17 Jun 2011

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!


  1. I think you would find these fun. I had the balancing pencil problem for one of my physics courses. I didn't get it before seeing the solution. http://www.physics.harvard.edu/academics/undergrad/problems.html

  2. @Scott

    Yes! I love those problems :). I've been trying to find time to do them but I only manage to get through one every month or so. David Morin also has a very nice classical mechanics book which features more zany problems (though all of them are physics related).

  3. How is temperature/humidity/air pressure defined on a zero-dimensional point?

  4. @Anonymous

    I'm not sure I understand what you mean. Could you clarify your question a bit (I want to make sure that I don't accidentally steer you the wrong way)?

  5. @Paul

    Well I'm having trouble with this:
    "Prove that there are two points directly opposite each other on the circle with the same temperature."

    Maybe I am mistaken, as I was thinking of temperature in terms of kinetic theory (http://en.wikipedia.org/wiki/Kinetic_theory_of_gases) where it is a measurement taken over a space with a definite volume, but how is temperature (or humidity for that matter) measured on a single zero-dimensional point?

    Is the assumption that temperature is continuous a good enough to assumption to make claims about the temperature of specific points on a circle?

    I guess I am more curious about the physics of the initial assumptions than I am about the math that follows.

  6. @Anonymous

    Ahh I see. In this post I explained that my temperature function applied to the macroscopic world only.

    As for the physics, we cannot define (or measure) temperature, humidity, or any of the physical concepts familiar to us on a daily basis because as you've stated, nature is discrete once we get to really small levels.

    However, when I say small levels I mean really, really small. Try pinching your fingers together. In the space between your pinched fingers, there are still millions of air molecules. So for any circle we can think of in the real world, this proof shouldn't be a problem.

    I hope that helps,

  7. This question looks extremely difficult.
    I'm confused already!

    1. Lol. You did this question. Like everyone else in class.