The Geometry of Roughness
Mandelbrot’s invitation to see mathematical form in clouds, coastlines, bark and lightning.
“Clouds are not spheres, mountains are not cones, coastlines are not circles.”
— Benoît Mandelbrot
Classical geometry gives us immaculate objects: circles without thickness, planes without texture and spheres polished beyond anything nature manufactures.
Nature speaks another dialect.
A coastline remains jagged as we inspect it more closely. A cloud has billows within billows. A tree branches, and each branch repeats something of the whole. These objects are neither shapeless nor smoothly Euclidean. Their order lives inside their roughness.
Mandelbrot’s great imaginative leap was to treat irregularity not as noise around the “real” shape, but as a shape worthy of mathematics in its own right.
Fractal geometry changes the question. Instead of asking how to smooth the mountain into a cone, it asks:
What law inhabits the mountain’s roughness?
This is a recurring pattern in discovery. What first appears to be disorder may merely require a language with enough imagination to describe it.
A little detour: For a comprehensive write-up on fractals — their fascinating history, families, and the ideas surrounding them — do visit Fractal Fair, by yours truly. :-)
Source note: Mandelbrot used the longer formulation in The Fractal Geometry of Nature; it contrasts idealised Euclidean forms with the rough structures of the natural world.