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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?

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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.

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  • $\begingroup$ 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. $\endgroup$ – JeopardyTempest Jun 19 '17 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 Jun 20 '17 at 18:23

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