The second is the Torus knot from Segerman’s new book Visualizing Mathematics with 3D Printing.

We started the project today by just talking about the knots. Comparing the two knots that are actually identical was useful in refining the language they used to talk about knots.

Next they wanted to try to compare the two identical knots by looking at their crossings. My older son had the idea of assigning a +1 to every “over” crossing and a -1 to every “under” crossing. My younger son noticed that this counting method should always produce a net 0 because we counted the over and under crossing for each crossing exactly once.

New we tried to compare Segerman’s torus knot to Taalman’s rolling knot. Here we used the “tangle” from Colin Adams’s book Why Knot?

One fun thing that came up by accident in this video is an amazing shadow cast by Taalman’s knot – that was a really fun surprise.

Unfortunately, it proved to be a bit difficult to get the tangle back together so we had to pause the video at the re-connect the tangle off camera. It is really neat, though, to watch kids try to make a copy of a knot.

Once we got the tangle connected we started the next video. Since the tangle can move around, it isn’t that hard to manipulate the tangle from the form Segerman’s knot to the form of Taalman’s knots. In fact, it happened more or less by accident!

As I mentioned above, it is actually a pretty difficult task for the kids to describe the features of the knots when they compare them – even with a knot as simple as the trefoil knot. I think one of the neat parts of this particular project is working on using more precise mathematical language.

So, a fun project. We have a new 3d printer and I’m really excited about using many more 3d printing ideas from Taalman and Segerman to explore math with the boys.