# 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