September 19, 2013

Crocheting and geometry

You might ask, what do these two subjects have to do with each other?

Most people probably think that math and crocheting don't have anything in common (well, besides the fact you count your stiches and such). However, the video below shows you a STRIKING connection between crocheting and hyperbolic geometry which has ONLY been found fairly recently.




Crochet model of hyperbolic plane by Daina Taimina
I found it surprising that mathematicians had thought for years that it wouldn't be POSSIBLE to build a physical model for hyperbolic geometry. Then... the mathematician Daina Taimina found that CROCHETING was the perfect model for it!

The essence of the hyperbolic space can be implemented with knitting or crochet simply by increasing the number of stitches in each row. As you increase, the surface naturally begins to ruffle and crenellate.

So... what is hyperbolic geometry?


You can draw several lines that are parallel to a given line and that go through a given point.

In a nutshell, it is geometry with this property: if you have a line and a point NOT on the line, through that point you can draw MANY lines that are parallel to the given line.

In contrast, in Euclidean geometry, you can draw exactly one line that is parallel to the given line and goes through the given point.


Then there is one more kind of geometry.

In spherical geometry, given a line and a point not on that line, you cannot draw ANY parallel lines through that point. Think of great circles as "lines" on our globe (equator is one, lines of longitude). Then pick a point outside your great circle. You CANNOT draw another great circle through that point that wouldn't also intersect the first great circle.




A certain kind of nudibranch.

Back to hyperbolic geometry

The fact that crocheting is the best way to model hyperbolic geometry is not the only interesting thing... they found that hyperbolic geometry DOES EXIST in nature as well! So, it is not a product of mathematicians' minds only, after all. Hyperbolic geometry is found in the DESIGN of various corals, sea slugs, flatworms, nudibranches, and some sea plants.

The effect is similar to what we see in lettuce leaves and certain types of kelp where the vegetable surface expands outward, generating a ruffled effect.

Yellow leather coral.


To me, learning all of this was really exciting. See, I studied hyperbolic geometry just a little bit back in the 1990s. I was teaching a course to future teachers where we touched on math history and other similar topics. We studied Euclid and the axioms of the Euclidean geometry. Then I found a little computer program titled NonEuclid that allowed me to contrast hyperbolic geometry with the Euclidean geometry and also with spherical geometry. I found these non-Euclidean geometries really fascinating back then!

crocheted coral reef

However, hyperbolic geometry was just an abstract notion to me at that time. So now, learning that hyperbolic geometry IS found in nature AND in crocheting is simply exciting! I wish I could have shown my teacher-students back then about these connections!

More reading

Crochet models of hyperbolic geometry

Geometries beyond Euclid — a complete unit of study about non-euclidean geometries

Hyperbolic Crochet Coral Reef: A Colorful Yarn of the Sea — Lots of pictures of crocheted coral reefs
Post a Comment