In the Stacks

The floor of my den/computer room is covered, in part, with stacks of books I don't have shelf space for.

With the abrupt return of warm weather, Buster and Gracie have been searching for new napping spots. The main bathroom is one favorite location; tile on the floor, porcelain in the fixtures - lots of coolth.

For some reason, though, Buster has taken to wrapping himself between and around the stacks of books in the den. It can't be cooler than the bathroom, but I suppose it does have the advantage of proximity to me (and hence early notice of my getting up, possibly to go to the kitchen!).

I can only hope that he refrains from knocking over the stacks. Domino effect, y'know?

Pairings

In my last post about mathematics, I was deliberately vague about my student's and my discoveries. I'm not going to be much more precise here, but I think it's fairest to be a little more clear.

Consider quadrilaterals. Two well-known (among students of Euclidean geometry) classes of quadrilaterals are the "equidiagonal" and "orthodiagonal" classes. In my classification, both are "rho-negative" classes. A quadrilateral is equidiagonal if its diagonals are equal in length; it is orthodiagonal if its diagonals are perpendicular.

If ABCD is a quadrilateral, its "bimedians" are the lines connecting the midpoints of opposite sides - the midpoint of AB to the midpoint of CD, or the midpoint of AD to the midpoint of BC. Here is the interesting phenomenon: a quadrilateral is equidiagonal if and only if its bimedians are perpendicular, and a quadrilateral is orthodiagonal if and only if its bimedians have the same length. In other words, each class has a characterization involving diagonals and one involving bimedians, and the diagonal characterization of each corresponds to the bimedian characterization of the other; and this is the phenomenon I was speaking of.

This sort of relationship happens all the time with first-order (never mind) rho-negative classes; what's unusual here is that it goes both ways. What stymied me before is that it doesn't go both ways most of the time; what pleases me now is that I know why it does go both ways for the four rho-negative classes of hexagons (two pairs), and I know when it will do so for evengons with more than six sides.

Still too vague, I know, but I hope it's a little clearer what I'm talking about.