Hilbert Tiles

  30. November 2020

There are several different but equivalent ways of defining the Hilbert curves: One way is “external” or “plotter-based”, given by absolute directions: up, down, left, and right. Another is “internal” or “turtle-based”, given by commands like “turn left” or “move forward” without any knowledge of absolute direction.

The Hilbert curve can also be defined by a set of tiles. The tiles were cut out and colored by hand afterwards. (Photo: Edmund Harriss)
The Hilbert curve can also be defined by a set of tiles. The tiles were cut out and colored by hand afterwards. (Photo: Edmund Harriss)

This is a tiling system, with only three types of tiles, that trace out the curve. The tiles correspond precisely to the instructions “turn left”, “turn right”, and “move forward”. These wooden tiles were created with a laser cutter and colored by hand. Below are some pictures of this process.

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).
It turns out that only three different tiles are necessary to define the Hilbert curve. (Photo: Edmund Harriss)
It turns out that only three different tiles are necessary to define the Hilbert curve. (Photo: Edmund Harriss)
Work-in-progress.
A close-up of the tiles.
Work-in-progress. 256 small tiles.
Work-in-progress.