Hyperbolic crochet

In 1997, professor of mathematics and crochet enthusiast Daina Taimiņa found a way to join those two passions in order to craft durable sections of hyperbolic surfaces.

Hyperbolic crochet
Margaret Wertheim, CC BY 2.0, via Wikimedia Commons

Today I was reading about the Diagram Prize, which each year is awarded to the book with the strangest title. The 2020 winner will be announced tomorrow. My money is on The Slaughter of Farmed Animals: Practical Ways to Enhance Animal Welfare, although Classical Antiquity in Heavy Metal Music would also be good. But then the title of a past winner jumped out at me: Crocheting Adventures with Hyperbolic Planes.

What is a hyperbolic plane, and why would you crochet one? Well, I’m not qualified to explain this complex branch of geometry very well, but I’ll give it a shot anyway. Let’s begin with parallel lines.

Draw a line, then draw two more lines that extend out from that line at right angles. Those two extra lines are parallel: they never meet. Euclid proposed this about 2300 years ago. It’s one of just five bedrock propositions out of which he built an entire system of geometry. Chances are good that this is the geometry you learned in school: lines, angles, parallel and perpendicular, curves, the Pythagorean theorem, and so forth. In the early 19th century CE, however, it became apparent that this was not the only way to think about geometry. In some places and some ways, Euclidian geometry doesn’t hold true. In some places and some ways, the lines drawn above are not parallel: they can and do meet.

Pbroks13 and Joshuabowman, CC BY-SA 3.0, via Wikimedia Commons

Consider those same lines drawn on the surface of a sphere. Rather than being parallel, they will always, inevitably, eventually meet. In fact, parallel lines cannot exist on the surface of a sphere. So this Euclidian model of parallel lines (usually termed the parallel postulate) doesn’t apply here. And because this tenet is one of the foundations of Euclidian geometry, altering that rule alters the nature of the system. You get other ways of thinking about geometry, other sets of rules and behaviour beyond the traditional modes we knew. These are the non-Euclidian geometries.

Still with me? Shifting that one assumption gives us new geometries. If the lines curve in towards each other and meet, like on the surface of a sphere, you get elliptic geometry. And if parallel lines curve away from each other instead, that’s hyperbolic geometry.

A sphere (elliptic geometry) is easy to visualise, but hyperbolic geometry is a little trickier. It looks a little like a saddle shape, or the frilly curve of coral growing on a reef. And that’s where we (finally) get back to Daina Taimiņa. This Cornell mathematician saw another professor struggling with flimsy paper models of hyperbolic space, and realised that there was a better way to illustrate this complex mathematical concept. Crochet.

In 2009 Taimiņa published a book demonstrating the intersection between crochet and non-Euclidian geometry: Crocheting Adventures with Hyperbolic Planes. It’s an amazing combination of mathematical illustration and practical application. The items pictured at the top of this post are physical manifestations of hyperbolic geometry in yarn – essential teaching tools to help us visualise a geography beyond the familiar.

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