Roger Antonsen

This is a 52-mirror labyrinth, with a “laser beam” that traces out the Hilbert curve.

The 52 small mirrors and a symbolic “laser beam” that traces out the Hilbert curve. (Photo: Edmund Harriss)

The labyrinth is made of different parts: The base is made of two pieces of wood glued together, and the top layer has slits precisely cut to hold the 52 mirrors in such a way that the back sides of the mirrors were on the appropriate diagonals.

Assembling the labyrinth. The top layer has slits precisely cut to hold the 52 mirrors in such a way that the back sides of the mirrors were on the appropriate diagonals.

The symbolic “laser beam” consists of a red piece of paper resting on a wooden platform of the same shape.

The labyrinth is made of different parts: The base is made of two pieces of wood glued together, where the top layer has slits precisely cut to hold the 52 mirrors in such a way that the back sides of the mirrors were on the appropriate diagonals.

This illustrates an internal point of view, as the “laser beam” has no knowledge of absolute direction, and the mirrors (or their absence) serve as the local commands.

A close-up of the mirrors. (Photo: Edmund Harriss)

Originally I wanted to use an actual laser beam to trace out the curve, but the mirrors absorbed too much light, and the beam was invisible after a dozen bounces.

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).