The first is, obviously, Borsuk-Ulam. The second you can get at quite a few ways, but I agree that your approach is much more natural.

But seeing as we’re on BB not math.stackexchange, I felt like throwing in a few extra examples while we’re on this topic. They’re related. Fixed point theorems are just cool in general.

Full appreciation of the coolness of fixed point theorems may require several years study of analysis and topology.

]]>Original source, better quality, no ads.

]]>I really think these examples “naturally” illustrate other theorems.

Edit: (so as not to be obscure) i.e. I feel the first follows most naturally from Borsuk-Ulam or IMVT depending how you state it, and the second from Banach CMT. If you consider the map and use Brouwer, it’s really not as nice a result (no unique fixed point, no construction, …).

]]>i don’t like the “it’s math. so don’t bother questioning it” assertion.

But that’s not what he said.

He said don’t go wasting your time *playing around with a hairy ball* trying to prove the theorem wrong, because the ‘Hairy Ball Theorem’ is math (i.e., it’s not a prediction of the physical behavior of actual hairy balls, it’s about vector fields in algebraic topology) – and it’s a theorem that’s already been proven.

So nothing you observe while playing around with an actual hairy ball is going to disprove it.

So you’d be wasting your time doing that. :-)

]]>Somewhere on the equator there are two antipodal points with exactly the same temperature. (Or barometric pressure, if you’d rather stick with air movement.)

Take two identical pieces of paper. Crumple one up and put it on the other. There is at least one point on the crumpled piece that is directly over the corresponding point on the flat one.

(You’d have to use transparent film with a numbered grid, or something, to see this for yourself.)

(Alternatively, take a map that includes your current location and drop it on the floor. At least one point on the map is directly over the point it represents.)

]]>I was going to add ‘in my defence I am drunk’… but that makes my predicament even more shameful. ]]>

One of the joys of pure maths is that once you’ve proven something *exists*, you mostly stop worrying about it.

However, the guarantee is of no HORIZONTAL wind; there could still be an updraft or a downdraft. And one could argue that the movement of wind on the real world is not particularly well modeled by a continuous vector field in the first place; ultimately, the atmosphere is highly discontinuous, being made of various teeny molecules with big gaps of vacuum between them.

]]>Don’t comb it. Just look at it.

]]>… I’m really sorry, but someone had to.

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