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A corollary of the Borsuk-Ulam theorem tells us that at any point in time there exists two antipodal points on the earth's surface which have precisely the same temperature and pressure. Pretty surprising! Here's the statement.

Borsuk-Ulam Theorem There is no continuous map f:S2->S1 such that f(-x)=-f(x), for all x.

Here S2 is the unit sphere and S1 the unit circle. The proof uses some algebraic topology.

Here's how to deduce the physical interpretation I mentioned at the beginning from the theorem. First of all the earth is a sphere! So in mathematical terms the result will follow if we can show that for any continuous map h:S2->R2 there exists x with h(x)=h(-x).

We give a reductio ad absurdum proof of this. Suppose that there is no such x. Then we can define a new function g:S2->S1 by g(x)=(h(x)-h(-x))/||h(x)-h(-x)||. By the hypothesis on h this function is well-defined, and it is obviously continuous. Note that g(-x)=-g(x), for all x. This contradicts the Borsuk-Ulam theorem, proving our result.

For another corollary of the theorem see the ham sandwich theorem.

This would be true if indeed the Earth is a mathematically perfect sphere.

But it's not. The Earth is slightly squashed like a flattened tennis ball, bulging outwards at the equator due to the spin of magma. In addition, there are the small problems of geography, coastal dynamics, and ablative heat transform (I don't know what that means, but it sounds impressive).

The Earth's shape is also affected (minimally) by the position of the moon, and the other solar bodies.

So unless this is true for any vaguely sphere like surface, the theorem doesn't apply.

Although the Earth is not a perfect sphere, a mathematical object representing its surface (a geoid) is homeomorphic to S2. Therefore, the theorem applies.

Except for the fact that, after we've located1 the aforementioned two points, and after transforming back to the geoid, the points probably won't be exactly antipodal anymore.

ariels says "One way to do it is to show that the Earth is star convex around its centre. Define a map R^3\{0} -> R^3 by x -> x/|x|. Then the map transforms the Earth says to a sphere, while preserving antipodality."

1located, somehow. We were given an existence proof, not a construction.

# 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 tA − tB = 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 xn to position xn + 1, the change from tn to tn + 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 tB and vice versa). The difference in temperatures is then tB − tA =  − 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.

Well, the above argument shows that when A and B switch places from antipodal points x0 to xt 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 pC and pD, and te difference between these is pC − pd = 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, pC and pD. Just as before, at some point in the path M, the pressures pC and pD 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.

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