(One unrelated minor point: Stranger mentions a requirement of differentiability, but the IVT doesn’t require differentiability, just continuity.)
You arent saying the antipodal points for temp AND pressure are in the SAME location are you?
He certainly is, and it’s true, too, by the Borsuk-Ulam theorem. I just don’t see how to derive it from the Intermediate Value Theorem alone, because of the gap I pointed out above.
Ahh, if there is a perfect correlation between temp and pressure I can see this. Not sure this is exactly true in the Earths weather system. Yes, I am aware of the old PV=nRT thingy.
Not saying I can prove this mathematically, but I can “intuitively see it”.
No, you don’t need any correlation between temperature and pressure for the result Chronos mentioned; the idea is, as long as temperature and pressure are both continuously varying, without requiring any kind of connection between them, we are still guaranteed that there are antipodal points which simultaneously have equal pressure and have equal temperature. Sure, if they’re exactly correlated, this reduces to simply getting them to have the same temperature, which is easier, but even if they’re not related in any particular way at all, you can still prove that it happens [as long as, as stated, they vary continuously].
Apologies to the OP for the whole extended hijack (but perhaps it can help illustrate what simple topology actually is).
I learned the Intermediate Value Theorem a month ago, and I’ve only just now realized some of the real-world applications of it.
I am tired and its past my bedtime, but I just drew a 2 D ? case where this aint gonna happen. Of course I could be in a sleepy fog at this point.
Good night all.
Damn you guys. I need to be asleep already !
I see it now. The equal temp antipodal points must form a ring that encircles the earth. The same goes for pressure. No matter how you distort or rotate those rings in relation to the earth or each other, they will have to cross each other at a minimum of two points. Where they cross is the answer.
I’ll probably now spend all night dreaming and fretting about this stuff.
As I’ve tried to explain several times, we use a topological representation of a road map for many uses in my work. It’s much easier for the computer to handle, and once the underlying topological structure is built into the format, it can answer standard questions about connectivity and adjacency automatically. And those questions come up constantly when processing map data.
I saw that, and I was very tempted to ask you to go into more detail about exactly what you do and how it works, but I didn’t want to be pushy-- sometimes one isn’t in the mood to post long, technical explanations on the web. If you’d like to tell me more about how it works, though, I’d love to hear it.
No, I think I can see it. (I definitely can’t prove it, but I can see it.) That is pretty cool. 
We do have a Doper named Topologist.
The second question is already addressed. For the first part:
Take a 2-d shape, such as a circle. It’s expressed by an equation, of the forum x2 + y2 = 1. The formula for a specific ellipse (oval) is (x/a)**2 + (y/b)**2 = 1.
Now, you can define a function that transforms the circle into an oval. Think of “time” t as a piece of the real-number line, the interval [0,1] as the domain. So you can define a continuous function f(t) so that f(0) = [formual for circle] and f(1) = [formula for ellipse]
Then setting f(t) = [x/((a-1)*t+1)]**2 + [y/((b-1)**t+1]**2.
Then, when t = 0, you have the equation of a circle; when t = 1, you have the equation of the ellipse, and the function f(t) is continuous over the interval [0,1]. So f(t) represents a continuous transformation of circle into ellipse.
That’s the pretty much easy one, but it gives you the idea.
My favourite application of topology is the demonstration of the (perhaps) surprising result that you can write a program that searches over an infinite set in finite time (a couple of seconds).
If anyone is interested, here is how I proved it to myself.
Imagine it is coldest at the south pole and warmest at the north pole. Now matter where you go in the east west direction (longitude) the temperature is the same if you stay at the same latitude (north south).
To find antipodal point pairs, we have to draw a line through the center of the earth, and where the temps are equal at both ends, you have a pair. With a little thought, its clear from that process and the earth’s temp as described above that the earths equator meets those conditions all the way around the earth.
So, you don’t have a antipodal pair. Its more like an antipodal continous ring. There is nothing special about the fact that in this case it turned out the be the equator. If we shift the described temp distribution around so that it is not perfectly aligned north/south, the antipodal ring will not be aligned with the equator but it will still exist.
So, we have this temp ring. We do the same sorta thing with pressure and we get another ring around the earth. With a few moments thought, its obvious that at the **very least **those two rings HAVE to cross at two points. There is your answer. And of course, at the opposite extreme they could also overlap each other perfectly.
But wait you say, what if the temp distribution is not anything nice like you describe above. This is harder to grok, but if you start with the distribution as describe above and slowly warp it to what you want, you will NOT break that perfectly circular ring. You will warp that ring into some other shape, but it will still be a loop/ring of some sort. The same goes for pressure. And no matter what shape those two loops end up as, they still have to cross at at least two points if they encircle the earth.
I could still be missing something here though.
Now THATS what we call a DO loop !
[rimshot]
I avoided all topology in grad school (math), because it looked complicated and useless, and so it’s only natural that I now use topology in my work. That work is database storage of large geographic data sets (as in** suranyi’s** electronic mapmaking business). I’m still not sure how that relates to the pure mathematical topology I so deftly avoided in school but they both arise from the same discipline, I think.
The way I have always described topology (in the database/mapping sense) is as a study of connectivity. So, within a given set of connected features (say a road network) we know certain things about how the number of intersections, the number of line segments and the number of “faces” interact. These are the 0, 1 and 2 dimensional elements of a 2D topology. I can say each line segment has two sides with one “face” on each side, for example, and I can describe a given face by the line sgements that make it up. This is non-rigorous applied topology.
So the reason a donut and a coffee cup are the “same” (in a 3 dimensional topology) is because they both have one outer ring and a one hole. That is about as simple as a topology gets.
An application of Topology (well, Algebraic Topology):
Suppose you have 100 computer processes that need to use a printer and there are 99 printers. (In general, n processes need to pick among n-1 things.) So one of them needs to be left out. This is easy to do with fancy memory ops or with processes waiting. But supposed no waiting is allowed (the unfortunate “out” process just gets told no) and the only shared memory operations allowed are read and write. (Fancier operations include read-modify-write or memory swap for example.)
The case for 2 processes is easy to show it can’t be done. But for more, it was a well known open problem for many years. It was finally solved and the people who did it (Herlihy and Shavit/Saks and Zaharoglou in two different groups) were given the 2004 Gödel Prize prize for it.
From the citation:
“The discovery of the topological nature of distributed computing provides a new perspective on the area and represents one of the most striking examples, possibly in all of applied mathematics, of the use of topological structures to quantify natural computational phenomena.”
In a nutshell: The initial state of affairs can be represented by a “surface” with a bunch of holes. The final state must not have any holes. But the operations allowed can’t remove holes. I once attended a talk by Herlihy on this and understood it for the remainder of the day. But then …
I don’t want to get into a lot of details, but it works like this:
At the bottom level, a map can be represented as a set of nodes, edges, and faces, each of which carries information on it. (For example, an edge can represent a road, but at the same time it can represent a county boundary, and the side of a park.)
Here is the topological information built into this bottom level structure: Edges are bounded by nodes, and faces are bounded by edges. There is a inverse relationship as well: nodes are cobounded by edges, and edges are cobounded by faces.
We have functions that can extract the boundary or coboundary of any one of these features. We also have functions that reset all the boundary and coboundary information whenever we change anything in the map, for instance if we add a new edge (e.g. a new road).
This way, in our work, if we need to write a program that (let’s say) finds all the roads that join ramps that lead to Interstate highways, we don’t have to figure out what roads are connected to each other – the information is readily available from the topology of the network. Similarly, if we want to find all parks bordered by a road that extends past the park boundary. Or any of thousands of other questions.
Not true, actually… unless you drill a hole through the side of the donut into the hole in the middle (remember, our nasal passages connect to our mouths, so we’ve got openings in 3 places per the doughnut’s two).
However, I would very much like to have a doughnut now.