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

[countrycousin.livejournal.com]

Neat!

[hornedhopper.livejournal.com]

That *new* thought moment where disparate facts suddenly align themselves in a startling but *right* frame is fabulous, innit?

You must be feeling so ready to start - congrats on the inspiration, and good luck with the writing!

[kattsune.livejournal.com]

That's fantastic! That click-y feeling is better than breathing sometimes (which, I think, is why I find that I've either been holding it during the entire revelation or failed to do it right and hyperventilated), but I'm curious... what do you *do* with this information? How will it be used?

I hope this isn't a dumb question. I passed Calc II and enjoyed it, but that was more because I like formulas and equations to find the *right* answer than I ever understood how it related to physical life. I'm an uninspired mathmetician, as you can tell.

[mmegaera.livejournal.com]

Congratulations. I'm not sure whether to be sad or relieved that I understood about one word in four of that explanation, but I am happy for you, anyway.

I got about as far as analytic geometry, and that was thirty years ago [g].

[ndrosen.livejournal.com]

Congratulations! More professional (and professorial) kudos for you.

[filkferengi]

Careful, professor; judging by the fashioning of your phrasing, your [Euclidean] foundation garment is showing. ;)

Congratulations; that's downright nifty!

W, SF, W!

[toraks.livejournal.com]

CONGRATS!! A bit late, but a breakthrough is worth celebrating anytime!!

That sounds really cool!!!