Recently a lot of videos have popped Up for me claiming that you can’t accurately measure the coastline of a landmass cause the smaller of a “ruler” you use, the longer of a measure you get due to the smaller nooks and crannies you have to measure but i don’t get how this is a mathematical problem and not an “of course i won’t measure every single pebble on the coastline down to atom size” problem”. I get that you can’t measure a fractal’s side length, but a coastline is not a fractal
In: Mathematics
A coastline has the same property that makes fractals problematic. The finer the details you measure, the longer the coastline will appear. Of course you won’t measure every pebble, but are you measuring in 1 meter intervals? 10 meter intervals? You’ll get very different answers.
The coastline paradox isn’t necessarily stating that you “can’t accurately measure a coastline” because making that statement would depend on a definition of “accurately.” Even if your definition of “accurately” was on the sub-atomic scale, then measuring a coastline would be difficult but, in principle, not impossible. (Though this is true about measuring anything.)
Instead, the coastline paradox says that as your definition of “accurately” changes, the resulting measure of the length of the coastline will change in unexpected ways. It’s not a paradox to say that greater accuracy will change the measurement in some way. We might expect that there is some “true” answer that inaccurate measures will only approximate. Sometimes they will be too high, and sometimes they will be too low. What’s unexpected about coastlines is that increasing accuracy will almost always *increase* the measurement. This has to be taken into account in a couple of ways:
* When comparing coastline measurements, it’s important to ensure they were both taken with the same “ruler”
* Unlike a case with symmetric noise, it’s harder to use statistical tricks to glean the “true” measure from several noisy measures. If the noisy measurements lay both above and below the most accurate measurement, you could take a bunch of noisy measurements and average them to get a good idea of the true measurement. This doesn’t work with coastlines, so that’s the sense in which you “can’t accurately measure a coastline.”
Hold up your fingers in front of your face with them held together.
Trace along the from the bottom of your pinky to the bottom of your index finger.
That’s one interval of measure.
Now do the same, but trace along the tiny gaps between your fingers.
That’s a different, more precise interval of measure.
With the larger interval, the small gaps between your fingers are too large to be measurable, with the smaller interval, the gaps becomes measurable and therefore add the total distance between each of your fingers.
That’s the issue with measuring coastline’s, the more precision you try to use the smaller features you have to measure and the greater total “distance” you get.
1. Coastlines are not static. Anything you do measure, will be instantly invalid to a certain degree
2. The length of something depends on how you measure it. The longer your measuring stick, the harder it is to approximate curves. You cant measure the perimeter of a circle with a straight line.
1. If you use your ruler to measure a diamond shape from the circle, you will get one length. Reduce your ruler, and now measure the octagon, you will get a new longer length despite the circle not changing.
It seems that once you get down to Planck units you ought to be able to come up with a reasonable answer, since any smaller unit is meaningless.
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