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Ways of Knowing

The World According to Sound

Season 2, Episode 5

Abstract Pattern Recognition … or, Mathematics

Sam Harnett: In any mathematics class you attend, irrespective of the topic, you will likely find yourself spending the majority of your time performing the identical task: mastering how to solve issues that someone else has already resolved ages ago.

Jayadev Athreya: Each challenge you tackle, up to, essentially through higher education, is one where you can be certain that someone already possesses the solution.

SH: Jayadev Athreya, a professor of Mathematics and the comparative history of ideas at the University of Washington.

JA: It would be akin to, for example, if the only activity we engaged in during English classes was merely repeating material that has already been presented. Never having the opportunity to compose an essay. Never being granted a chance to articulate your own ideas in a creative manner. Or if in an art class, you exclusively worked from paint by numbers. Or in a music class, you never had permission to perform a piece until you had spent 10 or 20 years playing scales.

SH: These instances are derived from an essay on math education by Paul Lockhart. His argument was that foundational skills are essential in mathematics, akin to their significance in art and music. But creativity holds equal importance. One must learn to tackle problems in unconventional, innovative manners. This is how mathematicians dedicate their lives, and excelling at it requires practice. Practice that Jayadev believes should commence significantly earlier in math education. In his classes, Jayadev occasionally assigns students challenges that he himself does not know how to solve or that remain unsolved by anyone.

JA: To be inventive, a certain level of proficiency must first be established. However, what Lockhart and many mathematicians contend is that learners must be given a glimpse of the imaginative and the unknown, a brief insight into the notion that they can introduce something novel to inspire them and cultivate that proficiency.

SH: He doesn’t anticipate students to resolve the problems. It’s about liberating them to think more broadly, to explore unorthodox avenues for solutions.

JA: Mathematicians are genuinely comfortable with uncertainty, feeling muddled about concepts, and experiencing moments of confusion. For a mathematician, engaging in mathematical research isn’t about seeking a singular correct answer. It’s about discovering the patterns that underpin, well, the world around us and also the realms we construct within mathematics.

[instrumental music plays]

SH: There’s a prevalent notion that math is the epitome of rational endeavor, even the antithesis of creativity. It’s entirely focused on precision and computation, excelling at adhering to rules. Again, this is how the majority of individuals spend their time in math classes. However, this isn’t the perception mathematicians like Jayadev maintain.

JA: So, one description of mathematics that I find appealing — although it’s not flawless, no definition truly is — is that mathematics is the language of recognizing abstract patterns.

SH: Mathematicians seek patterns across whatever they study, be it algebra, topology, or number theory, and endeavor to articulate those patterns in a graceful and insightful manner. In doing so, they often draw upon another concept that, like pattern recognition, may not seem particularly mathematical: metaphors.

JA: Often, the way to solve such a challenge is to ask, ‘What new patterns can we discern? Are there more profound underlying patterns beyond those we’ve already observed?’ It’s about correlating with another area of math or physics. And stating, ‘Hey, these entities appear to align similarly to how these other entities align.’

SH: Reflecting on significant discoveries throughout the history of mathematics, there are numerous instances where insights were gained through making comparisons or employing metaphors. For instance, calculus is partially founded on the principle of treating a curve as if it were composed of an infinite number of minuscule straight line segments. This analogy between two dissimilar entities — a line and a curve — unveiled a completely new method of interpretation and computation.

JA: I believe this perspective is an excellent way to conceptually approach mathematics. It involves creating these comparisons or metaphors between concepts by examining their structures. Thus, it’s metaphors at a fundamentally structural — at a profoundly structural level.

[instrumental music ends]

SH: To individuals who claim they are incapable of performing math, Jayadev would illustrate that we are consistently engaged in recognizing patterns and utilizing metaphors.

JA: Everyone is identifying patterns continuously, right? This is a fundamental aspect of our humanity. We endeavor to create and recognize patterns.

SH: Numerous individuals who could excel at mathematics are dissuaded from it by math classes that merely present the subject as a matter of memorization and repetition. Tedious. Restricting. Often punitive for those who tend to think more abstractly. Early in the educational process, there’s a propensity to categorize children into those who thrive in the arts and humanities and those who excel in the sciences and mathematics. In various respects, mathematics shares more characteristics with the humanities than it does with the sciences, especially regarding how challenges are approached. In mathematics, there isn’t the type of testing crucial to the scientific method.

JA: If you observe us, we’re often seated with sheets of paper, notebooks, laptops, and perhaps some chalkboards. We don’t require million-dollar labs, and frequently we’re engaging in thought experiments. We’re striving to identify these patterns and narrate engaging stories and develop compelling narratives about these patterns.

SH: And akin to the humanities, aesthetics play a vital role in mathematics. It’s not solely about resolving challenges, but uncovering beautiful or elegant solutions.

JA: The descriptors mathematicians employ concerning research outcomes frequently include “beautiful” or “elegant.” Or occasionally they’ll express, you know, I accomplished this proof, but I’m not entirely satisfied with how it appeared. I reached a conclusion, but I’m not fond of the method I employed to arrive there, so I continue searching for a more graceful route. Thus, aesthetics hold significant importance to us.

SH: The belief that math is entirely separated from the humanities and creativity does not solely deter individuals who don’t fit a particular mold. It also hinders research and innovations by dissuading methodologies that deviate from an established notion of what mathematics is.

JA: One of the major challenges we must confront as a field — one that some humanities disciplines have advanced further on — is reconciling the influence of power and the practice of mathematics: who gets to narrate their stories, who earns recognition as a mathematician, and what is deemed mathematics.

[instrumental music plays]

SH: Mathematical breakthroughs can emerge from various individuals employing diverse methodologies. A remarkable illustration is the recent identification of the “Einstein Tile.” Over decades, mathematicians

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I have been on the lookout for a form capable of creating an endlessly non-repeating design. This singular form could be employed to tile a restroom that extended infinitely, and would never recreate the same design. Numerous mathematicians attempted and failed to discover this shape. Some speculated that it may not be real. However, in 2022, the conundrum was resolved. An individual named David Smith developed a 13-sided figure that, from certain perspectives, resembled a silly hat. Smith was not a professional mathematician but an enthusiast. The 64-year-old had recently retired from his position as a printing technician, and during his leisure time, he enjoyed playing with shapes, cutting them out of paper and experimenting with a geometry software. That exploration led to the revelation of the renowned “Einstein Tile,” which had evaded mathematicians for numerous years.

[instrumental music fades]

SH: Jayadev is a strong advocate for experimenting with pieces of paper. This was part of his methodology for a recent breakthrough of his.

JA: The dilemma I tackled with my peers, one way we framed it is we dubbed it the anti-social jogger program. Essentially, if you inhabit one of these spatial forms with corners, like a cube, and you commence at one of the corners — t,. There exists a house at each corner, let’s say — . You begin at one of the corners and wish to go jogging. However, you’re irritable in the morning, so you prefer not to encounter anyone else. Additionally, you can’t really think clearly, thus you want to maintain a straight trajectory and return home. If you’re on a sphere, achieving this is quite simple. Regardless of the direction you take, as long as you go straight, you will return home.

SH: There are no corners to navigate through, so you simply run straight and you’ll find yourself back where you commenced. Jayadev and his associates aimed to ascertain whether it was feasible on a dodecahedron, which possesses 12 pentagonal facets.

JA: Regarding the dodecahedron, the query was, in fact, open.

SH: No one had succeeded in demonstrating whether it was feasible or not. Given its multitude of faces and corners, the dodecahedron presents a complex shape to traverse. Jayadev and his team attempted a different methodology towards a shape —– reducing it from three dimensions to two.

JA: Our approach was to essentially flatten everything.

SH: Utilizing sheets of paper, they tried various methods of cutting open and unfolding the dodecahedron. As they played around, they recalled a different sector of mathematics.

JA: What we realized was actually tied to something all of us had previously engaged with, akin to playing Pac-Mman.

[Pac-Mman noises play]

JA: If you exit the right side of the screen, you reappear on the left. If you exit the top, you emerge at the bottom.

SH: It’s as if the left and right of the square are interconnected, as well as the top and bottom.

[Pac-Mman noises fade]

JA: If you visualize the screen similar to a piece of paper, it’s like if the left and right sides are glued together, you create a tube. And if you adhere the top and bottom, it resembles a donut. That’s referred to as a torus. If you apply this to more intricate shapes, you generate more complex surfaces. Interestingly, by establishing a connection to these types of surfaces, we could tackle our original question.

SH: Jayadev and his colleagues applied this side-gluing concept to their flattened dodecahedron. This enabled them to generate a search query in a computational geometry program, which processed all the various possible pathways. The inquiry revealed that the answer is affirmative —, it’s indeed possible to be an anti-social jogger on a dodecahedron. There isn’t merely a single running route that meets the criteria, but at least 31.

JA: This represented a fusion of abstract mathematics, geometric principles, a method known as unfolding, and an extensive, detailed computer search. Thus, it combined several distinct mathematical elements. I doubt we could have resolved this problem a century ago. It was posed a century back by a duo of German mathematicians. However, we now possess both the mathematical insights and computational resources to accomplish it. It was tremendously enjoyable since it unified a multitude of sophisticated concepts, and ultimately, the theorem —– we could simply illustrate something on a sheet of paper.

[instrumental music plays]

SH: Mathematics, much like the humanities, necessitates learning to identify patterns and subsequently compose a compelling narrative surrounding them. There exists creativity both in how one seeks patterns and in how one articulates and communicates them. It’s an endeavor that frequently requires mathematicians to leverage the power of analogy and metaphor.

SH: Here are five references that will aid you in exploring abstract pattern recognition in mathematics as a means of understanding.

“A Mathematician’s Lament” by Paul Lockhart

SH: A groundbreaking critique of the manner in which mathematics is taught and potential alternatives.

“Mathematics as Medicine” by Edward Doolittle

SH: In this essay, Doolittle shares his mathematical journey as a Mohawk Indian and explores the relationship between Indigenous perspectives and contemporary mathematics.

“Weapons of Math Destruction,” by Cathy O’Neil

SH: A publication discussing how big data exacerbates inequality and jeopardizes democracy… which highlights the risks when we fail to critically evaluate our approaches to challenges, particularly 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 doctoral thesis, Harron fused mathematics with narrative in remarkably innovative ways.

“Mathematicians Report New Discovery About the Dodecahedron”

SH: An article in Quanta Magazine regarding the research conducted by Jayadev Athreya and his colleagues concerning the dodecahedron.

SH: Ways of Knowing is a production of The World According to Sound. This season focuses on diverse interpretive and analytical techniques in the humanities. It was created in partnership with the University of Washington and its College of Arts & Sciences. Music contributed by Ketsa, Nuisance, and our collaborators, Matmos. The World According to Sound is produced by Chris Hoff and Sam Harnett.

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