Scientists catalog ‘fractal dimensions’ of more than 130,000 islands

The “coastal paradox” helped define fractals, but coastlines themselves turn out to be less fractal than previously thought

By Alex Music edited by Sarah Lewin Frasier

In 1967, the mathematician Benoît Mandelbrot observed that Britain’s coastline is impossible to measure: its perimeter lengthens the closer it is measured. At that point, he had eight years left before creating the term “fractal”: a shape containing smaller parts, similar in shape to the larger whole, which become apparent as you zoom in, creating an infinite and infinitely complex repeating pattern. The so-called coastline paradox he observed is one of several known examples of fractals in Earth’s geography. But recent work has shown that coastlines may actually be much less fractal than previously thought.

The research, published on the arXiv.org preprint server and accepted in Geophysical research letters, brings together geographic data for more than 130,000 Earth islands to show that although landmasses behave like fractals in some ways, the extent to which they do so – their fractal dimension – differs depending on the geometric feature of the island you’re looking at. Researchers have found that coastlines come last (after, for example, surface rise). This finding deviates from current fractal models of the Earth’s surface, according to mathematician Matthew Oline of the University of Chicago, lead author of the study.

The fractal dimension of a shape tells you how far you can zoom in to make the fractal pattern repeat. The coastline of an island with a low fractal dimension may appear almost completely smooth, while a coastline with a high fractal dimension will have a visibly bumpy and complex border even if you zoom in further. This concept extends to other geometric features of islands, including size distribution, elevation, and volume. For example, small islands are far more abundant than large ones – a pattern of complexity that also makes the size distribution of Earth’s islands fractal. In typical models of the Earth’s surface, all features evolve with the same fractal dimension.

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Oline discovered, however, that some island features can handle zooming in on their fractal details more than others. He says this result is not remarkable, because existing fractal models look more like “toy models” than an accurate representation of Earth. What was unexpected, however, was the marked difference between the fractal dimensions of different geometric elements, especially for the surprisingly smooth coastlines. “The coastline paradox is one that people have heard about, but in reality, coastlines are the smoothest feature we see here,” Oline says. This finding is consistent with the idea that factors such as sedimentation and erosion reduce the complexity of a coastline more than, say, a mountain peak.

Andreas Baas, a geomorphologist at King’s College London who was not involved in the work, was also surprised by how smooth the coastlines were compared to estimates in previous studies. He says the authors’ approach to calculating fractal dimensions was “very rigorous,” but he is more cautious about assigning meaning to the differences. Their hypothesis opens “new avenues of research” to fill existing gaps between studies involving shoreline and surface models, says Baas. “It would be interesting to combine these models to see if they can reproduce the observed observations. [fractal] relationships. »