Abstracter and Abstracter
Jun. 18th, 2019 02:29 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe new stuff - group algebras, generalized polygons, and whatnot - works. It's clean, it actually simplifies the stuff in Taxonomy (once again, an arbitrary choice is revealed as natural), and all of the methods and concepts I've developed for the Taxonomy papers carry over (except sometimes some of them don't; it's predictable when they don't, and what happens then is also very neat).
I will continue writing the Taxonomy papers as intended, but I'm also going to start working out and writing up the new stuff. In theory I could scrap the old line and shift immediately to the new, but the old line has more concrete geometric interest and has already met with some acceptance, so...
I've already worked out the simplest new example, V-gons, where V is the Klein 4-group. Its elements are pairs of bits - (0,0), (0,1), (1,0), (1,1) - and they can be added mod 2 in each component. So (1,0) + (0,1) = (1,1), and (1,0) + (1,1) = (0,1). A V-gon is a set of four points, each labeled with one of the elements of V. There are nine first-order simple classes of V-gons. ("Simple" is a term of art, which I'm not going to define.) They come in three sets of three. First set: the line segments v(0,0)v(0,1) and v(1,0)v(1,1) bisect each other (or take either of the other pairs of "opposite sides"). Second set: those two line segments have the same length (triple as before). Third set: those two line segments are parallel (and again).
That one was easy to work out. The next two involve sets of eight points, and will require some serious symbol crunching, i.e., office time. Both of them look intriguing, for different reasons.
It's always more fun to find new stuff than to write up old. Must. Resist. Temptation. To. Neglect. Things....
I will continue writing the Taxonomy papers as intended, but I'm also going to start working out and writing up the new stuff. In theory I could scrap the old line and shift immediately to the new, but the old line has more concrete geometric interest and has already met with some acceptance, so...
I've already worked out the simplest new example, V-gons, where V is the Klein 4-group. Its elements are pairs of bits - (0,0), (0,1), (1,0), (1,1) - and they can be added mod 2 in each component. So (1,0) + (0,1) = (1,1), and (1,0) + (1,1) = (0,1). A V-gon is a set of four points, each labeled with one of the elements of V. There are nine first-order simple classes of V-gons. ("Simple" is a term of art, which I'm not going to define.) They come in three sets of three. First set: the line segments v(0,0)v(0,1) and v(1,0)v(1,1) bisect each other (or take either of the other pairs of "opposite sides"). Second set: those two line segments have the same length (triple as before). Third set: those two line segments are parallel (and again).
That one was easy to work out. The next two involve sets of eight points, and will require some serious symbol crunching, i.e., office time. Both of them look intriguing, for different reasons.
It's always more fun to find new stuff than to write up old. Must. Resist. Temptation. To. Neglect. Things....
