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Step Forward
Feb. 22nd, 2012 08:59 amThere is no pleasure like that of figuring something out. (There are greater pleasures, but nothing quite like this one.)
For various reasons, I have been thinking about quadrilaterals, and about a certain parameter - the ratio between the sum of the squares of the diagonals and the square of the distance between their midpoints. In investigating, I found, about a week and a half ago, that there was reason to believe that quadrilaterals for which that ratio is 4-to-1 were interesting. However, I couldn't figure out why they were interesting. (That sounds bizarre, I know; the "reason to believe" was based on algebraic evidence, but the interest itself had to be geometric, and I was having difficulty bridging the gap.) Last week I thought I'd figured it out, at least in part, but some experimentation on Friday revealed that I was mistaken.
Yesterday, while waiting for the bus, I started thinking about it again, from a different viewpoint. I set up a coordinate system and made a calculation; the result was suggestive, but gave only a blurry picture of what was going on. I shifted the coordinate system and redid the calculation: a slightly different result, still suggestive but unsatisfactory. A third set of coordinates suggested itself, but the computations looked more complicated than I could do in my head, and besides, the bus was coming.
When I got to the office, I fired up Mathematica and plugged in the data, using the new set of coordinates. The answer popped up immediately, simple and suggestive. More than suggestive: it implied a nice little theorem. I pulled up some previous calculations and checked them against what I had just seen.
Bingo. I don't know why the theorem is true, but that it is true is clear.
(The real importance is not just the theorem, but what it suggests in a broader context - not just quadrilaterals, but pentagons, hexagons, and general n-gons.)
:does happy dance:
For various reasons, I have been thinking about quadrilaterals, and about a certain parameter - the ratio between the sum of the squares of the diagonals and the square of the distance between their midpoints. In investigating, I found, about a week and a half ago, that there was reason to believe that quadrilaterals for which that ratio is 4-to-1 were interesting. However, I couldn't figure out why they were interesting. (That sounds bizarre, I know; the "reason to believe" was based on algebraic evidence, but the interest itself had to be geometric, and I was having difficulty bridging the gap.) Last week I thought I'd figured it out, at least in part, but some experimentation on Friday revealed that I was mistaken.
Yesterday, while waiting for the bus, I started thinking about it again, from a different viewpoint. I set up a coordinate system and made a calculation; the result was suggestive, but gave only a blurry picture of what was going on. I shifted the coordinate system and redid the calculation: a slightly different result, still suggestive but unsatisfactory. A third set of coordinates suggested itself, but the computations looked more complicated than I could do in my head, and besides, the bus was coming.
When I got to the office, I fired up Mathematica and plugged in the data, using the new set of coordinates. The answer popped up immediately, simple and suggestive. More than suggestive: it implied a nice little theorem. I pulled up some previous calculations and checked them against what I had just seen.
Bingo. I don't know why the theorem is true, but that it is true is clear.
(The real importance is not just the theorem, but what it suggests in a broader context - not just quadrilaterals, but pentagons, hexagons, and general n-gons.)
:does happy dance:
