Coastlines are highly fractal, leading to the coastline paradox: the length increases to surprisingly large numbers as your scale of measurement gets smaller.

Does the same thing apply to rivers, or is there some mechanism that keeps their lengths reasonably finite?


1 Answer 1


The same problem applies to all shapes that are complex and this includes coastlines, rivers and the perimeters of leaves. Most shapes derived by nature will have this issue. The unit of scale matters.

Only shapes that are composed of straight line segments and can be easily measured as such do not have this problem: triangles, rectangles, hexagons, etc. Such shapes are mathematical or created by humans.

  • $\begingroup$ Could imagine the more consistent/weaker nature of waves in rivers might reduce the variability a little, and if it did, it would reduce the degree of increase. However, flooding events exaggerate more, so perhaps not. But agree with your answer as a whole. $\endgroup$ Commented Jun 19, 2017 at 14:14
  • $\begingroup$ Checks houseplants, looks out window. At least locally, leaves have mostly non-fractal perimeters, that is, as you reduce your unit of scale, the perimeter is clearly converging to some length. $\endgroup$
    – Mark
    Commented Jun 20, 2017 at 18:23
  • $\begingroup$ However, rivers have thickness, with makes things easier, there's no point making your scale significantly smaller then the width of the river, and all the little curves at the riverbanks can be ignored. Imagine the river as a racetrack, and try to find the shortest path from start to finish, that is well defined. $\endgroup$ Commented Aug 10, 2022 at 5:38

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