Personal Thoughts on Math Research

I publish math research once in a blue moon, and always with co-authors. I've just finished some new research. There are a few things I like about it. (1) The result was surprising, and the work was challenging, but not too difficult. (2) It is a very concrete result in elementary geometry, that I can explain to most anyone. I think the result is nifty. (3) It was a family affair, as my co-authors are my brother Marshall and his son, Michael. It was a nice division of labor, with me doing the geometry part and Marshall and Michael doing the calculus part. Michael's TeX expertise came in handy, as well. (4) It is the first paper that I have contributed to that depended on mathematical software for its solution.

See http://arxiv.org/abs/0704.2716 for the paper, titled "Constructing a quadrilateral inside another one". Be sure to look at version 3, which is much improved over the earlier versions. It's only 9 pages, and a pretty easy read, as math papers go. We are at work on a new version, which should improve the exposition and simplify the proof of the main theorem a bit, but I think the current version is actually publishable quality. We'll be submitting the new version to The Mathematical Gazette.

This paper is an example of old-fashioned mathematics done with modern tools. There is nothing here that couldn't have been done in the 18th century, but without Geometer's Sketchpad I would never have come across the problem or have been able to obtain experimental evidence for our conjecture, now a theorem. And without Maple (or the reincarnation of Euler) we couldn't have done the calculations involved.

Technical production notes: Marshall wrote up the result using Scientific Workplace. Since I don't have a copy, I sent him my corrections as Word documents (using the MathType math editor), and produced wmf format graphics with Sketchpad. It would have really simplified our collaboration if we had the same software. What are other people using for collaborations?