Pentagon Jewelry by Diana Davis

clip-on and pierced stud earrings, dangly earrings, necklaces -- and more!

Samuel Lelievre and I are studying periodic trajectories on the regular pentagon. In the course of our investigations, we created a program (in Sage, using CoCalc) that draws pictures of these trajectories. They are so beautiful! So I decided to make them into jewelry to share them with the world. I make them on a laser cutter, with plywood or bass wood; see below.

I have given out about 100 pieces of jewelry, at the Joint Mathematics Meetings and at other conferences and talks. Those of you that have it, I hope you are enjoying it! Please let me know if you have any feedback.

Here is an info sheet, written for a general audience, about the mathematics behind the pentagons. I have been updating it as we discover new things about them!

There are countably infinitely many periodic paths on the regular pentagon. Here are the first 500 or so.

The earrings are hypoallergenic: The stud earrings are made out of surgical steel, and the dangly earrings are sterling silver.
Here are the patterns I have been printing for stud and clip-on earrings:

I make dangly earrings that are reversible, with a different pattern on each side. Just switch ears! Below are some of the pattern combinations I have been making.

Please note that the earrings in these pictures of dangly earrings are made with my early (raster) method, and now I make them in a better (vector) method that creates darker, more precise lines, as in the stud earrings above. Updated pictures to follow.

I am also developing a necklace pendant, but I have not finished optimizing my design yet. Early, triple-thick prototypes are shown below.

Here you can see a 24 x 6 sheet of bass wood with all three sizes (stud, dangly and pendant) pentagons, in process on the laser cutter.

Below is a video of the laser cutter engraving a trajectory. At the beginning, it seems "random," and then order emerges from chaos. The laser is drawing the lines in the same order as the path that a billiard ball would take in the pentagon, if it were shot in the direction of one of the lines on this pentagon. Video has sound (background noise of laser cutter and fan).

Diana Davis
Swarthmore College, Department of Mathematics and Statistics, 500 College Avenue, Swarthmore PA 19081