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1/ It's a lovely time of year to take a walk through the woods.
It's also a lovely time to be asking yourself:
"What is the fractal dimension of the trail I'm walking on?"
It's also a lovely time to be asking yourself:
"What is the fractal dimension of the trail I'm walking on?"
2/ During a recent 4-day trip on the Appalachian trail, I realized that Google maps was consistently underestimating the walking distance between waypoints, by about 30%.
It turns out that this is because Google maps only resolves the trail at ~1 point per 1/4 mile or so.
It turns out that this is because Google maps only resolves the trail at ~1 point per 1/4 mile or so.
3/ This is one of the defining properties of a FRACTAL: its size depends sensitively on the resolution with which you measure it
Its like the infamous "coastline of Britain" problem, where the coast length goes to infinity when the resolution goes to zero
https://rspatial.org/raster/cases/2-coastline.html
Its like the infamous "coastline of Britain" problem, where the coast length goes to infinity when the resolution goes to zero
https://rspatial.org/raster/cases/2-coastline.html
4/ Luckily for me and my curiosity, there is a great wealth of data about the Appalchian trail: it is mapped out in GPS coordinates at the ~10 meter level of resolution, and you can download the data publicly.
http://topofusion.com/at-gps.php
http://topofusion.com/at-gps.php
5/ Here's another look at the trail at various scales (in equirectangular projection). You can see the self-similarity as you zoom in.
6/ Here's how you figure out the fractal dimension: plot the number of segments, N, in the trail as a function of the segment length L.
A well-behaved curve would have N proportional to L.
But a fractal has a funny power N ~ L^x.
The AT has x ~ 1.08.
A well-behaved curve would have N proportional to L.
But a fractal has a funny power N ~ L^x.
The AT has x ~ 1.08.
7/ Here's what that means in terms of your length estimate.
The official trail length is about twice as long as the direct end-to-end distance. If you resolve the trail at one point per mile you'll be off by ~50%, and if you get one point per half mile you'll be off by ~30%.
The official trail length is about twice as long as the direct end-to-end distance. If you resolve the trail at one point per mile you'll be off by ~50%, and if you get one point per half mile you'll be off by ~30%.
8/ You can see from the previosu plot that the AT is "fractalish" at length scales between 0.01 miles (~20 meters) and ~100 miles. At shorter or longer scales it looks straighter-than-random. Which means that the typical size of a bend in the trail is about 10-20 meters ...
9/ ... and the typical horizon for large-scale course planning is ~100 miles.
10/10 Happy fall hiking, everyone!
