I was surfing the other night after a long day of working on the grant and ran across this announcement (hat tip infosthetics) of a new kind of city map, one you don't have to fold but can crumple up. It's made of Tyvek, a soft but durable waterproof material you can crease anywhere you want or just jam it into your pocket, any old way. They aren't available just yet but will be soon for London, Paris, New York, Tokyo and Berlin. I'm not sure what the market for city maps will be when everyone is carrying around smartphones with GPS, but it put me in mind of something completely different, a mathematical workhorse called Brouwer's Fixed Point Theorem.
Here it is in one of its simpler forms (courtesy MathWorld):
If g is a continuous function g(x) is an element of [a,b] for all x elements of [a,b], then g has a fixed point in [a,b].
What this has to do with crumpled maps in a moment, but first here's an amusing illustration one of the surprising consequences of this powerful theorem. From Jim Loy:
Question #1: I climbed a mountain, following a trail, in six hours (noon to 6 PM). I camped on top overnight. Then at noon the next day, I started descending. The descent was easier, and I made much better time. After an hour, I noticed that my compass was missing, and I turned around and ascended a short distance, where I found my compass. I sat on a rock to admire the view. Then I descended the rest of the way. The entire descent took four hours (noon to 4 PM). I thought that I remembered that there was a place on the trail where I was at the same place at the same time on both days. Can you tell if I was right?
Answer #1: Yes, there must be one "fixed point" on my trip. Here is a graph showing my height on the mountain versus time.
Regardless of the changes in rate of travel, it should be obvious that the two paths must cross at one point, at least.
(Jim Loy)
Back to crumpled maps. Here's a slightly different example than Loy's that uses a two dimensional version of Brouwer's Fixed Point Theorem (there are other fixed point theorem's besides Brouwer's, but this one is the most famous).
Suppose I am driving around the city and have a city map covering the same area I'm driving in and I do something that in fact I would never do: crumple up the map and toss it out the window (I am very persnickety about littering but this is a thought experiment). When the balled up map comes to rest somewhere on the side of the road, the Fixed Point Theorem says that there is a point on the map that is directly and exactly above the point in the city it is representing. Crumpling doesn't matter nor does the random nature of the toss. Wherever it comes to rest, there is a fixed point where the city and the map point agree.
For some people this is obvious, but for many more it is surprising and almost counter intuitive. Whichever camp you're in, it's true. At least Brouwer's Fixed Point Theorem says it's true.
Here are a couple of other examples, the first related to the crumpled map but making it more plausible. Consider an airport terminal map. Airport terminals can be complicated and confusing but thanks to the Fixed Point Theorem they can mark the place on the map that says "You are here." That's the place on the map where the map is located. More surprising is this. Spread sand completely over the top of a card table and then jiggle the table so that the sand moves around but doesn't fall off (maybe there is a lip around the edge). The Fixed Point Theorem guarantees that there is one grain of sand that doesn't move.
Or, for those of you who like a Martini before dinner, consider this:
In three dimensions the consequence of the Brouwer fixed point theorem is that no matter how much you stir or shake a cocktail in a glass some point in the liquid will remain in the exact same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, and that the liquid after stirring or shaking is contained within the space originally taken up by it. (Wikipedia)
These last two are quite counter-intuitive but both are consequences of what we might now call The Crumpled Map Theorem. Sorry Professor Brouwer.
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Off topic (or maybe Brouwer's theorem offers an answer). If I invest in Brazilian eucalyptus plantations, do I also get the girl in the thong who opens the advertisement?
Just make sure you don't get tears in the map when you are crumpling it. And be sure there are no discontinuities in the map. As in this one: http://www.arkansasgoodsams.com/usa-map.gif
@2: It actually doesn't matter that there are several connected components, as in your typical map of the USA. The theorem applies to the connected component you are standing in, so to speak.
@Revere:
"a mathematical workhorse called Brouwer's Fixed Point Theorem."
I'm not sure what you mean by workhorse but that's not a hard theorem in math. Even among the Fixed Point theorems, that one is pretty easy for many reasons. The main reason is that g: [a,b] -> [a,b] and both a and b belong to the Reals. Also, you did not define the concept "fixed point" prior to using it. Check out Banach's Fixed Point Theorem for a harder Fixed Point theorem. Basically put, Brouwer's Theorem is one on Euclidean spaces but mathematics has moved to more interesting, non-Euclidean spaces a while ago. Brouwer's is rarely taught in pure math classes today since you can deduce it from other, more complex, theorems.
Marc: A workhorse is something that does a lot of hard work. The mean value theorem is a workhorse but it isn't hard to prove. Even I know how to prove the FPT and have used it. It does duty in many, many places. I used it in work in ODEs but it's useful for a lot of things. It's a workhorse.
@Revere: Ah ok. Didn't know what you meant by workhorse. Thanks for the info. But also, math textbooks in general, especially Mathematical Analysis textbooks, define the concept of Fixed Point before presenting a theorem using it. This is important to do because depending how "fixed point" is defined affects the theorem and different FPTs use different fixed point definitions. The definition of Fixed Point varies when used for Brouwer FPT, Banach FPT, Kakutani FPT or Nielsen FPT (maybe there are also other FPTs that don't come to mind right now).
Marc: Yes, you are correct. That's why I was careful (most of the time) to refer to it as Bropwer's FPT. I used to call it the Contraction Mapping Theorem.
@Revere: Contraction Mapping Theorem usually refers to Banach FPT, but anyways. I'm interested though what kind of math do you use in epidemiology? You mentionned ODEs, do you also use Numerical Analysis? Analysis of Variance? I was thinking of doing some research there as well as in climatology. While I know what they use in climatology, I don't know about epidemiology.
@Marc: Yup, good remark about using the right fixed point for a specific FPT. Btw, I also saw your comments over at the RI blog where you tore apart some imbecile who was trying to convince others that vaccines cause autism. You said you're doing a joint degree in math and comp sci. I am as well. Few universities offer the joint. Where are you from? I'm curious. Btw, you don't need to answer if you don't want to. I'm just curious since it's kind of a rare program.
@Alex: Hey, Alex. I'm doing a joint honours in math and comp sci at McGill University. Yeah, the joint seems like a rare program. In most universities, ppl either get the major in math and a minor in comp sci or the opposite. And don't worry, I don't mind answering. The "imbecile" I was correcting over at RI is some guy called JB Handley who tried to defend an anti-vaxxer crank called Andrew Wakefield. He did some shitty study in 1998, published it in Lancet and convinced many people that MMR vaccines cause problems. Long story, you can check Orac's other post to learn all about it. Out of curiosity, where are YOU doing your joint?
@Marc: NO FUCKING WAY! I'm in the same program at the same univeristy! What classes are you taking? Maybe we're in some of the same?
@Alex: Alex, the ALEX, from Number Theory?/? Is this rly you?
@Marc: ROFLMAO. Of all the blogs in all the world... This is retarded. Get on MSN now. It's easier to talk there.
lol @ above
Marc: As an epidemiologist I teach ODEs for modeling of infectious disease (see this series) and of course I use the usual toolbox of statistical techniques (logistic regression, ANOVA, etc.) but for my research I am using methods from discrete math and computer science (Galois lattices (Formal Concept Analysis) and similar data mining techniques) and am especially interested in the analogy between the logic of questions in epidemiology and quantum logic (both give rise to non distributive lattices, i.e., non-Boolean logics). As for Banach vs. Brouwer, my recollection is a bit hazy (haven't looked at this in years and am too busy to go back and review it now) but I vaguely recall that Brouwer is a specialization of Banach in a less general metric space, but I may be mistaken.
@Revere: Very interesting. I've been having an interest in non-Boolean logics for a while. I was always more into applied math while being in pure math but didn't know exactly what those in applied math were doing. Brouwer FPT was generalized into Kakutani FPT sometime during World War II. Kakutani FPT belongs to the applied math field of game theory. GT is used in economics, biology, comp sci, etc. Basically, this shows the divergence between pure math and applied math. Banach brought up his FPT years before Kakutani and Kakutani knew about it. But, Kakutani's purpose was to find something more general than Brouwer but at the same time, maintain applicability (Banach FPT is not very applicable in practice).
@Marc: "I was always more into applied math while being in pure math but didn't know exactly what those in applied math were doing."
>> No he wasn't. That's a lie. Until very recently he was a good pure mathematician. He was corrupted by these evil applied mathematicians working in climatology. I tried bringing him back into the light but it seems like he's gone for good. =(
I know it as Satz vom Igel
hedgehog-theorem: you can't [kämmen] the hedgehog,
at least one quill will be vertical