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Comment by Waterluvian | Hacker News Reader

I had a bit of a brain burst the other day when I recalled from undergrad (GIS) that this was a real problem, but that area was a value that converged on a number the more precisely you measured, so presumably you don’t have to worry about massively different area values the more precisely you measure. But then realized that the coastline paradox must also exist for 3D surfaces (TINs or DEMs or whatnot) so asking “what is the area of a surface on earth” can be a very complex question if dealing with elevation.

That got me wondering if it’s generally true for any number of dimensions.

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Yes. In 2D, area enclosed by a noisy perimeter converges with more perimeter points, but length only increases. Adding more points to the perimeter can only increase length, because the shortest distance between two points is a straight line. But adding a point can increase or decrease measured volume.

In 3D, volume converges with more perimeter points, but surface area does not.

This line of thinking leads to sampling theory. Given a noisy analog signal, digitizing it at higher and higher resolution and sample rate can yield an arbitrarily large number of data points, as you track the noise. But the area of the envelope around the waveform converges.

Thank you, especially for that last paragraph that gives me a lead on what to begin digging a bit into.

>But then realized that the coastline paradox must also exist for 3D surfaces (TINs or DEMs or whatnot) so asking “what is the area of a surface on earth” can be a very complex question if dealing with elevation.

StandUpMaths did a great video on this recently, asking whether national land area includes elevation and which nations would gain/lose the most land area if it were/weren't: https://www.youtube.com/watch?v=PtKhbbcc1Rc

Another GIS-er here. Your comment made me think about the modifiable areal unit problem: https://en.wikipedia.org/wiki/Modifiable_areal_unit_problem

The coastline paradox and MAU effectively seem to be of the same class, i.e., change the rules for your measurement and you get different results. This hadn't occurred to me before. I wonder if there's other similar?

Well, there is a good way of measuring the area; instead of measuring actual area, measure the area of the surface on a perfectly smooth spheroid with an identical perimeter.

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