Differentiable = smooth = “micro-dead” = “micro-straight-and-still”
Fractal = thorny = “micro-living” = “micro-bumpy-and-wiggling”
Differentiable curves can be categorized in terms of order of contact
= smoothness of caressing/distributing/sectioning.
Fractals are “recursively alive” on each scale. If they repeat the same pattern on each scale they are called self-similar. If they repeat a sequence of patterns periodically, they could be called rational, in analogy with the correspondence between rational numbers and periodic decimal expansions.
Fractals are typically not self-similar (3Blue1Brown on Youtube):
Fractal Zoom (HD) to 6.066 e228 (2^760) Mandelbrot – (Last Lights On) :
3rd dimension cut 3D Mandelbrot set zoom:
Best Fractal Zoom Ever: