This is one of the things that, at first, sounds quite unbelievable. Then, when you think about the math involved, it becomes quite clear and almost trivial (well, in this case it's not exactly trivial, but in some other cases it is).

Similar (but slightly easier to prove) mathematical "tricks":

- if you take two pieces of paper, crumple one up and put it on the other, there is at least one point on the uncrumpled paper that is directly underneath the corresponding point in the other paper

- If you have a chair (or table) with 4 legs, and it wiggles because the floor is uneven, then you can always find a way to rotate it around the center so that it doesn't wiggle anymore

- there is no way to comb a hairy sphere ;-)

Stanisław Ulam was a great mathematician. Off topic, but I got the movie made from his book Adventures of a Mathematician from my local library and it was excellent.

Some personal history: Ulam was a good friend of my Dad and when I was a teenager (long time ago!) we rented his house for a summer while he was traveling and our family needed to be Bolder Colorado for some work my Dad was doing. I asked my Dad about Ulam last year and he started to laugh a bit and told me that Ulam used to flirt with my Mom, in a respectful way. Anyway, watch the movie Adventures of a Mathematition - excellent!

Cool. That's the principle behind why you can stabilize a wobbly table by turning it -- you will find a position where all four legs are on the ground.

Didn't know it had a name.

Interesting. In practice, what's the distribution of Borsuk-Ulam pairs across the globe? Something other than a Bingham distribution?

Very interesting question, I also asked this and have no idea. I will collect the data i get and plot it one day, but i expect the numerical method we use is not ideal in that strange thing happen when the BU pairs reach the poles.

my conjecture is these are uniformly distributed over a year, but that is a guess. other planets might have different andwers.

> Who made this?

> Colin Cotter (Imperial College, London) and Julius Ross (University of Illinois at Chicago, supported by NSF-DMS 1749447). You can provide feedback, or access the source code here.

I'd really like to know how these two people from different parts of the globe came to collaborate with each other. I know we live in the age of Internet and all but I'd really like to know how such people find each other and collaborate.

I missed out on this kind of collaboration when I was in college. But learning the answer to this question might be helpful to others who might want to do such collaboration but do not know how to get started.

Know each other's work through publications, connect via email (ostensibly to ask for clarification on a detail, or to request a preprint), maybe meet at conferences but not always. I've written a couple of papers with people who I've never met in person (and that's not uncommon).

We were graduate students together sharing an office, and have been friends since.

One of them might have been at the other's institution for a period before moving back or on. That's quite common in academia.

Neat demo. Although given the measurements here are discrete (and locations too, since float/double has finite resolution) the theorem does not actually apply so in theory there might not be any such points. Of course in practice with pressure and temperature only measured to few decimal places you will be able to find such points.

Super interesting. I thought Ulam was famous for Ulam sequences. I didn't expect this!