Math Art Project Reflections

 (1) Group reflection:

Our group recreated Seven Linked Surfaces by Eve Torrence. The piece is made up of seven crocheted knot surfaces that all have the same boundary of two unknots linked through each other twice. The one that we chose to recreate is homeomorphic to a torus with two punctures, as seen later.

For one of our recreations, we contacted the artist to get some help. She sent over the pattern that she used, and directed us to some videos showing the technique. Even with this help it was still difficult to recreate the original piece and we didn’t fully succeed. The crochet technique is more involved than we thought, but it was really fun to try! There’s definitely still a lot to learn about topological crochet. Re-creating the piece gave good insight about the shape of it, and the math behind it. Following the method that Eve used allowed us to understand more about the knot and how to go from a knot to a surface. 

We used different colours in our recreations, and one of the recreations followed a totally different technique from Eve’s. We thought it might be interesting to crochet a knot link similar to the boundary of the seven linked surfaces following a technique published by Shiying Dong. We made two chains of length 100 stitches—or 10m and 10n respectively where m=n=10 stitches—and marked each one with stitch markers at the crossing points (8m/2n, 3m/7n, 9.5m/0.5n, 4.5m/5.5n, 2.5m/7.5m, and 2.5n/7.5n) so that the pattern would be symmetrical. Then we did single crochet stitches along the chains, going either over or under the stitch marker points depending on the type of crossing. At the saddle points where the surface twists, we did double crochet stitches to get the desired twist effect. 

Another of the recreations that we made was done without any instructions, working only from the photograph. A picture of this is shown below this paragraph. We started by creating three ‘kissing’ chain loops (three chain loops joined at two points in a row). Then we worked double crochets into the chain loops around until we reached an intersection. At each intersection we then had to very carefully determine which of the three paths to continue onto based on the picture. In the end the result was surprisingly similar to the original piece despite us using a very different method to design and create it. This goes to show that there are often many ways to approach and solve a given problem and that different people’s intuitions will lead them down different paths.

We also made a similar piece that is much simpler in order to better understand the techniques at play. This piece is simply a torus with two punctures and no tangling on the edges. Making this piece was very helpful in understanding what can be done with the technique that we used and in understanding how the more complex structure can be formed. Making this we realized that we could make the more complicated structure by taking this twice punctured torus and introducing twists into four of the linear sections. This was the basis of our class activity.

For our class activity we had students create the same surface as in Eve Torrence’s piece except out of paper. This involved creating three loops of paper each with a 360 degree twist and then taping them together to form the final product. The specific instructions are below.

Making a Linked Unknot Surface Activity

Duration: 5-10 minutes (longer than expected).

Materials:

    • 3 paper strips (approx. 1 cm by 10 cm) per student

    • clear tape

Instructions:

Make a clockwise twisted strip:

1. Pinch one short end of your strip in your left hand and pinch the other short end in your right hand.

2. Flip the end of the strip in your right hand clockwise twice.

3. Tape the surface to itself.

Make another clockwise twisted strip.

Making a counter-clockwise twisted strip:

1. Pinch one short end of your strip in your left hand and pinch the other short end in your right hand.

2. Flip the end of the strip in your right hand counter-clockwise twice.

3. Tape the surface to itself.

Tape the first two of the loops together at a 90 degree angle.

Opposite where you joined the two loops, tape the third loop to the second loop.

(2) Personal Reflection

I found this project very challenging! Looking at the artwork we wanted to re-create, I never imagined just how complex and involved the process would be. I still have a lot of learning to do around topological crochet. My favourite part of this project has been my correspondence with Eve Torrence. I sometimes forget just how amazing the math community is, and how friendly people are when you take an interest in their research. It's so cool that you can look up math art online, send an email, and then start a collaboration with someone. This is something that I've been missing since completing my masters degree - collaborating, and just the community at large! 

I've certainly learned some math from this project - prior to this I had very little knowledge of knot theory, and limited experience doing topology. I've also refined some of my crochet skills, so that's been nice. As a teacher, I learned most from presenting the hands on activity. I always forget just how long it takes students to do certain tasks. I thought that it would take a lot less time to cut out the paper and twist them into the correct shape. It was a good reminder to always give more time than you think you will need. 

The most frustrating part of this project was absolutely the recreation. I still haven't managed to re-create the piece successfully, but I feel I've invested so much time, I don't want to walk away until I have. Doing this art project was very reminiscent of trying a hard math problem. You try, try, and try again and it's still not right, but each time you've uncovered a new dimension to the problem, and come away with a new understanding. 

In my own classes, mathematical art will always be a part of what I teach. Having visualizations and engaging with math in a hands on way is so powerful. It also works to deconstruct the narrative that math isn't creative and thus can only be done by 'left brained individuals'. Most mathematicians I know are highly creative, artistic individuals. I think the limitations of doing mathematical art as a class is that the materials have to be readily available, and the art can't take too long to create. I don't think it would be reasonable to assign a topological crochet project, but making the structure out of paper and tape is a great way to bring this idea to the classroom.