Mind-blowing explainer on fixed points


Originally published at: http://boingboing.net/2016/10/04/mind-blowing-explainer-on-fixe.html


Occasionally I’ll watch a video where someone uses an uncommonly spoken word (in my circles, anyway) like “idempotent”, and pronounces it differently from the way I thought it should be. I rush to look it up, and as often as not find out I’ve been wrong for years. Happily, this was not one of those cases.


This idea (but not its proof) blew my mind back in high school. I remember the illustration was pretty nice: a sheet of paper covered with numbers, bingo-style, with another copy crumpled up and dropped on the first sheet.


Why does he say that Brouwer’s Theorem applies to the coffee in a mug (one point in the coffee will always be back to where it started when you stop stirring), but doesn’t apply if you cut and glue paper?

Can’t stirring coffee be like cutting a gluing paper? Why does it have to be continuous? Can’t you get one point that is completely separated from its original neighbor, say, as your spoon slices between them?


By the way, although there’s a much easier way to think about the answer, can you also use Brouwer’s Theorem on that old puzzle of the monk who wanders up a mountain one day to meditate, and then hurries back down it the next morning, ask we ask whether there must be a point on the mountain that he would have been passing at the exact same time on both mornings?


You should imagine the spoon takes up no space.
Maybe “swirling” would be a better example.


The easier way you’re thinking of relies on the intermediate value theorem, which is equivalent
to the one-dimensional case of Brouwer’s fixed point theorem
(the proof in one direction is on wikipedia).


I think I found the fixed point of annoying.


Comes back together again. Your spoon does not separate it into two distinct pieces.


Yes, and coffee is made of a collection of discrete particles (mostly) and not a uniform medium filling a region diffeomorphic to {(x,y,z)|x^2+y^2+z^2<=1}.

However, it seems to be close enough to that sort of thing. These things are sometimes more robust than their hypotheses let on: I believe that one should be able to prove an approximate version of the Brouwer fixed point theorem for maps which are almost continuous.


Can you really say you have a museum if you only have one exhibit? Wouldn’t that rather be called a lame tourist trap?


My coffee is usually not spherical, no…


Not even when you add milk from a spherical cow?


Only if it’s a frictionless coffee cup.


It’s frictionless coffee with milk from a spherical cow in a massless coffee cup.


On an infinite plane coffee table, naturally.


Diffeomorphically, it is, because it has no holes. I’ve not studied topology, but if my rudimentary understanding is correct, the sharp edges of a cylinder make no difference.


How do you know I don’t drink out of a toroidal cup? Nothing goes better with coffee than donuts.


I know you do, so do I…


Why is it toroidal? Does it not have a bottom?