Borsuk-Ulam theorem

An illustration to the theorem (and why it doesn’t matter that Earth is not a perfect sphere). Heavily inspired by VSauce’s video.

Please note: This is the illustration to a mathematical theorem, so that means any physics-based objection to the theorem is most likely not applicable, because the real world is messy in ways (these) mathematics are not.

The Claim: At any one point, there are two places on the surface of Earth, exactly opposite to each other where temperature and atmospheric pressure are exactly the same.

Imagine placing two thermometers (A, B) at the Equator, but one placed at the 0° meridian and another placed at the 180° meridian. Measure the temperatures they record.

They will most likely be different. That’s all right, we can define this difference as t_A − t_B = δt. Now imagine moving those two thermometers so that they are always at opposite (“antipodal”) places from each other.¹ What happens when they switch places in this fashion?

Starting at position x₀ the temperature is t₀. Every time the thermometer goes from position x_n to position x_n + 1, the change from t_n to t_n + 1 is smooth, as in there’s never any sudden changes disproportional to the step size. The temperature changes continuously.

Now, after the thermometers have switched places, we record the temperatures. As expected, they are the same as before, but reversed (thermometer A records a temperature t_B and vice versa). The difference in temperatures is then t_B − t_A =  − δt

The difference of thermometer A minus thermometer B went from δt to  − δt. Since this difference itself also changed continuously, at some point it must have been exactly 0 (see Intermediate Value Theorem) If the difference of temperatures recorded by A and B is exactly zero, they must record the exact same temperature.

What about the atmospheric pressure?

Well, the above argument shows that when A and B switch places from antipodal points x₀ to x_t and vice versa, they have the same temperature at some point in their path. This argument was made without any specific path in mind, only with the condition that A and B be kept at antipodal positions at all times on transit.

Imagine the thermometers changing places, but placing down a pair of markers M at the point where their recorded temperatures are the same. Now, imagine doing another exchange, but through a different path, also placing down a new pair of markers. After trying every possible path, there should be a closed loop² around the Earth, made up of markers M representing the points where the temperature is the same at antipodes.

Now place barometers C and D on two antipodes on this loop. The air pressure they record is p_C and p_D, and te difference between these is p_C − p_d = δp.

Imagine moving the barometers just as we did with the thermometers (keeping them at antipodal points) along this loop M. After they’ve switched places, the pressures recorded have also switched.

Just as before, the changes in pressure are continuous and so is the change in the pressures recorded, p_C and p_D. Just as before, at some point in the path M, the pressures p_C and p_D must have been the same and, since the barometers are traveling along a path of equal temperature at antipodal points, these points on Earth’s surface must have the exact same temperature and atmospheric pressure at this point in time.

1. I realize that defining “antipodal points” in an irregular geoid is tricky, but they can be more or less approximated with lines of latitude and longitude laid in a very tight grid.

2. Not necessarily a Great circle, a circle or even a regular shape.