Crochet Projects: Hyperbolic Plane and Pseudosphere

In mid-July a friend sent me a link to the Hyperbolic Crochet blog. The blog author's projects awakened my long-suppressed inner math geek and I couldn't resist contemplating crochet patterns based on hyperbolic geometry. Hyperbolic planes are surfaces with negative curvature, opposite of the one of a sphere. And pseudospheres are the hyperbolic equivalent of a cone but with a high rate of increase they tend to form a spherical shape (hence the name). The mathematics involved can be a bit intimidating if you really get into it - so I won't go there (though if you're curious, here you go) - but hyperbolic geometric shapes like these are models of exponential tissue growth, which can be found in nature in coral reefs, polyps, lettuce and some other plants.

In theory a hyperbolic plane is formed when you start with a chain of stitches whereas a pseudosphere is formed when starting with a round/ring of stitches. However, in practice the two can often end up looking very similar.

While checking out some of the very cool hyperbolic projects on display across the interwebs, I stumbled across the Maine Reef which is "a satellite of the worldwide Hyperbolic Crochet Coral Reef Project". Then last night another friend sent me an article about the coral reef project, which reminded me that I still wanted to try to crochet a hyperbolic plane and/or a pseudosphere. So, I decided to take a break from my current super-secret projects and try one.

I chose to start out making a pseudosphere. For it I used a size F hook and some leftover odds and ends of yarn (I think it's all Red Heart Super Saver - shocking pink, bright yellow and a rusty brown).

For my pseudosphere I started with five sc in a ring, then repeated a pattern of working 1 sc in next st, then 2 sc in the next st (i.e. increasing 1 st every other stitch). This pattern means that the number of stitches in each round increases exponentially with a 3:2 ratio. The ring quickly started to curl up on itself into a ball-type shape, looking like a coral.

It was so much fun to feel this project grow and take on its unique shape that I decided I needed to make a hyperbolic plane right away too!

For my hyperbolic plane, I again used a size F hook and leftover yarn (bright yellow, royal purple and paddy green). I started with a chain of 15, and this one increases at the ratio of 4:3 (1 sc in next 2 st then 2 sc in next st - i.e. increase 1 st every 3 st).

It could have gone on for many more rows with either the plane or the pseudosphere to let them grow more complexly ruffled, but since I was using up leftover yarn, I just ended when I ran out of a few colors.

Personally, I found hyperbolic geometric shapes to be a lot of fun to crochet as a blind person, because even if you can’t see the entire project's shape, you can still feel all the neat curls, twists and curves that grow as you work, ever-increasing the project's surface area.

I'm not sure what I'll do with these two little projects, but I do think both pseudospheres and hyperbolic planes could make neat dish/wash cloths if made with the right (cotton) yarn. I also think a hyperbolic plane – if made on a much more grand scale than I did, with a larger ratio of expansion – could make a very interesting baby blanket. I'm really tempted to give that idea a try the next time I need to gift one.