In the long term, I intend to spend more time on these sorts of projects. Somehow I never seem to find the time.
I plan to write down a lot of ideas about the topology, geometry, and symmetry of knitting. There is some really interesting and surprisingly sophisticated mathematics behind knitting. Al and I wrote a not-quite-finished HTML version of our work on Mobius knitting.
I plan
to finish a paper about symmetries of parking lots. This is kind of
like
the classification of wallpaper symmetries into 17 types.
I gave a talk entitled Quaternions,
octonions, and beyond at the 2007 meeting of the Michigan Section
of the Mathematical Association of America. You can view the
slides by following the link.
I gave a talk entitled Hamiltonian circuits in
Cayley digraphs at the 2007 Michigan Mathematics REU
Conference. You can view the slides by following the link.
Do dogs need calculus?, preprint, 2007 (with M. Bolt).
A recent series of expository articles discusses
an optimization problem about a dog that swims and runs. We show
how to solve the optimization problem without any calculus. The
key technique is completing the square.
A cohomological viewpoint on elementary school arithmetic, Amer. Math. Monthly 109 (2002) 796--805.
This paper starts off with the notion of carrying in the sense of addition by hand of multi-digit numbers. From there, it goes on to develop the first principles of group extensions and group cohomology.
Shortest shoelaces, Math. Mag. 73 (2000) 60--61.
Suppose you've got a shoe with the same number of evenly spaced eyelets on both sides of the tongue. Take a shoelace and lace it through the eyelets in any order such that the lace passes through each eyelet exactly once and such that the lace begins and ends at the top two eyelets. In what order should the lace pass through the eyelets so that the shortest length of lace is used?
Mobius knitting, in Bridges: Mathematical Connections in Art, Music, and Science, ed. R. Sarhangi, 1999, 67--76 (with A. P. Petrofsky).
If you want to knit a seamless Mobius strip, it's not surprising that you have to use a stitch that has a lot of symmetry. It turns out that none of the standard stitches (such as stockinette, garter, or ribbing) has enough symmetry. So we invented a new stitch, and we knitted some examples. In fairness to Al, I'm using the word we very loosely in this paragraph. Basically, Al had all of the important ideas, and I helped him write the paper.
How to kick a field goal, College Math. J. 27 (1996) 267--271.
Suppose that it's third and goal on the three-yard line with five seconds left on the clock. You've got the ball, and you're just two points down. All you need to win the game is a field goal. You should go for the field goal and win the game, right? Or maybe it's a good idea to take an intentional delay of game penalty and back up five yards to give your kicker a better angle on the goal. If you watch football on television, you'll hear commentators make this suggestion all the time. However, a little bit of plane geometry proves them wrong.
Linear algebra on the gridiron, College Math. J. 26 (1995) 358--360.
When I was an undergraduate at the University of California, Berkeley, I helped to write some field show design software for the University of California Marching Band. This article describes some of the linear algebra that we used.