Go Figure

GO FIGURE: Overthrown

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.

I am a terrible knitter.

Despite patient lessons from my sister-in-law, Beth, and her 90-year-old mother, Maxine, when we were all together one Christmas holiday, the only thing I can knit are scarves that twirl in on themselves because I never equalize the surface with the proper combination of proper-sized stitches. Maxine also taught me how to crochet, and I managed to make a round doily-like dish cloth of which I was particularly proud. After I left the comfort of their tutelage, however, I could not pick up a crochet hook and a skein of yarn and get even the simplest design started. I had trouble counting the stitches, putting the hook in the right place, keeping track of whether I was working the front or the back of the piece. I ended up with lots of chain stitches that went nowhere, and piles of kinked up yarn that told the tale of my connecting it all together and then tearing it all apart.

Then I learned about Daina Taimina. A mathematician at Cornell University, Dr. Taimina teaches her students about hyperbolic geometry using crocheted models. One feature of her model is that it starts out flat and familiar, like a two-dimensional scarf or doily, but then quickly begins to curve in on itself, creating a sort of excess of surface area. Anyone who has made a kale salad has encountered this hyperbolic excess of surface—the leaves have these beautifully frilly ends that are, apparently, good for you. Hyperbolic geometry is common in nature, especially in the diverse shapes that form the world’s endangered coral reefs (see Dr. Taimina’s collaboration with Margaret Wertheim at the Institute for Figuring to learn more about their crocheted reef project).

It took me a while to get started, but I was emboldened by Dr. Taimina’s concept. Once I had my initial little flat doily-like circle, I followed the instructions for increasing the stitches, essentially adding not just a larger circumference to my doily, but an outer boundary that kept growing so that the piece began curving into a three-dimensional form. I had mastered crochet, at least when it came to making this one hyperbolic form called a pseudosphere:

Photo copyright Margaret Wertheim – the Institute For Figuring (from Wikipedia).

This new skill may not translate into scarves or mittens or sweaters, but it did teach me how to make the yarn curve so that I could attempt more complex forms. Plus, crocheting in this way feels wonderfully decadent, like I’m packing more yarn into the form than is allowed. At some point, the piece just erupts into this luscious, squishy, organic shape that is glorious to handle. I made hyperbolic pseudosphere Christmas ornaments and sent them to everyone, including Maxine.

2.

The concept of hyperbole is, of course, also literary—an overstatement intended to intensify meaning, e.g., Emerson’s “shot heard round the world” or Auden’s “I’ll love you till China and Africa meet.” Hyperbolic writers are those who pack more into the form than it seems capable of containing, only to have a new shape evolve out of the excess that productively overthrows reality. They include magical realists like Jorge Luis Borges, sci-fi celebrities Ursula K. Le Guin and Philip K. Dick, literary nonsense writers Edward Gorey and Lewis Carroll, and speculative fiction masters Margaret Atwood, Neal Stephenson, and China Miéville. While these latter three share skill in creating alternative realities with literary hyperbole, their work also exemplifies another key characteristic of the mathematical form—reflection.

A Quick Primer on the Hyperbola

Image courtesy Weisstein, Eric W. “Conic Section.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/ConicSection.html

The picture above shows the four conic forms, the curves formed when a plane intersects a cone or, in the case of a hyperbola, a set of cones. When the plane is parallel to the base of the cone, the curve is a circle; if the plane is at an angle to the base, it’s an ellipse that varies in eccentricity depending upon the angle of the plane. When the plane is parallel to the edge of the cone, the curve formed is a parabola. When the plane is parallel to the axis of the cone, the curve that results is a hyperbola. In a mathematical equation, a hyperbola always features two identical curves reflected across a boundary.

3.

Neal Stephenson’s novel Anathem is complex, lengthy, and known to spawn many political and philosophical arguments that I’m ill equipped to mediate. But I love this story for its complexity, its bombastic tone, and its appendix of “Calcas,” or mathematics lessons, Stephenson uses to explicate his extraordinary world and help us get inside the heads and hearts of his characters. Where other writers use hyperbole in a more careful, calculated way, Stephenson is a modern painter, using it in bold strokes and huge color blocks, and even painting outside the canvas and onto the frame.

Anathem is set on the planet Arbre, and starts with characters living in an enclave sequestered from the outside world. In this monastic setting, called a “Math,” the devout study mathematics and philosophy and are not allowed to use most modern technology. When one of the brothers of the Math breaks the rules and makes an astounding discovery, the plot explodes into a rollicking adventure that includes alien contact, perilous journeys, and parallel worlds. Near the end of the novel, Stephenson also splits the narrative along parallel tracks, sending the reader along one branch of the story only to curve back around and send her along another branch that features different outcomes. Stephenson makes Anathem about the potential of the multiverse—Arbre turns out to be a higher, parallel plane for Earth—by reflecting the narrative, like a hyperbolic curve, across a boundary to create a credible, co-existing story line.

There is similar parallelism in China Miéville’s The City & the City, where police inspector Tyador Borlú finds himself embroiled in a murder investigation complicated by an unusual jurisdiction problem. Besźel and Ul Qoma are two cities joined by proximity and a mysterious, shared history. Unlike other twin cities, Besźel and Ul Qoma are not divided by a boundary such as a river or a Cold War-era wall. These two cities are topologically the same: they occupy the same space at the same time. They are also somewhat schizophrenic—while each knows of the existence of the other, they are not cooperative personalities. To exist in Ul Qoma, one must navigate around Besźel, one must “unsee” Besźel as it exists all around. People in Besźel must do likewise, where their city streets are “cross-hatched” or altered by infusions from Ul Qoma. Citizens of one city are trained to unsee the elements of the other because these two cities are irretrievably riven. Passing from one to the other without going through proper, complicated diplomatic channels is considered a serious breach and interlopers are severely punished by a shadowy agency that functions outside the governments of either city.

I’ve read this story twice and written about it as part of a conference paper, and I’m still not sure if this story is fantasy or allegory: are Miéville’s cities truly physically sharing the same space and time, or are they built on well-constructed political and psychological boundaries? This uncertainty is what makes the novel hyperbolic in a new sense—the story propagates reflective narratives, co-existing on either side of some subjective threshold in my imagination. The trick in this funhouse is to figure out which image is real, which is the doppelgänger, and whether it matters. The story’s potential realities are superimposed just like Miéville’s cities, and one navigates them through a weird balance of acknowledgment and ignoring.

Margaret Atwood’s Cat’s Eye is the subtlest, but perhaps the most profound and personal of the three novels. Her protagonist, Elaine Risley, travels back to her hometown of Toronto for a retrospective of her art work; but Elaine is also going back home with a shared dread and desire to see Cordelia, a childhood friend who was equal parts compatriot and tormentor. Elaine, now middle-aged, is wrestling with her past, and the meaning of time: “You don’t look back along time but down through it, like water.” (p. 3).  Time is not a long, straight line, like a horizon, and the past is not long gone and out of reach: the past is right there in the reflection on the water’s surface.

At the end of the novel, Elaine visits the ravine where as a girl she was abandoned by her friends and saved by her hallucination of the Virgin Mary, where now she fantasizes about finally seeing Cordelia. She views the scene from her childhood with relativistic uncertainty: “There’s nothing more for me to see. The bridge is only a bridge, the river a river, the sky is a sky. This landscape is empty now, a place for Sunday runners. Or not empty: filled with whatever it is by itself, when I’m not looking” (p. 460). Past and present are no longer bound reflections of each other across an arbitrary line. Now time can be empty instead of filled, dynamic instead of static: you can travel back in time and change the past simply by no longer looking down into the water. While Stephenson and Miéville bust their stories open with an excess of possibility, Atwood collapses hers—she integrates Elaine’s co-existing past and present into a single, linear, unknown future no longer bound by the infinite reflection of what did or did not happen to Cordelia—just as it should be: physically, we move forward through time, we exist in the present, no matter how much we try to reflect and revisit and repair the emotional past.

However, like any good speculative fiction, there is a twist to these hyperbolic adventures; each of these novels uses language to expand, crack, and crenellate the surface of reality; and each employs a reflective, co-existing narrative within the original. But unlike the simplest mathematical hyperbola, with its matching twin curves riven and moving toward different infinities, these stories don’t reflect across an obvious boundary condition—there is no simple, congruent image on the other side, nor is there any clear definition of which is the primary and which is the derivative object. These boundaries are complex geometries consisting of more than the two points that draw a straight line: in the case of Cat’s Eye, Elaine herself is both the source of the reflection and the boundary upon which it is reflected. These are not equal and opposite forms, but reflections that intersect, interfere, and affect each other in demanding and chaotic ways, making us question what kind of mirror we are confronting.

Bloom Post End

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 copyright by ermell – Own work, CC BY-SA 4.0, Wikipedia.

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