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