By Tim Chartier, Davidson College

Turkish carpet-making, Renaissance art, M.C. Escher etchings, and modern work in 3D printing are a few examples of artistic work infused with deep mathematical ideas.  Math and art is an area of scholarly work and also suitable for the classroom.  Dr. Donna Molinek, Professor of Mathematics at Davidson College, regularly teaches a math and art course, which satisfies a necessary math credit for non-STEM majors.  In the past year, Donna taught an upper-level Geometry and Art course that springboarded into an independent study on Knot Theory and Art with senior math majors Grace Fahey and Anna West.  What mathematical ideas can be taught alongside art?  What art can be created?  With such questions, we turn to Donna, Grace and Anna to learn more about hands-on mathematical art. 

Tim Chartier: Donna, how did you get involved in mathematical art?  How did you decide to teach a class on mathematical art and how did you pick the topics for the course? 

Donna Molinek: I’ve always loved working with my hands as a contrast and complement to doing mathematics and believe that we get a different understanding if we can touch objects illustrating what we are doing mathematically. For instance, drawing a Mobius band pales in comparison to making one out of paper which in turn takes a back seat to crafting one from clay. I have always brought “props” into my classes to visualize math concepts and play. For example, a long time ago in a topology class when the set theory got a little overwhelming, we took a detour into knot theory and making Seifert Surfaces. Turns out college students like to use construction paper, tape, yarn, and glue! We have a 100-level topics math course which I taught as a math and art course. I thought about art I saw as mathematical which led to topics such as knot theory, symmetry, perspective, Sona drawings, complex dynamics, origami, fractals, and the list goes on. Last year I taught an upper level seminar on Geometry and Art where we could go into more mathematical detail while studying the art inspired by this mathematics.

Tim: Anna and Grace, what drew you to take such a course in mathematical art? 

Anna West: We took a Geometry and Art class last year, and loved seeing the intersection between what we were learning and the art we were creating. A lot of times we learn math and think to ourselves “when am I ever going to use this again” and when it is being applied to real life things, it gives more purpose to what we are learning, and makes the material much more engaging.

Tim Chartier: Anna and Grace, how was a class in mathematical art similar to other math classes you had taken?  How was it different?

Grace Fahey: We took Geometry and Art last year. This class involved the relationship between complex geometry and several mediums of art. I would say that this Knot Theory course is pretty similar. In this class we look at the relationship between the theory of knots and different art mediums. In both classes, we examined a mathematical topic, for example in Geometry we examined hyperbolic space and in Knot Theory we examined torus knots, and then we took these concepts and applied them to art. In Knot Theory, I made a ceramic torus knot on the wheel and am going to put a knot around it with yarn.

Tim: Donna, you had students producing mathematical art in your course. What was it like when the submissions began coming in?  What were your expectations for the assignments?

Donna: When the submissions came in, I was awestruck. Our students are so creative and talented. My assignments are pretty open-ended about the art part. I want the students to let the math and their own interests lead them. I do require a written part describing the connections to the math topic, history and cultural significance,  and an explanation of the mathematics behind what they’ve done.

Tim: Anna and Grace, what prompted you to propose an independent study following the course on math and art? 

Grace: We both love art and did it all through high school, but haven’t found as much time to do it in college. Creating this independent study was the perfect intersection of two things we really enjoy.

Tim: How did the three of you decide on topics for the independent study?

Donna: Anna and Grace came to me after the Geometry and Art class and said they’d love to continue to explore math and art. We met several times brainstorming ideas before deciding on knot theory and art. It’s been an area I always loved and do bring into classes whenever possible. After working with Anna and Grace in the Geometry class I knew they would bring creativity to anything we did. So, we got out a couple of texts, went through the table of contents and decided on what we wanted to learn.

Tim: What have you  done in the independent study?

Anna: In terms of art, Grace and I have created drawings, ceramics, laser cut pieces, intricate braids, and we are currently working on 3-D printing, and sticker cutting different knot designs. In terms of math, we have explored the history of knots, different types of knots, tangles, links, 3-colorability, Dowker coding, Reidemieister moves, Torus knots, satellite knots, hyperbolic knots, mirror curves, Lunda Designs, braids, and polynomials. 

Tim: Anna and Grace, can you each share an interesting piece you’ve created in your independent study and its connection to mathematics?

Anna: So far our larger projects have been in ceramics.

Grace: For the final, we are creating a concentration of several art pieces using different art mediums while also applying multiple concepts that we learned throughout the semester.

Anna: My favorite piece that I created was a clay model of the Borromean rings, which is an example of Brunnian links with three components. The idea here is that if you remove any one link, the whole piece falls apart.

Grace: My favorite piece that I have created so far has been the ceramic torus knot. I threw the shape on the wheel and then hand built a stand so that the torus could be vertical. I originally tried to put the 5-3 knot on using clay coils, but decided it would look cleaner if I added yarn or string around it after.

Tim: Donna, creativity inevitably leads to surprises. Can you share a surprise in the work you’ve done with Anna and Grace? 

Donna: Certainly; their individual talents being revealed was a pleasant surprise: Anna’s freehand carvings illustrating Celtic knots, Grace’s combined bowls showing a Hopf link rim, their enthusiasm and abilities using the maker space. Another was watching as they discovered how connected our world is to so many mathematical concepts and their surprise at seeing mathematics in their own childhood hobbies. Finally, I appreciated (not really a surprise knowing the two of them) their willingness to try new and, at least to me, seemingly impossible undertakings!

Tim: Donna, having led a course and independent study in math and art, what advice do you have on teaching a course to math majors?  What’s your advice in developing content that includes enough math while still having artistic topics?

Donna: My advice about teaching such a course is to try it! At Davidson College, we have the flexibility to try out topics courses as scheduling constraints permit. After having taught the 100-level course for non majors several times and with a gap in our upper level offerings, I proposed a 400-level Geometry and Art seminar. I used a standard Geometry text as a guide for course content and then matched with art topics. Having attended the Bridges conference on mathematical connections in art, music, architecture, and culture, I had no shortage of ideas connecting math and art. Their website and, in particular, the archived conference paper section is a treasure trove of amazing resources for artistic and mathematical topics. To make sure we learned the mathematical content, regular problem sets were assigned, and all had an opportunity for some art to be included. The majority of the art in the seminar class was done as group projects and students were encouraged to explore and share techniques for using different mediums. For this independent study course, we met regularly going over problems and discussing the mathematics. The art was done outside these meeting times and presented regularly to each other throughout the semester. We also had the great fortune to meet several times at my house to work on the pottery projects and had access to Davidson’s maker space for other projects.

Tim: Anna and Grace, there will be students reading this who haven’t studied mathematical art.  Where might someone begin learning?

Grace: Other than taking a class in mathematical art like we did with Dr. Molinek last year, I think that there are actually a lot of youtube videos and websites of mathematical artists that walk through their thought process and art pieces. In addition to a few textbooks, we have utilized the internet pretty heavily this semester.

Tim Chartier is the 2022-23 Distinguished Visiting Professor at the National Museum of Mathematics and the Joseph R. Morton Professor of Mathematics and Computer Science at Davidson College.