stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2019-08-23 05:05 pm
stoutfellow) wrote2019-08-23 05:05 pm
Entry tags:
Abstract Away!
There is a small piece of the classification problem that has been niggling at me almost from the beginning. It has to do with "magnitude forms", which are fairly simple first-order forms. Consider, for example, hexagons. The sum of the squares of the sides is a magnitude form. The sum of the squares of the short diagonals - if the hexagon is ABCDEF, I mean AC, BD, CE, DF, EA, FB - is another, and the sum of the squares of the long diagonals AD, BE, CF is yet another.
Unfortunately, these forms are not "invariant", which is a highly desirable property. The invariant magnitude forms for hexagons are... harder to describe, but there's an easy way to write each of the side/diagonal forms, above, in terms of the three invariant forms m1, m2, m3. Reversing that - writing the invariant forms in terms of the side/diagonal forms - has been a nagging problem. I've been able to work the reversal for everything up to hexagons, but a general reversal has eluded me.
Today, while contemplating magnitude forms on G-gons, I figured out the reversal in general for one set of possibilities, for another, for yet another... and it hit me that there was a definite pattern, and an easy one.
I now have a general technique for writing the invariant magnitude forms in terms of the side/diagonal forms, and it's almost the mirror image of the technique that runs the other way. I shoulda suspected...
[The song below is apropos, for I have slain a giant!]
Unfortunately, these forms are not "invariant", which is a highly desirable property. The invariant magnitude forms for hexagons are... harder to describe, but there's an easy way to write each of the side/diagonal forms, above, in terms of the three invariant forms m1, m2, m3. Reversing that - writing the invariant forms in terms of the side/diagonal forms - has been a nagging problem. I've been able to work the reversal for everything up to hexagons, but a general reversal has eluded me.
Today, while contemplating magnitude forms on G-gons, I figured out the reversal in general for one set of possibilities, for another, for yet another... and it hit me that there was a definite pattern, and an easy one.
I now have a general technique for writing the invariant magnitude forms in terms of the side/diagonal forms, and it's almost the mirror image of the technique that runs the other way. I shoulda suspected...
[The song below is apropos, for I have slain a giant!]