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Book: The Fractal Geometry of Nature

  • Nature consists of irregularly spaced entities which when taken together give rise to aggregate properties.

    • Ex: Density of a sphere of air fluctuates widely as we reduce the scale of the sphere
    • At molecular scales, it reaches extremely high values at some points while being 0 at most.
    • The same holds true for the universe, planets, solar systems, galaxies etc (i.e. at different scales)
  • Differential functions are the exception, not the norm in the space of geometries

  • If we plot the motion of a single particle in Brownian motion, we’ll see that it is topologically of one dimension (i.e. a moving in one dimension) but fractally 2 as it’s motion spans a plane.

  • Erudition and knowledge of a history of ideas is good for the soul

  • Rectifiable: Curves whose length rapidly approaches a limit as the smallest unit of measurement decreases.

  • Coastlines are non-rectifiable as their length increases indefinitely as the unit of measurement decreases.

    • Apparently there exists some range of measurement for which the length does not change much (Around 20cm to 20m)
  • Divergence syndrome: Some quantity that is commonly expected to be finite turns out to be infinite or vanish

Fractals, Fractal Dimension,