A 2024 study computed the fractal dimensions of several coastline segments before and after applying standard map projections, using divide-and-conquer algorithms on digitized images. The results confirmed what cartographers have long suspected but rarely quantified: projections do not merely stretch or compress coastlines uniformly. The Mercator projection increased the fractal dimension of Norway's western coast by measurable increments while leaving equatorial coastlines nearly unchanged. The distortion is not a scalar — it is a function of both position and geometric complexity.
This matters because fractal dimension encodes roughness. A smooth curve has a fractal dimension near 1.0; a jagged fjord-riddled coast approaches 1.3 or higher. When a projection alters fractal dimension, it is not just changing where things appear on the page — it is changing the apparent complexity of the geography itself. A coastline rendered in Mercator at high latitudes does not merely look bigger; it looks structurally different from its true shape. The projection manufactures texture that does not exist in the territory.
The usual conversation about map distortion focuses on area and angle — Mercator preserves angles but inflates Greenland, Peters preserves area but warps shapes. Fractal dimension introduces a third axis of distortion that neither framework captures. Two projections can agree on both area and angle for a given region yet disagree on the apparent roughness of its boundary. This is a distortion of morphology, not of measurement.
The broader lesson is that representation systems distort not only quantities but qualities. Any encoding that maps a complex object into a simpler space will sacrifice some structural feature, and the sacrifice is often invisible precisely because the representation looks complete. The map does not announce what it has altered. It simply shows a coastline — and the coastline looks plausible.