Breakthrough

[stoutfellow]

:bounce bounce:

As I've mentioned from time to time, my current research interest has to do with parametrizing polygons - coming up with quantities which somehow characterize them, distinguishing different (for various values of the word) polygons from each other. My main focus for the last while has been the classification of quadrilaterals up to affine equivalence. (Two quadrilaterals are "affine equivalent" if you can turn one into the other by applying a matrix and/or a translation.) I solved that problem, in one sense, quite a while ago; I have two parameters which do the job completely. (Two quadrilaterals give the same values for those parameters if and only if they're affine equivalent.) But, though I could compute the parameters for any quadrilateral, I had no clear idea of their meaning. Other lines of research gave tantalizing hints, but nothing I could really get a grip on.

The tide finally came in last week: I came up with solid and interesting geometric interpretations of both parameters. (One measures the failure of the diagonals to bisect each other; the other is determined by, on the one hand, the failure of the diagonals to bisect the quadrilateral, and on the other by the failure of the midlines - the lines connecting the midpoints of opposite sides - to do so.)

Tonight, a new thought struck me, and about forty minutes of computation gave me a second, completely different and equally interesting interpretation. Now I've got a link between two different properties of quadrilaterals.

Throw in those tantalizing hints I mentioned before, and I've got meat enough for my next paper.

:bounce bounce:

Comments

[stoutfellow.livejournal.com]

You've touched on something important, and often unrecognized. For many, perhaps most, mathematicians - and I'm one of them - the point is not utility. For us, mathematics is an art form; the point is the discovery - or invention, depending on your philosophy - of beauty. (The highest praise for a proof is for it to be called "elegant".) It's an austere kind of beauty, perhaps, best analogous to the sweeping lines of a great building, but it's there.

On the other hand, there's a famous cautionary tale. G. H. Hardy, of Oxford, was so adamant on this point that he insisted that the only good mathematics was that which had no application whatsoever. In one of his books - or perhaps in a lecture, but I think a book - he pointed to a particular theorem of number theory (which was his own field) as quintessential mathematics; it was utterly inconceivable, he said, that it would ever find application. This was in the teens or twenties of the last century. Sometime in the 1940s, Claude Shannon found an application of the theorem. (It's useful in cryptography and computer science.)

There are too many stories like that for anyone to be too confident that any given piece of mathematics will never find use. But I suspect, for various reasons, that mine will not find application in my lifetime. I just think it's pretty.

[kattsune.livejournal.com]

I wholeheartedly applaud that answer. Thank you.

[mmegaera.livejournal.com]

Wow. Now *that's* cool. Applied mathematics can be lovely (just look at your average pieced quilt), but pure mathematics as art form is just too nifty for words.

That light bulb over your head really is gleaming nicely.

[selenite.livejournal.com]

I suspect you're not familiar with archangelbeth's Illuminati University (http://www.sjgames.com/gurps/books/IOU/), so let me share a bit from the departmental descriptions:

College of Zen Surrealism:
Department of Inapplicable Mathematics
Since sufficiently advanced mathematics is indistinguishable from surrealism, the "pure math" people have ended up here. ("Applied Mathematics" lives over in WUSE*) In fact, advanced theoretical math is so disconnected from reality that surrealism is concrete in comparison.

There is no market for graduates of this department whatsoever, except for a few who become university mathematics professors.

A well-kept secret at IOU is the fact that the output of some of the departments here actually have practical applications. It is especially important to keep the professors in CZS from discovering this, lest they suffer from the Centipede's Dilemma.

*WUSE: The College of Weird and Unnatural Sciences and Engineering. Specializes in computers, gadgets, and explosions.

[stoutfellow.livejournal.com]

Sounds like fun; I may order a copy one of these days. (I just made an Amazon run a couple of days ago, so it'll have to wait a bit.)