Archive for the ‘Mathematics’ Category

The Tricki

There’s a wonderful new wiki (called the Tricki) dedicated to all the clever little tricks used in mathematical proofs. It’s a project started by none other than Timothy Gowers, author of this wonderful compendium.

Differential Forms

I’m always on the lookout for non-rigorous, ‘pictorial’ explanations of abstract mathematical topics. Because ones that are good, and also mostly correct, are difficult to find. And also I am not particularly fond of learning mathematics by plodding through theorems in a book. I find that only with the appropriate picture does a particularly daunting and abstract topic come into focus. And then plodding through the theorems is actually quite easy.

I found this short, but sweet introduction to differential forms, via the Googles. It must be good, because PageRank puts it on the first page.

Topology Apology

In lieu of actually delving into this, I’ll just give a short description and let you go look it up on Wikipedia.

There’s a way of classifying or generating seemingly complicated surfaces using a sort of surface (or shape, I guess) that isn’t really that complicated.

The way you do it is this: Take one vector of the polygon, say the top A vector, and glue it head-to-head and tail-to-tail with the other A vector. Now do the same with the B vector. What you literally get is a kind of weird, oriented, triangular surface. In topology-land, of course, you can continuously deform this weird pseudo-prism into something we all know and love, namely a sphere. Or, in the fancy terms, you’re utilizing a homotopy equivalence between the two surfaces. The same thing can be done for arbitrary surfaces, as well, provided they are also closed.

For example, here’s a torus:

This is a neat way to visualize certain surfaces, especially when they cannot be embedded in three-dimensional space without self-intersections (such as the Klein bottle, or the real projective plane).

Tensors

See here for a very friendly introduction to tensors (~25 pages).

See here and then here for a very friendly introduction to the Levi-Civita tensor (~3 pages).

Homotopy Classes and the Fundamental Group

If you imagine a hole on a plane—say, a circular one with radius \epsilon > 0—as being an infinitely tall column rising above the plane, you can determine how many holes exist on that surface in the following natural way:

Suppose that you take a string, tie it into a loop, and lay it on the surface. If you can pull the loop in a free manner to any location on the surface without breaking it, then it must not loop around any of these columns. Suppose that every conceivable loop on this surface has the same property. Then that surface must logically not have any of these columns—i.e., it must not have any holes.

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The Dangers of Visualizing Abstract Things

To follow on to my last post, a big part of visualizing abstract mathematics is, ironically, accepting that you can’t completely visualize it. Sometimes you have to settle with what you know. For instance, you can imagine points in L^2, which is the vector space of square-integrable functions. You just imagine E^2 or E^3 (the Euclidean spaces humans are most familiar with) and pretend that each point in that space represents a function. This isn’t technically correct, since the inner product for L^2 is quite different from the Euclidean dot product. But it is a useful mnemonic.

Maybe I’m just a visual person, but if I don’t have a picture.. any picture.. I just can’t understand anything. It’s much better, in my opinion, to have a picture that demonstrates or represents a single property of some abstract thing—even if all the other properties don’t really match—than to have no picture whatsoever.

Let’s enter rant mode for a second, though. Because I’m somewhat skeptical of the idea that these kinds of informal visualizations should be widely popularized à la Discovery/History Channel. I absolutely hate it when science popularization does not sufficiently caveat a technically incorrect analogy (which is not terribly difficult to do, I don’t think). It’s especially insidious coming out of textbooks. There’s a level of understanding required to fully grasp certain analogies and ways of thinking about things. Without that knowledge, you’re only getting a hint of a hint of what’s going on, at best. At worst, you get quantum-wave-energy-photonic-crystal-tao-healing.

Now, ‘technically incorrect’ could mean a lot of things as you attempt to make such a caveat. I imagine it’s real easy to be a snit about what constitutes ‘correct’, especially as you get to more and more complicated topics with very rigorous definitions and so forth. There’s a popular way of making such caveats that’s always satisfied me: “It’s correct to a first approximation”.

At the end of the day, asking the History Channel to do this sort of thing is probably a waste of time. Between all the science shows are the Jesus-UFO-Conspiracy shows featuring Ralph the Overly Dramatic Narrator. And you know they ain’t gettin’ no caveat. And even if they did, the only one apropos would be “I’M IN UR TV, MAKIN U DUMB”.

Which should literally pop up every time you turn to the History Channel, in place of the H. And then there should be a little animation of a guy pointing and laughing on an infinite loop, you mindless cretin.

Shame on you, forever.

Visualizing Abstract Things

I’ve heard variants of this joke for a long time, and despite being vague, it’s still kind of funny. And also completely true.

A mathematician and his best friend, an engineer, attend a public lecture on geometry in thirteen-dimensional space.
“How did you like it?” the mathematician wants to know after the talk.
“My head’s spinning”, the engineer confesses. “How can you develop any intuition for thirteen-dimensional space?”

“Well, it’s not even difficult. All I do is visualize the situation in arbitrary N-dimensional space and then set N = 13.”

(Volker Runde)

Of course, one wonders why the public lecture is on thirteen-dimensional space in the first place if a generalization to N dimensions already exists. Details!