FractalVision Lab · Interactive Demo
Coastline Paradox
Why does Britain's coastline have no definite length?
The Paradox
The coastline paradox, introduced by Lewis Fry Richardson and developed by Benoît Mandelbrot, states that the measured length of a coastline depends on the scale of measurement. Using a shorter ruler captures more detail — every bay, inlet, and promontory — making the total measured length longer.
This is directly connected to fractal dimension. A more irregular coastline has a higher fractal dimension, meaning its measured length grows faster as the ruler shrinks. The box-counting method used in FractalVision Lab quantifies exactly this scaling behaviour.
Interactive Ruler Demo
Measured Length vs. Ruler Size
As ruler size decreases, measured length increases — a signature of fractal geometry. The curve follows the Richardson power law L(r) ∝ r1−D, where D ≈ 1.17 is the fractal dimension of this coastline.
Connection to Box-Counting
The box-counting method used in FractalVision Lab is the discrete analogue of this ruler experiment. Instead of measuring length with rulers of different sizes, it counts how many boxes of each size are needed to cover the pattern. The fractal dimension D is the slope of log(count) vs. log(1/size) — exactly the scaling relationship this paradox demonstrates.