Thanks for the post Matthias! This is a beautiful example of a self-similar structure.
Unfortunately I only had time for a very quick test right now, but if we use more box sizes (a proper way would be to pad the image when necessary), we get a clearer picture about what is going on.
This river network is so dense that if we look at it from far enough, then it appears to fully cover the ground. It looks two-dimensional. But as we zoom in, the self-similarity becomes clear and the fractal nature emerges. Here's the scaling plot I got:
The horizontal axis is proportional to the number of boxes that the image was partitioned into. Thus a large number corresponds to a small box size.
There is a clear break in the middle of the curve. The slope of the curve is ~2 to the left of that break (i.e. for large boxes), which indicates that at this scale the object looks two-dimensional. The slope is ~1.5 to the right of that break (i.e. small boxes), which shows that it looks like a fractal at that scale. The power law is valid separately on both sides of the break.
To plot this, I was sloppy and simply cropped the image to {2580, 1810} to have many common divisors. I think a better way would be to pad the image with white pixels when it cannot be partitioned into an integer number of boxes. This way you can use an arbitrary box size.
If we could zoom in without limit, the exponent would become 1 when the rivers don't branch out anymore. They look like lines at this point. If we could zoom even more, the exponent would become 2 when the width of the rivers becomes visible.
