Hyperbolic Crochet

My name is Madison Boyer, and I am a Junior at Westfield State University studying mathematics with a focus in secondary education.

I have been crocheting for about a month, which basically means I’m an expert. One of my roommates was teaching me the basics about a month ago, and as I was single crocheting my chain kept curling in on itself because my tension was too tight. My other roommate, who is taking a Math Applications course, started to talk about Hyperbolic Geometry because that’s what she was doing in class at the time. So, I decided to investigate crocheting and Hyperbolic Geometry.

Before we get into the crocheting, let's talk a little bit about math. In your K-12 schooling, you’re taught what’s called Euclidean Geometry. In order for this blog to make sense we need to define what the terms line and parallel mean, and they’re not going to be what we traditionally think of. Line can be defined by the shortest path between 2 points. This is going to sound really weird, but this means that lines don’t always have to appear straight. Parallel can be defined as things that never intersect. This means that there can be situations where more than 1 line can be parallel to another line through a single point.

Euclidean Geometry has 5 postulates which are statements that are accepted as true without being proven. One of these postulates is called the Parallel Postulate which says that in a plane, if you are given a line, and a point that’s not on the line, then there is exactly one line, through the given point, that is parallel to the given line. Basically what this is saying is there's only one line that can go through this point that’s parallel to the other line.

Hyperbolic Geometry is geometry that deals with pseudospherical surfaces which are surfaces with constant negative curvature. In Hyperbolic Geometry the Parallel Postulate no longer holds true. On a Hyperbolic plane if you are given a line, and a point that’s not on the line, there are infinitely many lines that go through the given point that are parallel to the given line. My roommate’s Math Applications class, led by Alexander Moore, made this paper model of a Hyperbolic plane.

     On this model the blue line is the given line, and the blue point is the given point. The green and red lines are both parallel to the blue line. As you can see, each line is straight when you pick up the model and fold it along the line. So the green and red lines are parallel to the blue one because they are straight and never intersect the blue line. Now you might be asking yourself, “how are there infinitely many parallel lines? This is only showing 2.” My answer is, what is stopping you from filling the space between the red and green lines with lines? There are infinitely many possible lines that go through the given point, and are in between the red and green lines. And all of those lines are also parallel to the blue line!

Now the last super mathy thing I want to say is about negative curvature. Hyperbolic space has negative curvature because at any point on the plane if we were to place a flat plane that follows the curve of the Hyperbolic plane, the Hyperbolic plane would curve away from the flat plane in two different directions. Since any type of plane goes on forever this means that Hyperbolic planes have constant negative curvature throughout the whole plane.

    Ok now we can actually talk about crochet. In 1997 Daina Taimina, a Professor of Mathematics at Cornell University, was presented a paper model of a Hyperbolic plane similar to the one my roommate’s class made, to use in the geometry class she was teaching. The paper models are very fragile and can be broken easily, so she decided to crochet her own model that would hold up better than the paper model.

    It’s actually super easy to crochet a Hyperbolic plane. Start with a chain of single stitches about 15-20 stitches long. Then you can pick your favorite type of stitch out of single crochet, double crochet, or half-double crochet. Now you begin the first row, crocheting with your stitch of choice a set number of times. For the sake of simplicity let’s pick 3. So, you crochet 3 of your favorite stitches, and then you do what’s called an increase, where you crochet that same stitch twice in one hole. Now you repeat this pattern over and over again until your surface is the size you want it to be. When you get to the end of each row, you turn the piece, and do it all over again for that row and every row after. 

The reason that this constant rate of increasing in your stitches makes negative curvature is because each time you put an increase in your pattern it makes that row longer than the previous row. So, after your first row of single stitches, each row after that is longer than the one before. This causes the piece to curl in on itself, and creates negative curvature.

     Another model of hyperbolic space that’s super easy to make is a hyperbolic pseudosphere. You can make this by starting a chain of 8 single stitches, and then connecting it to make a circle by using a slip stitch. Then you stitch into the ring, instead of into a stitch, 1 single crochet, then 2 half-double crochets, and then 16 double crochets. After this you start stitching into stitches like normal. Each stitch should have 3 double crochets, and repeat this pattern until your pseudosphere is of the size you would like. Then to end it you do 1 half-double crochet in the next 2 stitches, 1 slip stitch in the 2 after that, and then you fasten it off.

     This is a wonderful way to see the beauty in math, so now go crochet your own. Investigate into different rates of increase, and how they impact the final product. You could also investigate the mathematical patterns in the other things you have crocheted or are going to crochet. There are endless connections between math and crochet!

Sources:

https://www.goldenlucycrafts.com/crochet-hyperbolic-coral/

https://www.theiff.org/oexhibits/oe1e.html

https://ncartmuseum.org/wp-content/uploads/2021/10/Reef-How-To-FINAL2020-copy-1.pdf

https://terrywhitmell.com/2021/08/13/crocheting-and-math/