It depends on the length of your ruler, and the closer you look, the longer it gets. Six interactive charts on the coastline paradox and fractal dimension, from a ruler survey of an island to the edge of the Mandelbrot set.
Two countries, one border, two lengths
In the 1950s the English mathematician Lewis Fry Richardson was studying whether wars were more likely between countries that shared longer borders. He hit an odd problem. Spain reported its border with Portugal as 987 km. Portugal reported the same border as 1,214 km.
Neither country was wrong. Portugal, the smaller country, had measured with a shorter ruler. A shorter ruler fits into more of the wiggles, so it finds more length. Try it on the island below.
- Ruler steps
- 0
- Coast length
- 0 km
- Slope of line
- 0
- Fractal dimension
- 0
Every point on the plot is one survey of the same coast. Shrink the ruler 10×, and this coast gets about 2× longer. The points fall on a straight line, so the growth never stops.
Rougher coasts grow faster
The slope of that line measures how rough the coast is. A smooth curve gives a flat line: past a certain point, a smaller ruler finds nothing new. A jagged coast gives a steep one.
Richardson found the slope was steady for each real coast. Benoit Mandelbrot later read it as a dimension. Take 1 minus the slope, and you get a number between 1 (a smooth line) and 2 (a curve so crinkled it fills the page). Set the roughness of the island and see if the survey recovers it.
Changes here redraw the map and plot in Chart 1. The survey estimate wanders a little from the setting because a real island only has a finite number of wiggles to find.
A coastline built from one rule
In 1904 Helge von Koch described a curve with a recipe. Take a line, cut it into thirds, and replace the middle third with two sides of a triangle. Repeat on every new piece, forever. Run it on the three sides of a triangle and you get the Koch snowflake.
- Sides
- 0
- Side length
- 0
- Perimeter
- 0
- Area
- 0
Measured against the starting triangle (side 1, area 1). Each step multiplies the perimeter by 4/3, so it grows without limit. The area settles toward exactly 1.6. A finite island, an infinite coast.
What “1.26 dimensions” means
Dimension tells you how a shape scales. Shrink a shape by a factor s, count how many copies N it takes to rebuild the original, and the dimension D is the power that links them.
3 copies rebuild it.
D = log 3 ÷ log 3 = 1
9 copies rebuild it.
D = log 9 ÷ log 3 = 2
4 copies rebuild it.
D = log 4 ÷ log 3 = 1.262
The Koch curve sits between a line and a plane. It is too crinkled to be a line, yet it covers no area. That in-between number is its fractal dimension, and it is the same number the ruler survey measures from the slope.
Real coasts, measured
Richardson's surveys, as Mandelbrot interpreted them in his 1967 paper “How Long Is the Coast of Britain?”, plus a later estimate for Norway's fjords.
Estimates vary with the map scale and the method used. Treat them as a feel for roughness, not as exact constants.
The coast that never ends
The Mandelbrot set comes from one line of arithmetic: start at zero, square the number, add a constant c, and repeat. If the result stays small forever, c is part of the set. The set itself is drawn solid, and every point outside it is colored by how fast it escapes, so the bands trace contours around the shore.
Its edge is the roughest coastline in mathematics. In 1998 Mitsuhiro Shishikura proved its boundary has dimension exactly 2, as crinkled as a curve can be. Zoom in anywhere along the shore and new bays and headlands keep appearing.
- Magnification
- 1×
- Iterations
- 0
- Center
- 0
This explorer computes in standard 64-bit numbers, so detail runs out at about 10 trillion times magnification. The real coast keeps going forever.
So how long is the coast?
There is no single answer. A coastline has a length only once you say what scale you are measuring at. What stays fixed across every scale is how fast the length grows as you zoom in, and that rate is the fractal dimension.
The same idea shows up in river networks, lungs, lightning, clouds, and the edge of the Mandelbrot set. Whenever detail keeps appearing as you zoom, length stops being the useful number and dimension takes its place.