There are many versions of topological fixed-point theoremata. I think what you are talking about here is that if you start with (something homeomorphic to) a non-empty, compact, convex subset of a Euclidean space, and map it continuously to itself, then there will be a fixed point. If the map is not continuous, or the domain is not compact or has holes in it then the argument will not work.
That’s pretty much what I said, a map of the US placed on the US. However I also believe that it works if the map is within the boundaries as well. The way I was exposed to the theorem was take a 10 x 10 grid with the numbers 1 to 100 on it, crumble it up and drop it on a copy of the grid. You are guarantied one number will be on top of itself.
Here’s a really bizarre one for computing pi:
And here’s why it works:
That was fascinating, thank you.
I’d say that by crumpling the map, but then treating it as a flat object in terms of its position above the un-crumpled map, you are effectively changing the scale of the map you are crumpling (not in any sort of consistent way, but you’re shrinking it)
Exactly. The scaling has no need to be consistent. Just don’t tear the crumpled map even a smidgen.
All this is, stretching a point, little more than the 2D topologically sophisticated version of the Intermediate Value Theorem:
Given a continuous real-valued function on a domain containing [x1 … x2] …
if y1 = f(x1) and y2 = f(x2) and (wlog) y1 < y2
then for every y1 < yn < y2 there exists some xn such that yn = f(xn)
Or in plain English: If you start at 2 and end at 5, you have to have gone through 4 (and 3, and 4.785, and …) to get there, no matter how much you zigged and zagged along the way.,
“Crumpling” sounds continuous. Whether the surface remain flat or smooth is not relevant.
Another application of the fixed point theorem is that at any instant there is at least one on earth that has no wind. The “You can’t comb the hair on billiard ball” theorem. If you could, you could find a fixed point free mapping of the sphere to itself by moving each point one cm in the direction the wind is pointing.
Another variation: At any moment, there must exist two points on the earth’s surface that are at the same temperature.
Merely saying it like that sounds pretty weak; applying the Borsuk–Ulam theorem to the surface of the Earth proves the “coincidence” that at any given moment there are two antipodal points with the same temperature and barometric pressure, for example.
Which was my reply when Mangetout said the same thing.
And of note, the Hairy Ball Theorem only applies to spheres of even dimension. It’s easy to find a “combing” for a one-dimensional sphere (that is to say, a circle), and it’s not too hard to do it for a three-dimensional one, either (i.e., one embedded in four-dimensional space).
I don’t know what this means. WHat corresponding point on your table? My table doesn’t have a map on it
I think I need more of an explanation what you mean.
Get a paper map of your town. Put it on your kitchen table. Somewhere on that map is the area that represents your lot. Within that area on the map is a spot on the map (probably a very small one) that presents your kitchen table in your kitchen in your house on your lot in your town.
Somewhere within all the molecules of that spot on the map representing your kitchen table, there’s at least one molecule where the map molecule is sitting exactly on top of the exact table molecule it represents.
thanks…but that seems pretty much standard. Any map of an area I’m in will have a representation of where I’m at. I mean, thats the primary point of maps, isn’t it.
Take any map of an area you are in, of any size and orientation. There is a point on that map that corresponds to exactly where you are standing.
“Where you’re standing” isn’t a point, though; it’s a blob. The theorem says more than that: there is a zero-size point on the map that corresponds to the same point on the landscape that it is above. And it doesn’t matter how crinkled or distorted the map is (as long as you don’t tear it).
That said, your basic intuition is correct. Assuming the map isn’t too distorted, and the scales are similar to real maps, you can find the point with a kind of divide-and-conquer approach, starting with where you’re standing. From there, your infinitely detailed map will picture a blade of grass exactly on top of the real one it corresponds to, and a cell in that blade of grass corresponding to the real one, and so on all the way down.
There must be some other condition for this to be true like the temperature vary continuously. Otherwise set temperature at 20 for every point in the Northern hemisphere and on the equator from 0W <= longitude < 180W and at 10 for every point in the Southern hemisphere and the equator from 0E < longitude <= 180 E
Holy SHIT!!!
If you aren’t impressed, you can just not say anything. The map is just an example; the generalized theorem covers many more situations (as noted).
53 EDO is pretty amazing when it comes to numerical coincidences, if you’re into music theory.
In that vein, there is the coincidence that 210 ≈ 103 , which to this day confuses people buying storage drives, especially when the free space is listed by the operating system using BSD-style abbreviations like “1.1T”, the mingling of binary and decimal notation being extra stupid.