These are Celtic knotwork patterns realized in paper. Paper is perfect for work that lies between 2D and 3D, and Celtic knots fits perfectly in this category: they are usually presented in a flat medium, but they also have depth, being alternating knots, in which the crossings alternate between being underpasses and overpasses.

The purple and green design is an “unknot” where one half is made from green paper and the other is made from purple paper. Since the strands are overlapping in the design, it is impossible to cut this design out from a single piece of paper. I therefore cut out the two parts separately, wrapped them around each other, and connected the ends. If you look closely at the bottom left and bottom right, you can see that this is exactly where the strands change color.

The red, orange, and yellow design is more complex. It is a 192-crossing 3-component Brunnian link. Each component has its own color and is equivalent to the unknot. It shares the property with the Borromean rings that if any one of the strands is removed, the other two are left unconnected. Because of the overlapping strands, I cut this from three different pieces of paper and then braided them together.

Paper is a fun medium, but also very fragile. I made some early versions of these curves with a paper cutter, but I quickly realized that a laser cutter was more appropriate and gave me more precision.

The overall patterns of both designs are different iterations of the Hilbert curve, a continuous fractal space-filling curve named after the German mathematician David Hilbert in 1891. This seemed like a good fit, since several Celtic designs are also naturally space-filling.

This is a part of my Hilbert curve explorations that took place during the Illustrating Mathematics program at the Institute for Computational and Experimental Research in Mathematics (ICERM). I believe that we can gain understanding by looking at an object – in this case the Hilbert curve – from different perspectives, so I set forth to play with the curve in many different ways:

- Celtic Paper Hilbert Curve
- Mirror Hilbert Labyrinth
- Hilbert Tiles
- Wooden Hilbert Curves
- Hilbert Curves

This project has also been documented on the ICERM project pages, and it has found it’s way to the Illustrating Mathematics book, published by the American Mathematical Society (AMS).

If you want to know more about the Hilbert curve, check out one of these books:

- Michael Bader, An Introduction with Applications in Scientific Computing, Springer-Verlag Berlin Heidelberg, 2013.
- Doug McKenna, Hilbert Curves: Outside-In and Inside Gone, Mathemæsthetics, Inc. (2019).