Go Figure

GO FIGURE: “Strange Loops,” Dancing Braids, and the Power of Curiosity

by Susan Sechrist

“Go Figure” is a new regular feature at Bloom that highlights and celebrates the interdependence and integration of math and literature, and that will “chip away at the cult of youth that surrounds mathematical and scientific thinking.”  Read the inaugural feature here.

1.

My interest in mathematics and literature started with Don Quixote.

A profoundly creative professor at Skidmore College gave a lecture on how the characters, plot, and narrative style of the Western world’s first novel resembled the structure of a revolutionary mathematical system yet to come—the Cartesian coordinate system.

On its surface, Don Quixote is an action-adventure—complete with scatological humor and sitcom misunderstandings. But, underneath the homage to the 16th century pastoral is the hidden machinery of the modern novel. Just as the Cartesian coordinate system related abstract algebraic equations to geometry, Cervantes juxtaposed storytelling with story structure—this intersection of narrative with narration forged a new relationship between author and reader, reader and text. Cervantes did for literary truth what Descartes would do for mathematical truth 30 years later using the same tools—concepts of zero and infinity.*

I’ve been looking for equally compelling literary examples of hidden numeracy ever since, asking people to think of a novel or story that they consider mathematical, geometric, or computational in some way.

This summer I had the distinct pleasure to meet Dr. Douglas Hofstadter—mathematician, physicist, and writer of the Pulitzer Prize-winning Gödel, Escher, Bach: An Eternal Golden Braid. Gödel, Escher, Bach (or, GEB) is an ambitious exploration of how “animate beings can come out of inanimate matter.” Hofstadter uses mathematics, music, art, and language to explore how meaningful patterns can emerge from systems of otherwise meaningless symbols. These meaningful patterns—what Hofstadter calls “strange loops”—give rise to consciousness and contribute to our concept of the self.

GEB features deeply philosophical and delightfully dense chapters about music and computers, figure and ground in art, coded messages, number theory, genetics, and artificial intelligence. Sandwiched between these chapters are dialogues between two characters, Achilles and the Tortoise, borrowed from Lewis Carroll (who borrowed them from Zeno’s famous paradox). Achilles and the Tortoise converse about the concepts presented in GEB, bouncing language around the ideas until they become simultaneously clarified and more complex. In one dialogue, Achilles introduces the Tortoise to the nature of meaning in haiku:

“Achilles: A haiku is a Japanese seventeen-syllable poem – or minipoem, rather, which is evocative in the same way, perhaps, as a fragrant rose petal is, or a lily pond in a light drizzle. It generally consists of groups of five, then seven, then five syllables.

Tortoise: Such compressed poems with seventeen syllables can’t have much meaning…

Achilles: Meaning lies as much in the mind of the reader as in the haiku…”

Careful readers notice, of course, that the last exchanges between Achilles and the Tortoise are both haikus—a clever, recursive nod to the ideas Hofstadter presents in the chapter that precedes this dialogue, a chapter about repetition, nesting, and self-reference inside all kinds of systems, mathematical and literary alike. Hofstadter’s dynamic spiraling of critical analysis and creative narrative doesn’t just propose to unite literacy and numeracy; it uses the power of curiosity to braid them tightly together.

When I spoke with Dr. Hofstadter, I asked him, with his unique perspective on cognition, language, and mathematics, could he recommend novels or stories that capture, codify, and clarify mathematical ideas? He thought about it and remembered one story he’d read as a boy. He went to his bookshelf and pulled out a yellowing science fiction anthology from August 1957. But he quickly pivoted to talking about other nonfiction authors that I should read, and I put aside the anthology to make notes.

2.

A month after my meeting with Dr. Hofstadter, I attended the Bridges Conference, an annual gathering of mathematicians, educators, artists, and writers. The conference is a celebration of the integrated mind in every sense: mathematicians present their artwork—from elaborate textiles to photographs of light in motion to sculptures of impossible curves—and artists talk about how experimenting with algorithms opens up new avenues in their art. Threaded through the plenaries and exhibitions are presentations and workshops on how to use these new expressions of mathematical beauty to teach math to students who are more distracted than ever, and who are told by their peers and even their math-phobic parents that they won’t need math in their future jobs as media moguls, models, or marketers. Teaching this difficult subject is a huge challenge: I overheard a conversation between two teachers talking about how educators have tried to make math relevant to students through real life applications, and how that approach has failed. One commented that while you don’t need to be an artist to appreciate art, or a musician to love music, our culture seems to insist that only mathematicians do math.

The Bridges Conference focuses heavily on the connections between math and visual art, but there are participants who represent other disciplines, including music, architecture, and literature. However, of the 100 papers that made it into the proceedings, only half a dozen of them had to do with mathematics and creative fiction: one explored an opera written about female French mathematician Émilie du Châtelet, and the other five were about poetry. One of these five stood out, Canadian poet Alice Major’s exploration of “numbers with personality.”

Alice Major opened her presentation with a personal story about her sister, who thinks about numbers as characters with very specific personalities. People with this rare type of synesthesia, called Ordinal Linguistic Personification (OLP), have detailed, even novelistic, perceptions about the nature of numbers: five is “female, a motherly figure who is ‘funny by accident,’” while one is “a responsible father who is nice but a bit tired.” Major posited that personifying numbers isn’t a useless or arbitrary exercise, a quirk of a uniquely wired brain: she showed how the Mayans did something similar when they created head variants, elaborate glyphs depicting the important numbers in their calendaring system. For the Mayans, the number nine was depicted by the face of a young man with a patch of jaguar skin on his cheek. In their spoken language, the number nine and the word jaguar sound similar, making the head variant glyph more than just a ceremonial decoration in the calendar: it is metaphor and code, giving number both literary and numerical power.

Numbers are not just “symbols of quantity,” says Major. “They are words, and as words they are massively connected in our brains to systems of perception and emotion, judgment and social relations.” Numbers, by nature, create strange loops that enable both counting and recounting—calculation and narration. Major ended her paper with three poems. Her poem “Four” shows how a number is more than just an abstract symbol that describes a set of objects: it’s a linguistic force that shapes reality. “Four is the god of corners, stocky and oblong as a bible. He is worshipped at the intersection of streets…”

That night, at cocktail hour—the great integrating moment of all conferences—I stood with four women: a mathematician, a physicist, and two math educators. We talked about how they each arrived at mathematics as a personal and professional destination. Their starting points varied from poetry to linguistics to anthropology, and their suggested interdisciplinary ways to teach math to reluctant students spanned just as broadly. But none of them had a novel or short story that was foundational to their understanding of or curiosity about math. I had yet to find someone who had had a Don Quixote moment.

3.

The next day, I participated in a workshop in “whole body learning techniques,” a method of teaching that helps students understand mathematical concepts through physical movement. After learning how to make our own braids and twisted rope by hand, the 16 of us in the workshop went outside and stood in a circle. Each of us was given one end of a colored ribbon attached to a fixture hanging above us. We were broken into eight pairs; we faced each other, and, to the sound of the workshop leader playing the squeeze box, chanted the following instructions as we started moving, one person going clockwise, the other counterclockwise: “pass on the left, pass on the right, pass on the left, pass on the right…” We sang, badly, to the tune of “Buffalo Gals (Won’t You Come Out Tonight).” After ten minutes of dancing and lots of good-natured giggling, we did indeed have a strong, multi-colored braided cord about a foot-and-a-half long woven by our movements.

I couldn’t help but think of the 16 participants as components of one of Dr. Hofstadter’s “strange loops”—we were the meaningless symbols that, through pattern, gave rise to a meaningful system. We took pictures of our creation, and we talked about it at dinner that night, how simple binary decisions (step forward, step left, step forward, step right) can yield something complex and functional and beautiful. Dancing the braid helped us better understand its geometry, and talking about the experience felt a lot like telling a story, an exercise in both counting and recounting.

4.

A couple of months after meeting Dr. Hofstadter, I received an email from him asking for my mailing address. He wanted to send me a hard copy of the science fiction short story he had read as a boy. A few days later I received the dark photocopy of the original story from that fragile and yellowing anthology: Les Cole’s “Tripod.”

“Tripod” is composed of three interlocking stories organized into two sets, each featuring a header that references the most human of strange loops—the categorization of time: “Yesterday,” “Tomorrow,” “Today.” The peculiar order of each set is the same, and each header tracks a particular strand in the intertwining braid: “Yesterday I” is about a group of aliens exploring an unknown planet and being chased by non-reasoning monsters; “Tomorrow I” features a group of humans landing on Mars and discovering that there was an intelligent civilization there, but it is mysteriously gone; “Today I” introduces us to John Travis, a man obsessed with changing the flow of time.

The second half of the story is structured with the same headers: “Yesterday II,” “Tomorrow II,” and “Today II.” In this set, the three story lines share the same artifact—a movie camera on a tripod. In “Yesterday II,” we learn that the aliens are exploring Earth during the Jurassic period and are being chased by dinosaurs. They find the camera and tripod and grab it up, returning with it to Mars and putting it in a museum. In “Tomorrow II,” the humans exploring Mars discover the aliens’ abandoned museum and the Earth-made camera. “Today II” reveals that John Travis has invented a time machine and sent a camera back to the Jurassic period, but he is perplexed by his inability to retrieve it. The camera is lost to time, but for a reason John Travis couldn’t possibly imagine.

“Tripod” is a simple story, but I understood why Dr. Hofstadter found its mathematical underpinnings compelling. The story highlights the ability of fiction to build complex and counterintuitive forms, how narrative choice can balance what is revealed and what is hidden to weave something surprisingly integrated: the three storylines are the different colored ribbons in the braid, while the reader is the dance, intertwining them to make a form that is larger than the sum of its parts. The reader knows what has happened across the measure of time, from the past to the future, but none of Les Cole’s characters ever will. “Yesterday II,” “Tomorrow II,” and “Today II” all end, rather extraneously, with a key character stating: “I’m afraid this is one mystery that’ll never be solved.”

5.

I found that last self-referential line from “Tripod” annoyingly prophetic. The threads of this essay continue to be very difficult to pull together. I have a personal, nascent idea of the connections and overlaps, but how “strange loops,” embodied mathematical learning, braids, “numbers with personality,” and the Don Quixote epiphany all intertwine is still a mystery to me—a wonderful mystery that I will continue exploring, whether or not it can be solved.

Bloom Post End

* Very special thanks to Professor Grace Burton, whose Don Quixote lecture at Skidmore College successfully integrated my left and right hemispheres. I hope to feature her ideas in a future Go Figure installment.

Thanks to Laura Pierson—a senior at Skidmore College double-majoring in math and English— who helped with the development of this installment.

Susan Sechrist is a freelance technical writer who is striving to better integrate her creative and mathematical sides. She is the Interviews Editor at Bloom and published her first short story, the mathematically-themed A Desirable Middle, both in Bloom and the Journal for Humanistic Mathematics.

Photo credits
Drawbridge, by Piranesi – Wikimedia Commons
Cartesian grid – Wikimedia Commons
Mayan head variant –  Project Gutenberg
Cover of The Magazine of Fantasy and Science Fiction, August 1957 – eBay

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