Warning: topologist in the hizzouse! (That’s how all topologists introduce themselves. Except for Lou Kauffman. He’s a rebel.)
So far, most of the discussion in this thread has been about a really neat consequence of the Borsuk-Ulam theorem. Stated carefully, what we’re all arguing about is the following:
This is a straightforward consequence of the more general theorem.
The connection to temperature, pressure and the Earth is as follows: Let the first (or x) coordinate of f(s) be the barometric pressure at the point s and let the second (or y) coordinate of f(s) be the temperature. Since both of these values vary continuously, the function f is a continuous function from points on the surface of the Earth into the plane. Hence at any time, there must be a spot on Earth where the temperature and pressure are exactly the same as at the antipode to that point.
(I feel compelled to point out at this point that while this is a neat little topological factoid, it isn’t actually true. For the interested reader, I offer this picture as proof.)
This property of spheres can be described in a much more hands-on way, which I think is easier to visualize. If you deflate a basketball and press it flat onto a table, there is always a spot where you can press a pushpin straight down into the table in such a way that (after you patch your ball) when you reinflate the ball the holes are exactly opposite to each other.
Now as to the question of whether you can prove this using the Intermediate Value Theorem, I expect that the answer is no. I’m willing to be pleasantly surprised, of course. For people who are attempting to do so, I would recommend recasting your arguments into the realm of the basketball to help see if they work. Sometimes it can be helpful to have a more concrete example in mind.
In the case of the basketball model of our result, the horizontal axis represents one value (temperature, say) and the vertical axis represents the other.
Of course, there’s also the question of what it means, mathematically, to be able to prove something using a particular theorem. Does that mean using just that theorem, and not any of the other postulates of the field one’s working in? Surely not. But if you have the postulates, then you can prove any theorem at all, regardless of how similar it is to the theorem you’re starting with.
Yes, it’s not a very concrete notion, as you rightly point out.
Still, I think we can make somewhat clear the sense of what we want, even if it must be, ultimately, a subjective judgement. Basically, the hope would be to provide a proof such that the general feeling is “Oh, yes, that’s clearly following the rough proof outline we all saw before, where you first employ the IVT to get a path of zeros for one function, then employ the IVT again to get a zero for another function along the path. To the extent that this fully developed proof goes beyond that outline, it just consists of technical, fairly direct filling in of the handwaving/rigor-holes in that outline.”
[Ok, that’s a bit stronger than saying “A proof using only the IVT” (one could imagine “only using the IVT”, but in another way), but this is, I think, what the discussion has basically become about. Though I guess I would also be thrilled to see any proof which rigorously established the result and which could be presumed grokkable by anyone familiar with the IVT, without requiring much further mathematical sophistication (no invocation of theorems or machinery in topology not already deeply familiar to laymathematicians). I know; subjective judgement calls up the wazoo. C’est la vie.]
Eh, it all boils down to this: You all know what I mean.
I’m pretty much useless on the math, but I did 3D modeling and animation for many years, and many 3D modeling systems use topological representations. The groundbreaking Symbolics S-Geometry system represented objects as an n-dimensional “tree”. I never worked in the system, but I got a demo of it, and here’s what I recall:
A cube is not just 6 square faces, joined at the corners (12 triangles in most systems), but a “tree”, with each corner consisting of a 3-way “branch” that links to 8 other branches.
An advantage of this representation was being able to quickly and easily bevel the corners of the cube by moving along each branch and branching again. In this way, a cube could be smoothed into a sphere with enough iterations. But the major advantage was that it was very easy to move back to the root of the “tree” and produce a more simple representation to move around in animation, only dealing with the more complicated version when working on the model or rendering.
By comparison, the systems I worked on - Digital Arts DGS, Lightwave, and 3DS - all used meshes of triangles composed of vertexes in 3D space. There was no easy way to return to a more simple version of the model. Just a “bounding cube” that wouldn’t show a good approximation of the object.
