A Different Kind of Geekery

[stoutfellow]

Usually, I get my mathematical epiphanies in the shower. Today, though, it happened in the kitchen, as I was making a Gouda-and-ham sandwich.

The concept of Varignon duality, which I discovered in connection with "rho-negative" classes of evengons (polygons with an even number of sides) and which describes, for example, the relation between orthodiagonal and equidiagonal quadrilaterals, turns out to be, with a slight tweak, applicable to *all* classes of polygons with any number of sides, even or odd. (Well, almost. It doesn't apply usefully to some simple classes of evengons, such as trapezoids, but those are a small minority.)

This has Implications. Unfortunately, this is the last week of classes, so I can't give it much attention for another week or two (finals, and grading) - but I'll spend a good bit of Christmas break seeing where it leads.

(I still have to write Taxonomy II; this stuff will be in the revamped Taxonomy III.)

Comments

[graydon]

I cannot escape a sense that if I understood this stuff I would write much better sorcerers.

[ndrosen]

Congratulations on your epiphany. Perhaps I’d get that kind of feeling more often if I were an academic, instead of a federal bureaucrat; on the other hand, some of us may not be destined for that kind of greatness.

[11011110]

Do you happen to know if there is more published about Varignon duality anywhere? Because Wikipedia doesn't have much (besides mentioning it in the equidiagonal and orthodiagonal cases) and I would be interested in adding more, but only if it can be properly sourced. I assume the basic concept is that two quads are Varignon dual iff their Varignon parallelograms are dual in the usual sense, but I'm not sure what that implies about the geometric relation between the two Varignon-dual quads, or how the generalization to higher order works.

[stoutfellow]

To the best of my knowledge, the general concept of Varignon duality is original with me. The paper I'm currently working on ("Towards a Taxonomy of Polygons II: Isolated Classes") will discuss it a little; the next paper, "TaToP III: Central Weighted Transforms and Varignon Duality", will go into it in greater detail. So, no, I don't know of any published work on the subject.

Varignon duality, as I conceive it, is a relation between classes of polygons, not between polygons themselves; the class of orthodiagonal quadrilaterals is dual to the class of equidiagonal quadrilaterals. I know of some reasonably simple examples involving classes of hexagons and of octagons, and my recent epiphany suggests that there are many more examples.