Antipodal Temperature and Pressure

Assume the earth is a perfect sphere. Show that at any given time, there exist two antipodal points (points on exact opposite sides of the Earth) with the same temperature and air pressure.

Hint

This puzzle is much harder than the previous one. There is a mathematical theorem you can use to solve it, but it makes the problem almost trivial.

Expand to see what the theorem is

Look up the Borsuk-Ulam theorem.

Solution

Answer: At any given time, there does exist two antipodal points with the same temperature and air pressure. However, to prove this requires a more powerful tool.

The Borsuk-Ulam Theorem

The Borsuk-Ulam theorem states that any continuous function from an n-dimensional sphere to n-dimensional Euclidean space maps at least one pair of antipodal points to the same point.

For the 2-sphere (the surface of the Earth), this means: any continuous function $f ⁣ : S^{2} \to R^{2}$ has antipodal points $p$ and $- p$ such that $f (p) = f (- p)$ .

Let’s apply this to our problem.

Define a function $f ⁣ : S^{2} \to R^{2}$ that maps each point on Earth to the pair (temperature, pressure) at that point. Since temperature and pressure vary continuously across the Earth’s surface (no sudden jumps), the function $f$ is continuous.

By the Borsuk-Ulam theorem, there exist antipodal points $p$ and $- p$ such that $f (p) = f (- p)$ . This means both the temperature and pressure are equal at these two antipodal points. Our proof is done.

Antipodal Temperature and Pressure

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