Applying Brouwer's Fixed Point Theorem

Suppose we choose two points, $A$ and $B$ , that are antipodal. We measure each of the temperatures there with thermometers. Now suppose we switch the thermometers between $A$ and $B$ , keeping them antipodal along the way. Here's a visual representation of what happens while we switch the thermometers:

Note that no matter where $A$ and $B$ are and what temperatures they are, at some point while we are switching the thermometers the temperature has to be the same. In other words, the graph of the temperatures must cross at least once. Not only that, but since the thermometers can take any path as long as they switch places in the end, there is a continuous "band" of points around the earth with the same temperature.

Suppose we label two antipodal points along that band as $C$ and $D$ . We measure the humidity at $C$ and $D$ with a hygrometer (device used to measure humidity). Suppose we switch the hygrometers between $C$ and $D$ along the continuous band of equal temperature . Here's a visual representation of what happens while we switch the hygrometers:

Note that no matter where $C$ and $D$ are and what humidities they are (as long as they are along the band of equal temperature), at some point while we are switching the hygrometers the humidity has to be the same. In other words, the graph of the humidities must cross at least once.

So, we have now identified a pair of antipodal points, along the band of points with equal temperature, that have equal humidity also.

We know that the above always works since $A$ and $B$ , the points we started with, can be anywhere as long as they are antipodal. Therefore, at a given time, it is $\boxed{\text{Certain}}$ that there will be two antipodes with the same temperature and humidity.

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The result we achieved above is part of the Borsuk-Ulam theorem , which actually implies the Brouwer fixed point theorem .

Wait, you got this from the video https://www.youtube.com/watch?v=csInNn6pfT4