## Wooden Hilbert Curves

November 30, 2020

These are versions of the Hilbert curve made with a laser cutter in wood. Laser cutting opens up an enormous space of possibilities by its extreme precision. After a couple of minutes of laser cutting, you have created a piece of wood that it would take months or years to create by hand, if even possible.

The following pictures all illustrate the recursive steps from one level to the next.

This picture shows three iterations of the Hilbert curve, illustrating the recursive step from one level to the next. (Photo: Edmund Harriss)
This picture shows four iterations of the Hilbert curve, illustrating the recursive step from one level to the next. (Photo: Edmund Harriss)
This illustrates the recursive step from one level to the next. (Photo: Edmund Harriss)
A curve suspended. (Photo: Edmund Harriss)

It is illuminating to look at the inverses. The inverses turn out to be plane-filling spanning trees.

This shows the inverses, the spanning trees, of four levels of the Hilbert curve. (Photo: Edmund Harriss)
This shows the inverses, the spanning trees, of four levels of the Hilbert curve. (Photo: Edmund Harriss)
This shows the spanning tree for the most fine-grained of the curves. (Photo: Edmund Harriss)

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:

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).