**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.

I gave out about 100 pieces of jewelry at the Joint Mathematics Meetings in San Diego. 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 will need to update it soon, as we have recently discovered more than what is explained there.

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

I am printing pentagons in several standard sizes:

**Tiny size!** So far, I have identified seven trajectories that look good:

These are, clockwise from the top: 100-p (top), 11-p, 30-pp, 1000-p (bottom), 12-p, 101-p; 33-pp (center). If you'd like one of these, you can either tell me by its label (e.g. 100-p) or its location in the picture.

Think of all the things you could superglue a tiny pentagon onto! Below are three possibilities.

**Medium size!** I make these into dangly earrings for pierced ears. Personalization is available!

I put a trajectory on each side, so that they are reversible; just switch ears. You can choose a trajectory for each side. Below are some of my choices.

If you like these choices, you can tell me by their location in this picture.

**Large size!** I make these into necklaces. They are triple-thick with two quarters inside to weigh them down. You can choose a trajectory for each side. This is also a great size for gluing over the branding on your laptop cover, or hanging from your name tag at a conference.

**Custom size!** Anything is possible.

**Choosing your trajectory:**

As above, anything is possible, but for a fixed size of pentagon, only certain trajectories will look good. (In particular, if it looks mostly black on the screen, it will probably look bad.) Here are the first couple hundred options. Please, choose one that looks good. Click on the pentagon you want, and tell me the file name. For example, for this trajectory you would tell me that you like 102-p. If your trajectory is not symmetric and you care about which corner is up, specify that, too. You can also tell me multiple trajectories. There are so many nice ones!

**Learn some mathematics.**

Samuel Lelievre and I are investigating periodic billiard trajectories on the regular pentagon billiard table. In so doing, we made these pictures of periodic trajectories. They are so beautiful that I decided to make them into jewelry to share them with the world. I make them with a laser cutter, with plywood. Here is an info sheet. Check back for more mathematical explanations.

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

ddavis3(at)swarthmore.edu