Fact:
There is at least one pair of point on earth surface which are diametrically opposite to each other and have exactly same temperature and pressure.
Image 1: Credits [Antipodes II]
I found this fact very amazing because Earth circumference is about 40, 000 km in length. The above theorem suggests that there are two points situated 20, 000 km away (along the surface) from each other and have EXACTLY same temperature and pressure.
Interestingly, unlike most of amazing science facts, this fact is not an experimental result and can be proved with very simple logic.
Proof:
- Terminology used:
- Antipodes: It refers to a pair of points located diametrically opposite to each other on the surface of a sphere. In this case, The Earth.
- Consider yourself at point A in Brazil and me and other point B in Philippines. They are almost of diametrically opposite to each other but let’s say A and B are chosen such that they are exactly diametrically opposite to earth.
- Also, let’s assume that temperature at point A is 0 Celsius and temperate at point B is 25 Celsius.
- Now, I will start walking towards the point A (you) to in an arbitrarily chosen path and you do the same but in a way that we always stay diametrically opposite to each other.
- As we move, we also keep a record of the temperature for every unit of distance we travel.
- And finally, after we swap our positions, we plot the how the temperature varied with the distance we traveled on the same graph. What would it look like?
Image 2: Credits [Borsuk-Ulam Theorem (The global version)]
- Yes, you will find that there is no way for us to swap our temperatures without these graphs crossing atleast once.
- At the point this graph crosses, we were in same temperature and because we have always been diametrically opposite to each other. We just found a pair of point on earth which is diametrically opposite to each other and has exactly same temperature.
But wait, we were supposed to find a pair of point not only with same temperature but same temperature and pressure.
Proof continuation:
- The above proof starts with me walking in an arbitrary path from B to A but there are infinite such paths, which implies that for every path we should be able to find a new Antipode with same temperature. So, there are infinite such points.
- Interestingly, those infinite point cannot be sprinkled over the Earth’s surface but has form a continuous curve. Why?
- Because if they don’t form the continuous curve, I can choose an arbitrary path carefully has I don’t step over any of those points and not get same temperature throughout the journey, which we already know isn’t possible.
- So the infinite number of antipodes with same temperature should look like this:
- Now, you can do the same exercise for pressure but me following the dotted lines instead of any arbitrary path and find that there will be at least one pair of point on dotted line which are diametrically opposite of each other and have same pressure.
Image 3: Credits (Borsuk-Ulam Theorem (The global version))
- But all the diametrically opposite pair of point on this dotted line has same temperature as well.
- Cool, you just found a pair of point on Earth which are diametrically opposite to each other and they have same temperature and pressure.
BTW, the same logic can be applied to any surface and for any continuously changing variable. This generalized version is famous as Borsuk–Ulam theorem.
Hope you had fun reading this!
Credits: