How long is a coastline? 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. No sign-up and no libraries.
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Two countries, one border, two lengths: walk a ruler (1 to 60 km) around a generated island and watch the measured length grow as the ruler shrinks. A log–log plot of length vs. ruler gives the fractal dimension from its slope (D = 1 − slope).
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Rougher coasts grow faster: set the island's roughness (dimension 1.05 to 1.5), generate new islands, and check that the survey recovers it.
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A coastline built from one rule: step through the Koch snowflake. The perimeter grows by 4/3 at each step while the area settles at 8/5 of the starting triangle.
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What "1.26 dimensions" means: the scaling rule N = s^D for a line, a square and a Koch curve.
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Real coasts, measured: Richardson's data as interpreted by Mandelbrot (1967), plus Norway.
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The coast that never ends: a zoomable Mandelbrot set. Click to zoom in, shift-click to zoom out, drag to pan, scroll to zoom, or jump to a preset. Three palettes: Lagoon, Sunset and Survey chart. It uses 64-bit floats, so zoom is capped near 10¹³×.
Plain HTML, CSS and JavaScript, with no dependencies and no build step. Just open index.html. Everything is drawn on canvases.
- The app logic is in js/coastlines.js, its styles in
css/coastlines.css. - Three color themes (dark, light and blue) are shared with my other projects through OMG-Themes. Run
npm run sync:themesto refreshcss/core.css,css/densities.css,css/themes/*andjs/omg.jsfrom it. Project tweaks go incss/overrides.css. The canvases read their colors from the theme. - Your theme is kept in the browser's local storage.
The island starts as a lumpy 12-sided outline. Each segment's midpoint is pushed sideways by a random amount proportional to the segment's length, repeated 11 times (24,576 points). Each level doubles the pieces and scales them by r = √(1 + 4g²) / 2, so the curve is self-similar with dimension log 2 / log(1/r).
The length is measured with the divider (Richardson) method: from the current point, step to the next point on the coast exactly one ruler length away.
- B. Mandelbrot, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", Science 156 (1967).
- L. F. Richardson, "The problem of contiguity", General Systems Yearbook 6 (1961).
- J. Feder, Fractals (1988).
- M. Shishikura, "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets", Annals of Mathematics 147 (1998).
Infinite-Coastlines is open source at GitHub with MIT license.
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You may also be interested in my other projects Sunflower Golden Angle, Mandelbrot and Julia and Snowflake Grower. See them all on Generative Art.
Copyright (c) 2026 Olivier Giulieri.