... where the temperature and pressure are exactly the same as on the opposite side of the earth?
This happens at every point in time, throughout the history of the earth.
And what's weird is, if stand on that point and heat it up, then this magic point simply moves to another location. There becomes a different location where the temperature and pressure are exactly the same on opposite sides of the earth.
This weird fact is a consequence of the Borsuk-Ulam Theorem which says that any continous function from a sphere to a plane must send some pair of opposite points to the same value.
Intuitively, it's actually pretty easy to explain in lower dimensions.
Imagine you're standing at the equator of the earth and measuring the temperature difference between the point you're standing and the point on the opposite side. Let's say you currently measure +5°C difference. That means when you stand on the opposite side, the difference would be -5°C.
Well, now start walking along the equator to the opposite side. Since you're walking from a temperature differential of +5°C to -5°C, and since temperature varies continuously, at some point you'll pass a point where the temperature differential is 0°C, which means the two sides of the earth have exactly the same temperature.
It works with any continuous measure too. So not only is there a point where temperature and pressure are the same on the two opposite sides, there's also a point where the temperature and humidity are the same, or a point where pressure and humidity are the same. Or any measurement which is continuous in space that you can think of, even something like the average distance to the nearest human.
What if it's not exactly a sphere and the measures are not exactly continuous?
The measures need to be continuous but it doesn't need to be a sphere. It can be a warped or stretched sphere as long as the space is also continuous.
What if the Earth is flat or toroidal?
Flat definitely wouldn't work. Using the walking around the equator logic... well that logic breaks if there's no continuous loop.
Not sure about toroidal.
My intuition says you'd maintain equivalence of one measure, but maybe not two.
Wtf I thought this was ~math!?
(╯° - °）╯︵┻━┻
In any case this detail messes with me. Since the earth is turning in relation to the sun, I would think this point is always changing.
Does it have other implications besides cool magic (ahem-- math!) stuff?
Perhaps, so mysterious.
The theorem only tells us the point exists... it doesn't tell us where it is...
Not any more. The cat is securely out of the bag now, and I'm guessing it's doing its best trying to stay away from this scary Ulam portal. A word to the wise ....
I feel like this theorem would make for good scifi / fantasy. Any continuous measurement will have this mysterious Ulam point, including things like psychic potential, hidden evil, quantum entropy, ....
Mix in some historical fiction and you've got yourself a best-seller
Didn't know dude was on the Manhattan Project. The mystery deepens...
You joke, but I seriously have always been awed by math. I'm logical enough to appreciate it, but only after a good dose of handwringing. I like your explainers. They make sense to me.
Not pursuing math professionally is a minor regret of mine. I say minor because I don't know if I actually would have succeeded at it.
Is this something that's been proven?
yes, although I don't know if anyone has ever tested it and found two such points on the actual earth
why are you saying yes if you're unsure if the thing that would be the proof has not been done?
It's mathematically proven
wat
weird, right?
Does the Borsuk-Ulam Theorem extend to n-spheres?
Yes, any sphere in N dimensions mapped onto a N-1 dimensional plane will have at least two antipodal points with the same values.
But if you are walking to find a spot where the temperature matches the opposite won't the pressure likely have changed by that point?