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Diana Davis’s Beautiful Pentagons – Scientific American Blog Network

Swarthmore Faculty mathematician Diana Davis, whom I’m fortunate sufficient to name a pal, is multitalented. Her mixed love of math and running led to an earlier appearance in this blog, the place she investigated the query of whether or not, in case you run a marathon at, say, a 7:00 minute per mile tempo, should you have got run any mile in precisely 7 minutes? Not too long ago, she has been making jewellery, clothes, coasters, and different objects primarily based on her analysis involving billiard trajectories in pentagons. I requested her about her analysis and the gorgeous issues she has constructed from it. You’ll find her creations in particular person at conferences together with the Joint Arithmetic Conferences or explore her website.

Diana Davis stands in front of a sign that says "Math is Beautiful" and behind a table with pentagonal jewelry and toys on it
Diana Davis peddles her pentagonal wares on the Joint Arithmetic Conferences in January. Credit score: Diana Davis

First, introduce your self. The place are you from, the place did you go to high school, and the place do you’re employed now? Aside from math, what are you captivated with?

I am initially from New Hampshire. I went to Williams Faculty for undergrad and to Brown College for my Ph.D. I am at the moment a visiting assistant professor at Swarthmore Faculty. Aside from math, I am captivated with good instructing, equity in all issues, and long-distance working.

What’s your analysis space, and what are among the math questions you want to consider?

My analysis is in dynamical methods, primarily in billiards, which is the examine of a ball (or level) bouncing round in a form (normally a polygon, and infrequently a pentagon). Folks have understood billiards on the sq. for over 100 years, and that is about it: as people, we do not perceive billiards on every other form, besides possibly a few particular triangles (equilateral, isosceles proper, that kind of factor). So I wish to know what occurs with billiards on tables which are different polygonal shapes. 

Diana Davis peddles her pentagonal wares on the Joint Arithmetic Conferences in January 2020. Credit score: Diana Davis

The query that drove me for a few years was: suppose you have got a ball on an everyday pentagon billiard desk, and also you select a route to hit your ball, and you already know that the ball’s path goes to be periodic: the ball will bounce round for some time, after which return to the place it began and repeat. What number of occasions will it bounce earlier than it repeats — how are you going to get this data from the route? For the sq., if the slope of the trail (assuming the perimeters of the desk are horizontal and vertical) is rational, then you already know the trail will likely be periodic, and if the slope is p/q in lowest phrases, then the ball will bounce 2(p+q) occasions earlier than it repeats. It is an attractive consequence, and I wished to generalize it to the common pentagon.

What’s so nice about pentagons? Did you’re keen on them earlier than they obtained to be such a giant a part of your work, or did your love for pentagons develop as you researched them extra?

5 has at all times been my favourite quantity. This isn’t why I studied the pentagon. The primary cause to check it’s that it is, in some sense, the “next-simplest” common polygon after the sq. — in spite of everything, a sq. has four sides and a pentagon has 5. Arguably, although, the common octagon is somewhat easier, as a result of it is only a sq. with the corners reduce off. Certainly, John Smillie and Corinna Ulcigrai had achieved some work on the common octagon, and my Ph.D. advisor instructed me to learn their paper concerning the octagon, and see if I may use the identical methods to know the common pentagon. Certainly I may! In order that’s how I obtained began with the pentagon. In that case, I used to be really finding out a floor constructed from two pentagons, which you’ll be able to see in my viral dance video.

Cutting Sequences on the Double Pentagon, explained through dance from Diana Davis on Vimeo.

Then I used to be at a convention in Oberwolfach in spring 2014, and Samuel Lelièvre got here as much as me. He instructed me that he appreciated my video concerning the double pentagon floor, and that we should always work collectively to know billiards on the common pentagon billiard desk. We’ve been working on it ever since.

How did you get the concept to start out making fairly issues primarily based in your work, and what fairly issues do you make proper now?

In summer time 2017, I used to be working in Moon Duchin’s analysis cluster at Tufts College. She allow us to take day trip to get skilled on the gear in Tufts’s Makerspace. Earlier than that, I had no real interest in laser cutters or any sort of Makerspace actions, however after I discovered methods to use them, instantly I wished to strive all types of issues. The primary issues I made have been coasters — plastic pentagons about 5 inches throughout, engraved with periodic billiard trajectories. However though coasters are straightforward to make, there’s actually low demand for coasters. I think about that many individuals have about 5 or ten occasions as many coasters as they really use. 

Illustrations of billiard trajectories on pentagons. Credit score: Diana Davis

At some point, I had the concept to make some earrings, in all probability only for my roommate — I haven’t got pierced ears, myself — and I made a few tester pairs. As quickly as I had that concept, and the take a look at pairs turned out effectively, I spotted I used to be actually onto one thing. You see, nearly everybody wears earrings. I did not notice that once I began this venture — once more, I haven’t got pierced ears — however I’ve discovered that about 90% of (ladies) mathematicians have pierced ears. And, not like with coasters, individuals are at all times completely satisfied to have one other pair of earrings.

The beauty of earrings is that they’re a great dialog starter. I’ve a tough time speaking to new folks. So my new technique is that if there’s somebody I wish to speak to, I stroll as much as them, open my field of earrings, and provide them a pair. The earrings are lovely, so the particular person is usually completely satisfied about this provide, and issues work out nice. What’s extra, folks normally ask follow-up questions, like “what are the patterns?” and “that is your analysis?! What’s your analysis about?” which, to place it calmly, shouldn’t be the response you normally get once you inform the particular person subsequent to you on the airplane that you’re a mathematician. My attain objective is to place stealth arithmetic into widespread tradition. I might love for a traditional retailer to hold my earrings, the way in which that they carry Alex and Ani bracelets and that kind of factor, and folks would purchase them as a result of they’re lovely, and nobody would even know that they’re mathematical. Possibly that might be a rumor that got here out later, within the tabloids. Alternatively, I named my earring firm “Math is gorgeous,” which is stealth promoting for math, as a result of when folks discuss my fledgling firm, I’ve tricked them into saying “math is gorgeous” out loud.

As of late I’m making a few issues. For the earrings, the prettiest factor I’m doing is utilizing translucent coloured plastic, and reducing out the detrimental house, the place the trajectory isn’t. I additionally take sheets of strong wooden and engrave a trajectory on one facet, after which some details about the trajectory, comparable to ‘101 brief trajectory’ on the opposite facet. I give these away earlier than my talks, after which folks ask, “what’s 101? what does ‘brief trajectory’ imply?” and I say, “come to the speak!” Then folks have a cause to concentrate to the varied components of the speak, in order that they’ll perceive the data on the again, and so they additionally get one thing to take dwelling with them. [Readers can check out Davis’s and Lelièvre’s paper about these trajectories here.]

Davis’s wood pentagon pendants engraved with billiard trajectories. Credit score: Diana Davis

Unrelated to my analysis, I’ve been reducing out tiling lizards, primarily based on Escher’s woodcut from 1943. I preserve them on my desk, and this can be very satisfying to place them collectively like a puzzle, as they snap into place with a satisfying click on. My college students love them, and it helps them to have one thing to do with their arms after they speak with me. 

How did you study what you wanted to study to really begin making issues, and the way did you get entry to the correct gear?

I discovered the fundamentals of laser reducing at Tufts’s Makerspace. Then I began working at Swarthmore a number of months later, and I used to be delighted to seek out that there was additionally a laser cutter right here. I did all of my experimenting with supplies and strategies right here at Swarthmore, and as with most issues, I discovered by attempting issues, having failures, and attempting one thing else. 

If you wish to laser reduce issues, it is essential to have entry to another person’s laser cutter, reasonably than proudly owning your individual. A laser cutter is the scale of a chest freezer, prices about $40,000, must have direct venting to outdoors (so it will probably’t be in a basement or an inside room), and might simply begin fires or explode if left unattended. In addition they want frequent upkeep. For these causes, it’s nice to have the ability to use another person’s machine.

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