Tessellation and Space-filling Curves

Hooray for in-flight internet! Unfortunately the guy in front of me has decided to put his seat back and take a nap, so I’m typing with the lid down, trusting that there won’t be any typos.
The plane is taking me back to Northfield for the final push—one short week before our show opens this Friday. There’s a bit of printing left to do, which is nerve wracking, but fairly straightforward. (Note: This post was started a week ago—that short week was too busy for writing blog posts. Now the show has opened and I am much more relaxed and ready to ruminate. What follows is an expanded version of my artist’s statement, printed in the gallery.)

Here I’d like to explain a little more of the theory behind the work. As has been clear thus far, one of the themes of my work is process, lots of it. Being a math major, though, contributes another inevitable theme. I drew most of my inspiration for this work from two different mathematical concepts—tessellation and space filling curves.

I’ll start with tessellation, which is a bit more familiar and approachable. A tessellation in two dimensions is a tiling of shapes which fit together without gaps or overlaps to fill the page. For example, the squares on a chess board fit perfectly with each other, covering the entire board. Similarly, the hexagons created by chicken wire or a bee’s honeycomb fill space without overlapping. In the art world, the work of M. C. Escher has made famous these and other more complex forms of tessellation.

Lizard Tessellation

A tessellation of lizards by M. C. Escher.

The concept can also be carried into three dimensions where the technical term is “honeycomb“. In three dimensional space there aren’t as many obvious examples apart from cubes, the three-dimensional analogue of two-dimensional squares. Though I hope to expand to other shapes next semester, for this show I’m sticking with the simple and familiar—squares and cubes. The bronze cube tessellates three-dimensional space and the printed squares it creates tessellate the two-dimensional plane.

The second concept is a space-filling curve. The basic idea here is that a one-dimensional line fills two-dimensional space. This is a rather strange concept. Certainly it is easy to take a Sharpie marker and completely cover a small square with black. However, a marker draws a thick line. Mathematically, a line is defined as a path that has no thickness. Drawing just vertical lines of this sort across a square, it would take an infinite number to cover every point in the square. Thus, it has long been thought impossible for objects to bridge dimensions, for a one-dimensional object to become, in some sense, two-dimensional. In 1890, however, the Italian mathematician Giuseppe Peano proved otherwise with what is now called the Peano Curve.

Peano Curve

3rd iteration Peano Curve

Hilbert Curve

First four iterations of the Hilbert Curve

The curve I used in my work is similar, developed by the German David Hilbert a year later. Both of these curves operate on the principle of iteration: the shape of the curve’s path is defined as a series, where each step is created by making the same modification to the previous step. At right you can see the first few steps of the Hilbert Curve. Notice that each new step consists of four smaller, connected copies of the previous step. In each step the line gets longer and more intricate, covering more of the square. The actual Hilbert curve is defined to be the limit as this sequence approaches infinity. In other words, we have created the Hilbert Curve once we have made an infinite number of steps. Of course this is impossible to draw with pencil and paper, or even a computer. The true space-filling curve is solely a mathematical ideal defined by an equation (sorry if that’s disappointing).

Three Hilbert Curves

An example path of printed faces, showing the three types of end points.

However, lower iterations can be visually interesting and can still suggest an approach to the ideal. On the cube I have used fourth iteration Hilbert Curves. (The fourth and most complicated box in the diagram above.) This brings up another theme of the work, that of continuous line. All of the sides are connected by one continuous loop which traces over each face in turn. Similarly, the printed faces connect at their edges, linking into one long line which rambles through the print. The three types of faces needed (straight, left turn, and right turn, as shown at right) are all present in the cube in both a positive relief, which prints a colored line, and a negative, which prints a white line with a colored background. In the final print, each color I used traced out one continuous line across the page, using different faces of the cube to wind its way back and forth.

Hope this helps to give an insight into some of my methods and thought processes. If something doesn’t make sense, or you are curious about something, please post a comment! The show is done and up now, so I won’t be doing more with this work, or writing more about it, for a little while. On to the next project!

2 Responses to “Tessellation and Space-filling Curves”

    • wastvedt

      Yes, sorry about the character-to-pixel ratio. I’m afraid that photos wouldn’t be much help in explaining the math though. The most useful perhaps (aside from the mathematical diagrams) would be photos of the finished cube and print. I’ll work on getting some of those, though very nice ones will have to wait until after the show is down and I can get the pieces under controlled lighting.


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