Borsak-Ulem Theorem simplified

A mathematical theorem proves that at any given moment on the surface of the earth, there exists at least one pair of points that have the exact same pressure and temperature.

Borsak-Ulem Theorem defines antipodal points as specific points on a sphere that are located in exactly opposite directions from the centre of the sphere, or situated on diametrical end points along the sphere’s diameter.

“If you place a piece of bread on the ground somewhere on earth, and another one on that point’s antipode – well, you’ve made yourself an earth sandwich,” said Michael from Vsauce, who has a great video that explains this theorem along with some other interesting information about fixed points.

On earth, most points on land have antipodes located in water – since most of the planet’s surface is covered with oceans. The Pacific Ocean is so large, in fact, that it contains its own antipodes.

“There are places in the Pacific Ocean where you could float and know that even if you dug a hole straight down through the centre of the earth and emerged on the other side, you would still be in the Pacific Ocean,” Michael said.

To explain Borsak-Ulem Theorem more clearly, Vsauce encourages you to imagine two thermometers located on opposite ends of the earth. While the recorded temperatures at these two locations will likely be different, if you swap their locations – keeping them on opposite sides of the planet at all times – their temperature readings will flip.

“In order to swap, their readings are going to have to cross at least once,” Michael said. “No matter how we swap these always antipodal thermometers, a criss-cross will have to happen at least once, somewhere.”

He added that the same could be said of atmospheric pressure – by swapping barometers between the two points, just like thermometers, you’ll find both points will measure the same value, eventually.

“Even though weather is chaotic and always changing, and even though the other side of the world is very, very far away, there must always be at least two places on opposite ends of the earth where the temperature and pressure are the same,” Michael said.

Please be aware, that there is an Andy Warhol art piece, that is on the moon, at the start of the video that may not be suitable for those under 18. Please skip to 9min and 30 seconds for the Borsak-Ulem Theorum

In fact, any two functions that vary continuously across the surface of the earth will be equal at two antipodal points. Antipodal points form a continuous, unbroken band around the earth’s surface, dividing two opposite regions – otherwise, you’d be able to swap them in a way without their readings having to meet, which would be impossible thanks to the Borsak-Ulem Theorem.

However, note that this concept only applies in theory – because the earth’s temperatures and pressures are constantly changing at variable rates, it’s impossible to physically prove the Borsuk-Ulam Theory in practice.

Still, this concept has been around for almost 90 years – Borsak-Ulem Theorem first appeared in 1930. The first proof is given in 1933 by Karol Borsuk with credit for the formulation of the problem going to Stanislaw Ulem. A number of alternative proofs have since been published.