Testwiki:Reference desk/Archives/Mathematics/2017 August 30

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August 30

Coastline length problem: does a width help?

In real-world cases, can the coastline paradox be solved by taking the limit of the area that is within some distance r of the coastline divided by r, as r approaches zero, and defining that as the length? Or does that limit also fail to be finite? NeonMerlin 01:34, 30 August 2017 (UTC)

That procedure does not converge and consequently does not define a coastline length. Bo Jacoby (talk) 06:13, 30 August 2017 (UTC).

The insolubility of the coastline paradox is best understood in the simplified context of the Koch snowflake. Since one can have a well-defined area encased inside an immeasurably-long perimeter, the shape doesn't need to be regular, as in the Koch snowflake, just fractal in nature, as the coastline is. --Jayron32 12:36, 30 August 2017 (UTC)

In mathematical cases, fractals (of dimension >1) really have no natural length. Your procedure can't magically change that.

In real-world cases, you can't let $r\to 0$ (if for no other reason than Heisenberg's uncertainty principle). What you can do is... not let $r\to 0$. That is, pick an r that is relevant for the real-world application (say, 1km), and use that. You will get a finite, well-defined value for the length, useful for practical applications. -- Meni Rosenfeld (talk) 20:28, 31 August 2017 (UTC)