Penrose Tilings

First project

The Penrose tiling has fascinated me since I first saw it in Martin Gardner's January 1977 column in "Scientific American". This article describes a way found by Roger Penrose (an Oxford Professor) to tile a flat area using two simple shapes, but with no overall symmetry.

kd base
At first glance, it looks like a random arrangement. After some study, symmetrical patterns emerge at different scales. As the article points out, there is a rule that turns a pattern into a larger (or smaller) scale pattern. What this means is that all patterns are the same if you look over a large enough area.

kd base
One shortcomming of the article was that there was no picture that showed a large area of the pattern. I thought that if I could see a larger portion of the tiling, I could better understand the overall pattern. (Even though the article points out that there is no overall pattern).

My first Penrose based needlepoint project set out to fill this need. I generated this view of more tiles.

kd colors
I also did not like the coloring scheme used in the article, as it seemed to detract from the real patterns latent in this tiling. My coloring scheme emphasized the partial symmetries. It allowed me to see what symmetry there was, but also to see where the symmetries broke down, and bumped up against each other. Here is the design for my first Penrose project, showing the color scheme I picked.

This shows the ideal design, as first realized on the computer. Click on the image to see an enlargement.

kd needlepoint
And here is the actual needlepoint piece. I think this shows that the limits of needlepoint construction can be overcome in a complex design.

Second project

My next project was based on a curved line that can be drawn on top of the Kites and Darts.

kd red

Martin Gardner's article showed two different ways to do this. One is called the "Red" curve.

kd blue

The other is called the "Blue" curve.


I explored both, but to me the blue curve is by far the most interesting.

My goal then became to show the symmetries, and partial symmetries of this curved line. I wanted to see it on several scales at once. I have never seen any design like it, even though it is in some sense a "fundamental" arrangement, like a honeycomb or grid of squares.

Here is the basis of my second Penrose project.

curvey needlepoint
It ended up looking like this.

Third project

colored curves For my third Penrose project, I colored the curved lines, so I could follow the connections, and closures of the lines. Here one of the many interesting properties of this pattern needed to be dealt with. It turns out that the curved lines close up to form an infinite number of different shapes. What is worse, all of these different shapes "touch" each of the others somewhere in the design. If you don't want two different shapes to have the same color, and "touch", you must use infinitely many different colors.

Fortunately, I only needed to color in a limited area. I could do that and worry about the problem of the infinite colors another day. The project ended up looking like this

Fourth Project

I was still not satisfied, so I beefed up my computer program and did a fourth Penrose based project, which you can see and read about on the Project: Hortus page.

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