by Jonathan Kujawa

As we know from the Law of Small Numbers, coincidences happen. Indeed, Ramsey's Theorem tells us they are downright unavoidable. Unfortunately, not all can be happy coincidences. In the first two weeks of July we lost three remarkable women of mathematics: Maryam Mirzakhani, Marina Ratner, and Marjorie Rice.

The most famous was Maryam Mirzakhani. She passed away on July 14th. This was reported in the New York Times, the Economist, and across the internet (including, of course, here at 3QD). Her widespread fame was in large part due to the fact that she was both the first woman and the first Iranianian to earn the Fields Medal. As we discussed at the time, Mirzakhani worked in geometry. More specifically, she worked with moduli spaces: these are geometric spaces where each individual point is itself a rich geometric object.

That is, imagine you are interested in studying geometric objects of a certain type (say, circles). Rather than study them one by one, you could study them as a collection. If you say two circles are "close" to each other when their radii are close in size, then you have a way of measuring distances in the space of circles and this lets you unleash 2000 years of geometric tools on the problem of circles. Of course, Mirzakhani studied moduli spaces of much more complicated gadgets than circles, but the principle remains the same.

Loyal 3QD readers saw this approach in action a few months ago. Thanks to the work of Canterella, Needham, Shonkweiler, and Stewart, we know it is possible to match the points on the sphere with triangles in a way where nearby points on the sphere match up to nearby triangles in Triangle Space. If we wanted to be fancy (say, at a geometers' cocktail party), we could have said that they showed that the moduli space of triangles can be identified with the sphere.

Their primary motivation was to give an answer to one of Lewis Carroll's bedtime problems [1]. But thinking geometrically also gives us entirely new insights. A big one is that with geometry we have the ability to talk about the shortest path between two points. But the shortest path depends on the ambient geometry. As we all know, the shortest distance between two points in the plane is given by a straight line. However, on a sphere, the shortest path is given by a "great circle".

When I flew from San Francisco to Taipei this summer, Alaska was on the way:

Shortest path from SF to Taipei.

In a general, these shortest paths are called geodesics. As you might guess, as the geometric space gets more complicated, the geodesics can act in unexpected ways. For example, we all know that in the plane the triangle made by the straight lines connecting them (aka the geodesics connecting them) will have angles adding up to 180º. On the sphere, you can use the equator and a couple of lines of longitude to make a triangle with angles adding up to as much more than 180º. At the other extreme, if you have a saddle shaped hyperbolic space, you can draw triangles whose angles add up to less than 180º. But once you are over the strangeness, you can study geodesics in your favorite geometric space.

As a real world example, we would like to understand how a protein folds. This difficult problem is fundamental in modern biology. Well, imagine you have made a moduli space of all possible configurations of your favorite protein. Then a journey through this space is nothing more and nothing less than a movie of the protein folding or unfolding. In particular, the geodesics in this space are the shortest paths from one configuration to another. Since nature likes to be efficient, it is reasonable to expect that when a protein folds it will do so as directly as possible and, hence, follow a geodesic. Understand the geodesics of Protein Space and you'll understand how your protein is likely to fold.

You can also use geodesics to understand the nature of the geometry you are studying. For example, on the plane if you start at any point and head off in a straight line in any direction, you go off forever in a straight line. You never return and never even travel near most of the other points in the plane. On the other hand, on a sphere, if you start walking and follow a great circle (for example, the equator), you will eventually come back to your starting point and start retracing your steps.

Geodesics on a doughnut [2].

Things become really interesting as the geometry gets more complicated. Say your world is shaped like a doughnut. In certain directions, your journey will simply make a circle and you repeat the circle over and over. For example, if you follow the blue path I've drawn. In other directions, you will loop around and around forever. The red path is the beginning of such a journey. You will never close the loop. in fact, your path will be so complicated that you will eventually pass arbitrarily close to every point on the doughnut.

On July 7th we lost Marina Ratner. Like Mirzakhani, she also was a geometer. Ratner's most famous theorem describes these long walks along geodesics in any homogenous space. A homogeneous space is a geometry which is extremely symmetric. So much so that the geometry looks identical at every point. Circles, spheres, and doughnuts are all homogeneous spaces, while squares, triangles, and cubes are not [3]. Homogeneous spaces frequently appear in math and physics (for example, physicists usually assume that any point of empty space is like any other and, hence, that empty space is homogeneous).

Ratner's Theorem proves that long walks in a homogeneous space are an either/or situation just like for the doughnut. Either they repeat over and over, or they go on forever and eventually get arbitrarily close to every point [4]. In particular, this shows that even though there are infinitely many, these long walks can be put into a short list of easily described families. Ratner's Theorem is a widely used tool in the study of homogeneous spaces and has had a huge impact on our understanding of their geometry.

So important was Ratner's work, mathematicians have long dreamt of establishing similar results for non-homogeneous spaces. Indeed, one of Mirzakhani's most acclaimed results was her work with Eskin, Mohammadi, and others to obtain Ratner's type theorems for certain moduli spaces. So surprising and powerful was their result that it is sometimes referred to as the Magic Wand Theorem.

One example of how the Magic Wand Theorem can be used is in the study of billiards on polygon shaped tables. An active area of research is to understand the long walks of a billiard ball: what are the possible paths if you hit a ball and let it go forever? We discussed this problem a little bit here at 3QD. If you are an eccentric billiard player who likes a challenge, you might build yourself a billiard table with an unusual polygonal shape. On such tables, it is a difficult problem to say which trajectories will eventually repeat themselves, which will go on forever, and to describe which parts of the table are traveled by the ball during its journey. While it might sound silly, billiard ball trajectories have close connections to a wide variety of mathematics and it is an active area of research.

In the best case scenario, your table will have the same either/or dichotomy of the doughnut: trajectories either repeat themselves, or go on forever and eventually go arbitrarily close to every point on the table. Just like Ratner's theorem for homogeneous spaces. Unfortunately, since they have corners we know polygons aren't homogeneous spaces. Ratner's theorem can't help us here!

In fact, such optimal billiard tables are incredibly difficult to find. Right now we know of only eight families of triangular tables which are optimal! In a recent application of the technology of the Magic Wand Theorem, Eskin, McMullen, Mukamel, and Wright discovered two new families of quadrilaterals which are optimal. These are the first new optimal quadrilaterals in over a decade.

From Quanta Magazine.

There is a great article in Quanta if you'd like to read more about their work. The folks at Quanta put out fantastic articles. I cannot recommend them highly enough!

Sadly, polygons bring us to our third mathematician. On July 2nd we lost Marjorie Rice. In 1975 she was a housewife in California who happened to read a column by Martin Gardner in Scientific American magazine [5]. In it, he explained the problem of tiling the plane with polygons. Imagine you have an infinite supply of tiles, all of the same polygonal shape. The challenge is to tile an infinite floor in all directions without leaving gaps. For tiles shaped like triangles and rectangles, it is easy to see that you can always tile the infinite floor. We also know that there are exactly three convex hexagons which tile the plane and there are no convex polygons with seven or more sides which do the job [6].

Things become vastly more interesting once you have a convex pentagonal tile. We've known for a long time that some pentagonal tiles definitely can't tile the plane. In 1918 the first pentagonal tile was found. At the time of Gardner's column, folks knew of eight different pentagonal tilings, but it was an open question if there were more and to find them. Rice had a natural talent and curiosity for mathematics, despite never having gone to college, and was captivated by the problem of finding pentagonal tilings. As she later wrote,

I became fascinated by the subject and wanted to understand what made each type [of pentagon] unique. Lacking a mathematical background, I developed my own notation system and in a few months discovered a new type.”

In the end, Rice discovered no less than four new pentagonal tilings of the plane! One thing I love is that mathematics is one of those rare fields where anyone can add to the frontiers of knowledge. With cleverness, a homemaker can do as much as an army of professional mathematicians.

By 1985 we knew of fourteen pentagonal tilings, but then the search stalled. For the next thirty years people searched and searched, but only met failure. No doubt the less optimistic (or pragmatic!) decided that there were only fourteen pentagonal tilings and moved on with their lives.

But losing hope is not a proof it can't be done. With the endless optimism of treasure hunters, a few continued the search. In 2015 lightning struck! Mann, McLoud, and Von Derau found a fifteenth pentagonal tiling! But before you rush to your desk to find your fame and glory in the sixteenth tiling, I should say that just this April, Michael Rao used computers to complete an exhaustive search of all possible pentagonal tilings [7]. The verdict? The fifteen known tiles form the complete list. Sadly, Rice suffered from Alzheimer's in the last years of her life. She never knew we were able to complete the list, or that she alone found nearly a quarter of all that was possible.

One of Rice's pentagonal tilings [8].

[1] I say "an answer" rather than "the answer" because this is a case where different reasonable interpretations of the question can lead to different answers. See the discussion at the end of the 3QD article for a sampling of other reasonable solutions. Carroll himself gave multiple, different answers.

[2] From Wolfram Demonstrations. They have a free app here if you'd like to play with geodesics on the doughnut.

[3] Because in contrast to circles, spheres, and doughnuts, these have corners and the geometry of the corners is sharply different than at other points.

[4] Not surprisingly, Ratner actually proved a much stronger and more general set of results. For a taste of her more general setup, let me just mention that for her a long walk which goes on infinitely long doesn't necessarily go arbitrarily close to every point. But the points it does approach themselves form a smaller homogeneous space. Just knowing this still tells you a lot about the geometry.

[5] We talked about the marvelous Martin Gardner here at 3QD.

[6] A convex polygon is one in which every corner is less than 180º. If a polygon is not convex, then the problem becomes vastly more difficult.

[7] You can read Rao's paper at his webpage.

[8] Image from Quanta.

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