Mathematical Art Activity Reflections

Group Reflection:

EDCP 342

Danielle, Ben, and Hannah

Math/Art Group Blog Post

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.

Personal Refection:

I really enjoyed this process (learning new crochet techniques, learning some new math, writing a class activity), however I've struggled to connect the heart of the project to my conception of my teaching in mathematics. One positive takeaway is that I really like the wall of math art that we have in the classroom and I would love to have something like that in my own classroom to display student art. 

Frankly, I would really love for more of a direct approach to learning math pedagogy, and one that is more grounded in the grade levels that we are going to be interacting with. Currently it seems like we are doing a lot of playing around with fun math concepts, which is something that all of us here have done quite a bit over the course of our math degrees. I'd really like to learn more about the ways in which teaching math is different from teaching other subjects and what kinds of pitfalls we are likely to face.