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

= 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? Neon Merlin  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.  -- Jayron 32 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 $$r\to0$$ (if for no other reason than Heisenberg's uncertainty principle). What you can do is... not let $$r\to0$$. 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)