天美影院

Ways of Knowing

The World According to Sound

Season 2, Episode 5

Abstract Pattern Recognition 鈥� or, Mathematics

Sam Harnett: In any math class you take, regardless of the subject, you鈥檒l probably spend most of your time doing the exact same thing: learning how to solve problems that someone else figured out a long time ago.

Jayadev Athreya: Every problem you work on, up to, basically through college, you know that someone knows the answer to it.

SH: Jayadev Athreya, professor of mMath and the comparative history of ideas, at the 天美影院.

JA: It would be like, for instance, if the only thing we did in English classes was only sort of regurgitate stuff that was already done. Never get a chance to write an essay. Never get a chance to express your own thoughts in a creative way. Or if in an art class, you only did paint by numbers. Or you only did music class, you were never allowed to play a piece of music until you played scales for 10, 20 years.

SH: These are examples from an essay about math education by Paul Lockhart. His point was that basics are valuable in math, just as they are in art and music. But so is creativity. You have to learn how to approach a problem in counterintuitive, novel ways. That鈥檚 what mathematicians spend their lives doing and to be good at it takes practice. Practice that Jayadev thinks should start far earlier in math education. So in his courses, Jayadav sometimes gives students problems that he doesn鈥檛 know the solution to or problems that no one has figured out yet.

JA: To be creative, you do have to have some kind of proficiency and build that. But I think what Lockhart and many mathematicians would argue is that you have to allow people a glimpse of the creative and a glimpse of the unknown, a glimpse of the fact that they can contribute something new to motivate them, to build that proficiency.

SH: He doesn鈥檛 expect students to solve the problems. It鈥檚 about freeing them to think more expansively, to search unexpected places for solutions.

JA: Mathematicians are actually very comfortable in a space of not knowing something, about being confused about things, about feeling like they are kind of at a loss. And math for a mathematician, doing math research, isn鈥檛 about, 鈥極h, I want to find the one right answer.鈥� It鈥檚 about trying to find the patterns that underlie, well, the world around us but also the worlds we create in mathematics.

[instrumental music plays]

SH: There is a common perception that math is the ultimate rational endeavor, even the opposite of creativity. It鈥檚 all about precision and calculation, being good at following rules. Again, this is what most people spend their time doing in math classes. That鈥檚 not how mathematicians like Jayadev think of it.

JA: So, one definition of math that I kind of like 鈥� of course it鈥檚 not perfect, no definition is 鈥� is mathematics is the language of abstract pattern recognition.

SH: Mathematicians are searching for patterns in whatever they鈥檙e studying, be it algebra, topology or number theory, and then trying to explain those patterns in an elegant and insightful way. To do that, they often turn to another thing that, like pattern recognition, may not feel very mathematical: metaphors.

JA: And the way you are often able to solve such a problem is to say, 鈥極h, what are the new patterns? Are there some deeper underlying patterns beyond the patterns we鈥檝e observed?鈥� It鈥檚 to try to make a metaphor with some other piece of math or a piece of physics. And say, 鈥楬ey, these things kind of look like they fit together the same way these things fit together.鈥�

SH: If you look back at major discoveries in the history of math, there are many examples where the revelation came from making a comparison or metaphor. Take calculus. It is built in part on the idea of treating a curve as if it was made up of an infinite number of tiny straight line segments. This comparison of two unlike things 鈥� a line and a curve 鈥� opened up a whole new way of understanding and calculating things.

JA: I think this is a really, really excellent way of thinking about mathematics. It鈥檚 a way of making these comparisons or these metaphors between things by observing their structures. So, it鈥檚 metaphors at kind of a structural 鈥� at a deep structural level.

[instrumental music ends]

SH: To people who say they can鈥檛 do math, Jayadev would point out that we are practicing finding patterns and using metaphors all the time.

JA: Everyone is recognizing patterns all the time, all right? This is what we do as humans. We are trying to create and recognize patterns.

SH: So many people who could be really good at math are driven away from it through math classes that present the subject just as memorization and repetition. Dull. Confining. Often punishing for those who tend to think more abstractly. Early on in education, there鈥檚 a tendency to divide kids into those who are good at art and the humanities, and those who are good at science and math. In some ways, math has much more in common with the humanities instead of the sciences, especially in terms of how problems are approached. In math, there isn鈥檛 the kind of testing that is essential to the scientific method.

JA: If you see us, we鈥檙e sitting with pieces of paper and notebooks and laptops and maybe some chalkboards. We don’t need million-dollar labs, and we鈥檙e doing stuff in our heads a lot of the time. We鈥檙e trying to find these patterns and tell compelling stories and develop compelling narratives about these patterns.

SH: And just like in the humanities, aesthetics are a major part of math. It鈥檚 not just about solving problems, but finding beautiful or elegant solutions.

JA: The adjectives that mathematicians will use about pieces of research are 鈥渂eautiful鈥� or 鈥渆legant.鈥� Or sometimes they鈥檒l be like, you know, I proved this but I don鈥檛 feel great about how it looked. I got to the end but I don鈥檛 like how I got there, so I鈥檓 still looking for a better way to get there. So, the aesthetics are something that are really important to us.

SH: The view that math is completely divorced from the humanities and creativity doesn鈥檛 just drive away people who don鈥檛 fit a certain profile. It also slows research and breakthroughs by discouraging approaches that don鈥檛 conform to a preconceived notion of what math is.

JA: One of the big things that we have to reckon with as a profession 鈥� that some of the humanities fields are way ahead of us on 鈥�, is reckoning with the role of power and the practice of mathematics: who gets to tell their stories, who gets to count as a mathematician and what counts as mathematics.

[instrumental music plays]

SH: Mathematical breakthroughs can come from all kinds of people working in all kinds of ways. A great example is the recent discovery of the 鈥淓instein Tile.鈥� For decades, mathematicians have been searching for a shape that could make an infinitely non-repeating pattern. This one shape could be used to tile a bathroom that stretched on forever, and would never repeat the same pattern. Many mathematicians had tried and failed to find this shape. Some believed it doesn鈥檛 exist. But in 2022, the problem was solved. A man named David Smith devised a 13-sided shape that looked from some angles like a goofy hat. Smith was not a professional mathematician, but a hobbyist. The 64-year- old had recently retired from working as a printing technician, and in his spare time, he liked to fool around with shapes, cutting them out of paper and messing around in a geometry computer program. That play led to the discovery of the famed 鈥淓instein Tile鈥� that had been eluding mathematicians for decades.

[instrumental music fades]

SH: Jayadev is a big proponent of messing around with pieces of paper. It was part of his process on a recent breakthrough of his own.

JA: The question that I worked on with my friends, one way we framed it is we called it the anti-social jogger program. Which is, if you live on one of these spaces of shapes with corners, like a cube, a. And you start at one of the corners 鈥� t,. There鈥檚 a house at each of the corners, let鈥檚 say 鈥� . Yyou start at one of the corners and you want to go for a run. But you鈥檙e grumpy in the morning, so you don鈥檛 want to see anyone else. You also, like, you can鈥檛 really think so you want to follow a straight path and come back home. If you鈥檙e on a sphere, it鈥檚 pretty easy to do this. No matter what direction you go in, if you go straight, you鈥檙e going to come back home.

SH: There are no corners to pass through, so you just run straight and you鈥檒l end up back where you started. Jayadev and his colleagues wanted to see if it was possible on a dodecahedron, which has 12 pentagonal faces.

JA: With the dodecahedron, the question was actually open.

SH: No one had been able to prove it was possible or not. With all its faces and corners, the dodecahedron is a complex shape to try and navigate. Jayadev and his colleagues tried to take a different approach to a shape 鈥斺�� to simplify it down from three dimensions to two.

JA: How we thought about it is we kind of flattened everything out.

SH: Using pieces of paper, they experimented with different ways of cutting open and unfolding the dodecahedron. As they fooled around, they were reminded of a different area of math.

JA: What we realized was this was actually connected to something all of us had worked on before, which was something like playing Pac-Mman.

[Pac-Mman noises play]

JA: If you go off the right side of the screen, you come out the left. If you go off the top of the screen, you come up the bottom.

SH: It鈥檚 as if the left and right of the square are connected, as are the top and bottom.

[Pac-Mman noises fade]

JA: If you imagine the screen just like a piece of paper, it鈥檚 like if the left and right de sides are glued, you get a tube. And if you glue the top and bottom, it looks like a doughnutnut. That鈥檚 called a torus. If you do this with more complicated shapes, you get more complicated surfaces. And it turns out, by making a connection to these kinds of surfaces, we were able to solve our original problem.

SH: Jayadev and his colleagues applied this gluing of sides idea to their flattened dodecahedron. That allowed them to devise a search query in a computer geometry program, which crunched through all the different possible pathways. The search showed them the answer is yes 鈥�, it is possible to be an anti-social jogger on a dodecahedron. There isn鈥檛 just one running path that satisfies the conditions, but at least 31.

JA: This was this combination of abstract math, abstract geometry, a procedure called unfolding and then a really serious, deep computer search. So it brought together a bunch of different pieces of math. I don鈥檛 think we would鈥檝e been able to do this problem a 100 years ago. It was posed 100 years ago by a couple of German mathematiciansmathematics. But we literally have both the math and computational technology to do it. It was incredibly fun because it brought together lots of sophisticated ideas, and in the end, the theorem 鈥斺�� we could just draw something on a piece of paper.

[instrumental music plays]

SH: Math, like the humanities, requires one to learn how to identify patterns, and then come up with a compelling story about them. There is creativity both in how one searches for patterns, but also how one explains and communicates about them. It鈥檚 an endeavor that often requires mathematicians to draw on the power of comparison, of metaphor.

SH: Here are five sources that will help you learn more about abstract pattern recognition in mathematics as a way of knowing.

鈥淎 Mathematician’s Lament鈥� by Paul Lockhart

SH: A seminal critique of the way math is taught and the possible alternatives.

鈥淢athematics as Medicine鈥� by Edward Doolittle

SH: In this essay, Doolittle recounts his experience in mathematics as a Mohawk Indian and the relationship between Indigenous thought and contemporary math.

鈥淲eapons of Math Destruction,鈥� by Cathy O’Neil

SH: A book about how big data is increasing inequality and threatening democracy 鈥� which shows the danger when we don鈥檛 think critically about how we approach problems, especially when numbers are involved.

Piper Harron’s Ph.D. thesis on the Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields

SH: In her Ph.D. thesis, Harron combined mathematics and narrative in quite unusual ways.

鈥淢athematicians Report New Discovery About the Dodecahedron鈥�

SH: An article in Quanta Magazine about the work Jayadev Athreya and his colleagues did on the dodecahedron.

SH: Ways of Knowing is a production of The World According to Sound. This season is about the different interpretive interpretative and analytical methods in the humanities. It was made in collaboration with the 天美影院 and its College of Arts & Sciences. Music provided by Ketsa, Nuisance, and our friends, Matmos. The World According to Sound is made by Chris Hoff and Sam Harnett.