Math Circle – Celtic Knots

On Saturday at (MC)^2, Harini and I led a session on Celtic knots. Many of the ideas for it were taken from these great articles on the NRICH website. Here’s the description we sent out ahead of time:

It’s St. Patrick’s Day this week! We’ll look at some ornate Celtic knots, observe and explain some patterns, and make some knot drawings of our own. We’ll also discuss the difference between knots and links, ways to identify each, and some other basic topological ideas. Come shamrock some great math with us!

Here is the warm-up sheet we gave to people on Saturday as everyone arrived and got settled.

How many bands are each of these designs made up of?

How many bands are each of these designs made up of?

We started with some introductory remarks and showing a few examples of Celtic knots:


Here are all of the examples we showed.

Then we played Vi Hart’s video “Snakes + Graphs”:

examples curveAfterwards we focused in on the first doodle game Vi shares. We drew and talked through a simple example on the board, then game folks time to try some of their own or to construct the over and unders on this example curve.

Next we shared a second, much more restrictive method of drawing knots. It produces images like the A and B weavings pictured above. We showed Alison’s video on this NRICH page twice and let folks follow along with this sheet (or work on whatever they liked).

Then I shared a third method for drawing knots that I learned through these lecture notes. Here was my first try, which I made in preparation for the session:


Here’s the handout of dot paper that we passed out for people to try on. You’ll need to tilt the page at a 45° angle.

We showed this photo of a project Harini did at a Bridges math art workshop a couple of summers ago, as inspiration for them to take next steps on their own at home.


Much of the 45 minutes of the session was giving attendees time to try out these methods on their own and giving them support and answering their questions as they arose. As a result, it was a much quieter math circle than what I’ve been used to, but still a lot of fun. People seemed pretty happy, and lots of great knot drawings got made.


4 responses to “Math Circle – Celtic Knots

  1. I absolutely love this! I sat in a on knot theory class as an undergrad, it always captured my imagination. A huge thanks for sharing.

  2. Justin – this is wonderful. I can’t wait for the opportunity to use it. And thank you for sharing so completely all the materials and procedures.

  3. Justin – great post. I learned how to draw knots following a process very similar to the one you shared. There is a great book by Adian Meehan that describes the process. A while back I was playing with ways to write programs to generate these patterns, and one technique I tried was using tiles that could be assembled together into a knot pattern. Some info on this approach is found here:

    — Dan

    • Hi Dan,
      Thanks for the comment and the link! I’m looking forward to thinking more about your knot tiles. I just got a George Bain book in the mail; the Adian Meehan book(s) may be next steps—thanks for the lead. I hope to see you in the next few days!

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