Equal temperatures at two opposite points across the earth

This is a neat question I got in my first year calculus course:

Imagine a circle anywhere in the universe. (For example, draw a circle on a sheet of paper, or imagine the equator is a circle.) Prove that there are two points directly opposite each other on the circle with the same temperature.


Yes. It is true. And not just with temperature either. The above fact is true if you replace "temperature" with "humidity", "air pressure", or any other continuous phenomenon. Coincidentally, this is also why nobody is ever as cool as they think they are, since there could always be someone right on the other side of the world who is just as cool (ok, that was completely false and un-funny, but you can't blame a guy for trying).

If you're a big boy and you want to skip to the proof, see my next post. If not, let's settle down and get some intuition as to why this surprising fact is true.

First of all, lets stop thinking about temperature as an actual physical thing and start thinking about it like a function, perhaps some function that looks like this:

For our question, you may imagine the y-axis as temperature, and the x-axis as some indicator of where we are on the circle (e.g. an angle). Additionally, there are two properties that are unique to the function we are describing in this problem:

1. It is continuous. We don't expect temperature to "jump" from one temperature to another without hitting all the temperatures in between (at macroscopic levels at least).
2. It is periodic (think sin, cos, etc.). If we go from 0 radians to $2\pi$ radians, we'll have gone full circle and ended back at where we started. So we know that the temperature at 0 radians and $2\pi$ radians are the same.

These properties should drastically change what our function looks like. Instead of the mess of a function above, our temperature function should look something like the graphs below:

Yes. That last graph is periodic (crazy isn't it?). If you look closely, all graphs have a period of $2\pi$ (i.e. they are all zero at 0, $2\pi$, $4\pi$). Unfortunately, if you were to plot temperatures on a circle around the Earth, it'd probably resemble the third graph more than the first or second graph. Don't worry though, the shape of the graph doesn't complicate things as long as its periodic.

Let's get back to the original question now. We need to find two equal temperatures at two locations directly opposite to each other on a circle. This means that the two temperatures are separated by an angle of $\pi$ (or $\pi$ units on the x-axis above). This means we can verify that there exists opposing points with the same temperature by simply drawing a horizontal line of length $\pi$ and moving it around until the ends hit two points on our graph (do you see why?). What the question says is that if you draw a horizontal line of length $\pi$ on your temperature graph, you can always find endpoints for the line which lie right on the temperature function. For $\sin{x}$, this horizontal line has endpoints $(0,0)$ and $(\pi,0)$ (or $(\pi,0)$ and $(2\pi,0)$ if you prefer). The same goes for the triangle wave graph. For the third graph, the line is drawn below:

I'll leave you with a little puzzle. Although we only drew two lines above, we could have drawn two more lines corresponding to the two we just drew. Where are the other two lines? How many unique solutions are there?

Now that we've gained some intuition to this problem, we are ready to proceed with the proof (in the next post).

Update: This post has gotten quite a bit of attention from reddit! Thanks to all who took the time to read and spread this post!