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In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
Formally: if f : S n → R n {\displaystyle f:S^{n}\to \mathbb {R} ^{n}} is continuous then there exists an x ∈ S n {\displaystyle x\in S^{n}} such that: f = f {\displaystyle f=f}.
The case n = 1 {\displaystyle n=1} can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space.
The case n = 2 {\displaystyle n=2} is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space.