The Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891. I believe that we can gain understanding by looking at an object – in this case the Hilbert curve – from different perspectives, so here are different perspectives and versions I experimented with at ICERM: computer generated curves, paper versions, wooden tiles, laser cut wooden curves, and a Hilbert curve mirror labyrinth.
Media
The Hilbert curve in two different versions, curvy and straight.Two iterations of the Hilbert curve with different colors. This was made by laser cutting two sheets of paper. (Photo: Edmund Harriss)Celtic knot renditions of the Hilbert curve made with paper. I have previously explored Celtic knots, which are naturally space-filling, and I wanted to use them to explore the Hilbert Curve. (Photo: Edmund Harriss)Caption: Closeups of the Celtic versions. These knots are usually presented in a flat medium, but they also have depth, being alternating knots, in which the crossings alternate between under- and overpasses.These are versions of the Hilbert curve made with a laser cutter in wood. The left picture shows three iterations of the Hilbert curve, illustrating the recursive step from one level to the next. The right picture shows 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)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)It turns out that only three different tiles are necessary to define the Hilbert curve. (Photo: Edmund Harriss)Work in progress with 256 tiles.This is a mirror labyrinth, made with 52 small mirrors, with a symbolic “laser beam” that traces out the Hilbert curve. (Photo: Edmund Harriss)Originally I wanted to use an actual laser beam to trace out the curve, but it turned out that mirrors absorbed too much light. The above illustration shows what I was aiming for.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.The mirrors themselves were then placed in the slits. The symbolic “laser beam” consists of a red piece of paper, cut with a laser cutter, resting on a wooden platform of the same shape.
I want to thank Edmund Harriss, Judy Holdener, Laura Taalman, and Carolyn Yackel for inspiration and assistance while making these pieces. Judy provided the mirrors, Laura, Carolyn, and Edmund were the expert laser cutters, and Edmund took most of the above pictures.
References
Michael Bader, An Introduction with Applications in Scientific Computing, Springer-Verlag Berlin Heidelberg, 2013.
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