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

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

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)