## Equations or Exploding Teapots?

August 16, 2007 — Daniel Lichtblau, Scientific Information Group

This is not something I generally admit, but I have a couple of teenagers. It’snot that I’m embarrassed about having them, or what it says about my age, but rather that they are a bit embarrassed about having me. If you have some of your own, you understand.

My point is this: I am a certified bore, and I have kids to prove it. With such a background it is understandable that I sometimes talk about arcane things like computational math. But (and this is important), even folks who live in that world need something to keep their attention. That is to say, just because a bunch of us are interested in the same topic does not mean we won’t bore one another to tears.

What might this have to do with Demonstrations and technical publishing? (I am boring you already? So sorry.) It goes like this: I give a talk about something computational, perhaps geometric in nature. I try to describe it in words. And I fail miserably.

I can try to draw it–except I can’t draw. Or I can show it in a `Manipulate`.

I will describe a recent scenario. I was speaking about “Implicitization via the Groebner Walk” in the 2007 Applications of Computer Algebra (ACA) conference session on Approximate Algebraic Computation.

In this setting one might come across such exciting phrases as “cone traversal,” “weight vector,” “parametric patch,” and “hybrid symbolic-numeric computation.” Now some of us need a quick run to the restroom when we hear that last one, but for most people the geometric topic of interest right then is finding the shortest path to the exit.

Clearly I need to show pictures *tout de suite*. And if those pictures move, then all the better. Enter the Tempestuous Teapot (click to download the Demonstration and source code).

A classic from computer graphics lore, Newell’s Utah Teapot can be utilized in service of pretty much anything one wants in a computational conference (except, sadly, tea). My purpose was to, well, be the first to do something a bit obscure with it. But showing it counts for more than just showing a bunch of equations and timings.

I tell people that by expanding at the seams one can get a visual idea of why some of the patches are less tractable, for my purposes, than others. That’s all banter. What I really want is to show the thing exploding outward–now that makes for a nice

Demonstration.

I will also mention that that was hard to get right (and required help from in-house graphics guru Yu-Sung Chang). It involved making Bézier surface patches move in just the right way.

And I’m happy to say we did get it right. So people saw the Demonstration and stayed awake at least until the end of the talk.

I get to chalk it up as a victory.

This was not quite a first. I had used a Demonstration once before in service of a technical talk.

This Demonstration citation has now made its way into a conference proceedings article. But, truth be told, the situation was grim indeed: that one was more for me than the audience.

The scenario it depicts is relatively simple. Five points form the vertices of two regular tetrahedra, joined on a common face situated in the horizontal plane. There are exactly six cylinders containing all of those five points (never mind why). Now vary the uppermost point, enforcing that the cylinders also vary so as to always contain all five points. Eventually the cylinders all vanish. (Where? Into complex space, if you must know.)

The math told me what was going on. But I simply could not picture it. Over the years I had folded paper and even glued toothpicks to get at this, all to no avail. The Demonstration has cured me of these ailments (both the lack of comprehension, and the need to attack it with toothpicks). As it seemingly did no harm to the audience, it

clearly qualifies as an overall gain.

Alas, cylinder and teapot Demonstrations notwithstanding, I still (bad pun coming…) spout computational math. And my teens point out that I remain a bore. But at least the Demonstrations allow me to wax less ineloquently than others of similar proclivities.

Not a bad deal, really.