Some "Eurekas" of Mathematics - Part Eleven

This blog will use some topological terminology, but I try to keep everything at a basic level in terms of understandability.

The big result that I want to lead up to is this:

Some topological terms that will be used, such as homeomorphic, are mentioned in this blog. Check it out for more background information! And if you're unsure what the word antipodal means, this refers to antipodes, that is opposite points on a sphere, e.g., the North Pole and South Pole.

A huge result from the early 20th century by Luitzen Egbertus Jan Brouwer, arguably the most important of his life's work, deals with fixed points. Today we call it the Brouwer fixed-point theorem. Brouwer initially proved this using topological invariance of degree. Personally I'm more familiar with it as a corollary to the Lefschetz fixed-point theorem, but for simplicity sake we'll bypass proving this statement in this blog. Here's the statement of the theorem:

That is, if you continuously map a compact set like say a disk onto itself, there will always be at least one fixed point. Supposedly Brouwer received inspiration for this when he was mixing his coffee, which allows for a nice analogy: there's always a motionless particle on the surface when you stir your coffee.

As stated above, the interesting application we are focused on in this blog is how at some pair of antipodal points on Earth, the temperature is identical. In the image below, we see an example of two antipodal points that have the same temperature. I found this image online and I'm honestly not sure if the data is real or not, but let's just go with it and say it is.

One important thing to note is that this will rely on the fact that temperature is "continuous", which I will handwave for now. I'm no meteorologist, so I don't know whether temperature discontinuities are possible, but a quick Google search told me that temperature is continuous. Plus it feels intuitive, so we'll go with it.

Now onto the proof that it is true. This proof won't be incredibly detailed as some of the details may be too technical. We'll also be operating under the assumption that Brouwer's fixed-point theorem is true, as we mentioned above. Proving that falls a bit out of the scope of this blog.

We will approach this by employing a proof by contradiction. Thus, by means of contradiction, we assume that there exists a continuous map

What do you think? I personally find some aspects of this fixed-point theorem to be intuitive, such as giving a simple stir to a coffee mug, but there are other aspects that seem more shocking. This concept of antipodal points can be applied also to other phenomena such as wind speeds or air pressure.