Cover stories: Making the graphene quasicrystals cover

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Science  24 Aug 2018:
Vol. 361, Issue 6404, eaav1395
DOI: 10.1126/science.aav1395

Cover stories offer a look at the process behind the art on the cover: who made it, how it got made, and why.

As described in a Report in this week’s issue, an unusual geometric property arises in a bilayer graphene structure in which one layer is rotated 30° degrees with respect to the other. Tiles of triangles, rhombuses, and squares can be mapped over this grid in a pattern called Stampfli tiling. While looking at a figure from this Report (Fig. 1), I thought, “No problem. I’ll fire up Illustrator, draw the three different shapes, and get to work.” It did not take me long to discover, however, that this tiling is not as trivial as it first appeared. Nervous excitement (read: mild panic) set in as I realized that I could not determine the pattern. I needed more information and leapt to the internet. That is when the fascinating and beautiful world of tiling opened up to me.

Fig. 1

An image from the Ahn et al. Report (figure 1F in the paper) shows Stampfli tiling mapped onto a false-colored transmission electron micrograph of a graphene bilayer.

Credit: S. J. Ahn et al.

The structure of a single layer of graphene is a periodic, hexagonal grid. But when a second hexagonal grid rotated by 30° is placed atop the other, a more complex, quasi-periodic grid is created. This quasi-periodicity is the reason I could not see a repeatable unit. Rhombuses, squares, and triangles appear when one connects the points at the centers of every polygon of the double-hex grid (Fig. 2). This is virtually impossible for a human to do accurately. Thus, it is necessary to write an algorithm that generates the tiles.

Fig. 2

The double-hex grid is shown in black and gray. The “dual”—that is, the grid created by connecting the centers of polygons that arise from the intersection of the grids—is shown in red. The lines have to be adjusted to produce a tiling with squares, rhombuses, and equilateral triangles.

Credit: V. Altounian/Science

I learned all of this and more from a blog I found, written by none other than the namesake of this tiling himself: Peter Stampfli. I reached out to Peter to thank him and ask to use the Processing code he generously makes available to generate the tiling. Excited to have connected with an early pioneer of generative art, I asked Peter if he would share some insight into his interesting career and what he is working on these days.

–Valerie Altounian

My grandfather, Oskar Stampfli, taught me about Platonic solids and tilings of the plane when I was about 10 years old. It was fascinating, and I subsequently became interested in geometry and symmetric images. Later, when studying mathematics and physics at the ETH Zurich, I read the “Mathematical Games” column of Martin Gardner and Douglas Hofstadter in Scientific American and thus discovered the beauty of fractals, self-similar images, and the geometric art of M. C. Escher. I tried to create my own images and even built a “home-brew” computer. But this was the 1970s, and the capabilities of my computer were much too limited to accomplish anything significant.

In the 1980s, I was working at the Freie Universität Berlin. The quasi-periodic Penrose tiling had been discovered, which was very exciting. Hans-Ude Nissen of the ETH Zurich showed me his images of a quasi-periodic structure of 12-fold rotational symmetry in Ni-Cr particles. I noted that we could superpose two periodic hexagonal lattices to create a new quasi-periodic tiling with this symmetry. You can see it on the current cover of Science. But this work had more scientific than artistic value, because again my equipment was very limited.

After encountering this hurdle, I became frustrated and didn’t try to create images with a computer for a long time. Instead, I took to drawing on paper (see a sampling of my sketches in Figs. 3 and 4), thinking about tilings, and dreaming about really powerful computers. I collected books on geometry, such as The Symmetries of Things by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss. Although I admired the images, I noted that I didn’t really understand their geometry if I couldn’t reproduce them. Being able to manipulate and modify these images would have been very helpful. However, this is very difficult to do on paper.

Fig. 3

Metamorphs. This sketch (a bit unusual because it has no handwriting) is a summary of tilings with hexagonal symmetry and their grids. It shows how the tilings can be transformed into each other by changing the weight of the grid lines.

Credit: P. Stampfli
Fig. 4

Tiling intersection—a typical sketch depicting tilings and their duals. The notes refer to calculating the intersection of lines, which results if one superposes two grids. The calculations are necessary for a program that generates the Stampfli tiling.

Credit: P. Stampfli

Around 2010, I noted that our home computers finally had all of the capabilities I needed. Additionally, I had access to Processing, a programming language that makes it easy to create images. So I had a new start. I begun creating tilings and publishing them in my blog I felt that it would not be enough to simply show the final tilings and discuss how they were made. This would leave a big gap: the computer code needed to create the images. Because this code is quite complicated, I decided to publish it in my blog as well. Thus, the tilings became open source. Now others can more easily do their own experiments with these software tools and gain a better understanding of the tiling geometry. Also, I hoped that they would create more interesting images than I do! An excellent example is this week’s Science cover image, done by Val Altounian.

Apart from tilings, I create images using fractals and other ideas. Currently, I am programming kaleidoscopes that create periodic images in curved space. All rotational symmetries become possible, such as the fivefold rotational symmetry of a decoration of a sphere with icosahedral symmetry. For more details, have a look at my new site In particular, it contains a kaleidoscope browser application (Fig. 5) that you can use to create your own images. At the time being, it is not fully touch-enabled, so you should use it on a laptop or desktop computer.

Fig. 5

A screenshot of the kaleidoscope app. Its result is shown at left. Part of the input image is visible at top right; controls are at bottom right.

Credit: P. Stampfli

I think that creating images with the computer is much more rewarding than playing computer games. Have a go at it—my code repository might be helpful.

–Peter Stampfli

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