The Coastline Rule

—

the log–log proof · slope = D

Ask how long a coastline is and you get a paradox: measure it with a shorter ruler and you find more length, without limit. The honest answer isn't a length — it's a dimension. Cover a shape with boxes of side ε, count how many it touches (N(ε)), shrink the boxes, and watch how fast the count grows: D = log N(ε) / log(1/ε). A smooth curve gives D = 1, a filled region D = 2 — and a fractal lands in between. This bench doesn't draw the dimension; it measures it, and proves the measurement against the known answer.

The set

Detail / generation

iterations · particles 5

seed 7

The ruler

box scales swept 14

show the counting grid (at one ε)

grid ε preview 1/24

Measured

Box-counting (Minkowski–Bouligand) dimension. Self-similar sets hit a closed form exactly; DLA uses the mass–radius law M(r)∝r^D. A self-test proves every measured D against its known value — and that a smooth curve reads 1, a region reads 2, and a single point fakes nothing. · part of the Workbench.