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 nonrectifiable 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
 backlinks
 Fractals, Fractal Dimension,