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Knit Theory

Hyperbolic space is an unimaginable concept, unless you're a Latvian mathematician who's handy with needle and yarn.
By David Samuels
Photography By Richard Barnes
DISCOVER Vol. 27 No. 03 | March 2006


All photos courtesy of Richard
Barnes and The Insitute For Figuring.

Daina Taimina's crocheting skills turn a
simple hyperbolic plane (the hyperbolic
equivalent of a strip of paper) into a
work of art. These models are part of a
traveling exhibition created by the In-
stitute for Figuring in Los Angeles.

On a Thursday night in Ithaca, New York, Daina Taimina, an ebullient blond mathematician at Cornell University, sits at her kitchen table with her husband, David Henderson, a Cornell professor of geometry. In front of her sits a big Chinese bowl filled with crinkled forms made of gray, blue, red, and purple yarn. Reaching into the bowl, Taimina pulls out a woozy multicolored surface, the likes of which would have delighted Dr. Seuss.

"This is an octagon with a 45 degree angle at every juncture," she explains, displaying a familiar eight-sided shape outlined in white on the curvilinear surface of the wool. "And when you put it together," she continues, folding the material together so that the opposite corners of the octagon touch, "you get this." Just as the ends of a flat piece of paper can be joined to form a cylinder, so the ends of Taimina's woolly octagon can be joined to form a double cylinder. In front of my eyes, the octagon has been transformed into something that is also familiar but even stranger—a pair of hyperbolic woolen pants.

Holding the purple folds of wool in her hand, Taimina pauses to shoo her cat away from a dish of half-eaten vanilla ice cream, then reaches into the bowl again. She pulls out a colorful piece of knitting that looks as if it's been through the wash a few dozen times too many—and transforms it into something like a woolen cube with tubes protruding from the corners. She says it is four joined hyperbolic hexagons; a three-headed creature might be tempted to wear it as a sweater. "If you are a creature living on this plane, let's say an ant, it's still the same hyperbolic space. I can attach four more, like this," she says, assembling a miniature hyperbolic universe before my eyes.

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Originally posted by Chippewa

Oh no! She has opened a crocheting rift into another dimension, spinning a mathematical yarn that might allow these creatures, into our space!

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