although they are like 1 and 2 dimensional objects in regards that they do not happen in reality, I cannot comprehend their possible usage in modelling some problems. For 1 and 2 dimensional objects we can think of them as a limiting case of 3 dimension. For example if we keep x and y coordinate constant and just move along z direction we get the one dimensional motion. I cannot imagine such a thing for a fractal. If they are not just mathematical entities, how is it possible to use them to describe a physical phenomena in a 3 dimensional world? For example how would it be possible to walk on the edge of a fractal?

MOON
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    Are you aware of Mandelbrot's claim (in The Fractal Geometry of Nature) that fractals are better models for physical objects than conventional geometric figures? Certainly, as he says, clouds are not spheres. The closer you look at them, the more complicated their boundaries appear. – MJD Oct 27 '14 at 13:22
  • "We live in a $3D$ world" is something we actually don't know. According to Einstein's general relativity we live in a $4D$ world. – mfl Oct 27 '14 at 13:24
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    One of the defining properties of a fractal is self-similarity. In the concrete, physical world there's a limit to how far down you can take that, and that is the atomic level. Apart from that, there are many things that look like fractals on our scale. Coastlines are often mentioned. Break a stone, and the surface will be as close to a fractal as the rock type will allow. – Arthur Oct 27 '14 at 13:41
  • @Arthur. If coastline are like fractals, why do they have integer dimension, they are one dimensional. – MOON Oct 27 '14 at 13:51
  • I think there are more than one definition of dimension. I consider dimension as the number of parameters needed to locate a point. In 3 dimension you need 3 parameter to do that. – MOON Oct 27 '14 at 13:59
  • Do you mean $\mathbb R$eally? –  Oct 27 '14 at 14:01
  • That is a problematic definition. Due to the various constructions called plane-filling and space-filling curves, you never really need more than one parameter. Ever. – Arthur Oct 27 '14 at 14:02

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