stoutfellow | Reorganizing

I've been stymied, for the last few days, by the first part of Taxonomy III. The division into three sections still seems right to me. The opening section defines the general concept of "cyclic transform", a procedure for turning an n-gon into another n-gon in a "nice" way. The next section discusses the interaction of cyclic transforms with translation: if you have an n-gon P and apply a cyclic transform to it, producing Q, what happens if you shift P a particular distance in a particular direction? In general, Q also moves in that direction, but not necessarily the same distance. This section, then, defines "vector transforms", which do not move Q, and "weighted transforms", which move Q in exactly the same way that P moved. Weighted transforms are geometrically more appealing than general cyclic transforms. The third section is more complex. An n-gon is a list of vertices, with a specific starting point and in a specific order. The key question here is, what happens if you reverse the order of the vertices of P? If the order of the vertices of Q is likewise reversed, but with the same starting point, the transform is "central"; if the order reverses but has a new starting point, it's "quasi-central"; and if the order reverses, with the same starting point, but Q is also rotated 180 degrees around a certain point, then it's "anticentral".

The sticking point is the opening section. In moving for greater abstraction, I made the classic algebraist's error: I failed to tie the algebraic analysis to the geometry. This morning, I realized that that was the problem, and saw how to rectify it. The algebra had me making a particular arbitrary choice; bringing the geometry in early shows that that choice is not only not arbitrary, it's required. Once I reframed it in those terms, everything else fell smoothly into place.

I was trained as an algebraist; I work as a geometer. Keeping my balance between the two isn't always easy, but it's always a great relief when I find the fulcrum point.