Closed-path temperature

[barnabas_truman]

Here's an interesting example of how certain ideas from math apply to other fields in ways that might not be expected.

Suppose you draw out a loop path on a map. Any map you want; any shape path you care to draw, as long as it starts and ends at the same point.

Is it possible that every location along the path you drew is a different temperature? Or must there be at least two locations on that path that are the same temperature? Why?


(Clarification: when I say "location on the path" I mean locations in the actual place represented by the map--but that doesn't actually matter; if we were instead discussing the temperature of different parts of the map itself, it would still be fundamentally the same question.)

Comments

[dushai.livejournal.com]

If the normal laws of thermodynamics are followed and temperature is not discontinuous, it is not possible. The initial location is a temperature T_0. Now move some small distance around the loop. Either this new location is at the same temperature, violating the condition instantly, or it's at a new temperature T_1, either a bit higher or a bit lower than T_0. Assume T_1 is a bit higher than T_0. You know that by the time you continue all the way around the loop, you'll be back at T_0, so -- since there are no discontinuities -- you had to pass back through T_1 at some point, and therefore there are two points at T_1. I forget the name of the theorem, and it's Google-resistant, but it basically says that for a continuous function, if you're going from A to B, and C is in between A and B, then you have to pass through C... (Hmm, is this it?)

[barnabas-truman.livejournal.com]

Ding! Ten points!