Foolproof 3D Quasicrystals

Growing Perfect Decagonal Quasicrystals by Local Rules

Hyeong-Chai Jeong

PRL 98, 135501 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e135501

Perfect Penrose Tiling (PPT)

Penrose developed a set of rules which allow a perfect tiling of the 2D plane in a non repeating pattern. These are called quasicrystals. His rules allow such a tiling, but they do not guarantee it. At the edges, there are legal attachments that introduce defects and prevent you from tiling the entire plane. However, another guy called Onoda showed that if you start with a defect in the center, then local growth rules will fill the rest of the plane with a PPT. The only defect is the seed at the center.

To extend 2D tilings to 3D crystals, people thought the best you could do was stacking 2D planes on top of one another. This would introduce a line defect where all the decagons overlap. The authors of this letter developed an algorithm that starts with two defects, but their vertical and lateral attachment rules allow you to fill the rest of space --- "the bulk" --- with PPTs.

A decapod has 10 edges, and the way these growth rules work is to assign an arrow to each edge. Since the arrow at each edge can point in either of two directions, there are 1024 possible decagons. Symmetry under reflection and rotation reduces this number to 62. I'd like to see how that counting works, because 2x31 seems a difficult number to get from 2^10! Anyway, there are 62 unique decagons that will fill a plane with local growth rules. Of these, only one can be filled in: the cartwheel.

Using a special decagon at the bottom and a cartwheel on top of it, the authors were able to add a vertical growth rule that would overcome any dead zones. As a result, they are able to grow a 3D quasicrystal from a single point defect.

Who cares about local growth rules? Well, Nature for one. If I had a set of Penrose tiles (which I'd love to get my hands on), I could sit and meticulously place them one by one until I used up my bag, with a perfect tiling. However, Nature might not be as attentive as me. She'd probably take a tile and try to fit it at an edge. If it would stick (i.e. satisfied local growth rules), Nature would be happy and move on to the next tile. After enough of this, she'd eventually end up with nothing but dead zones and defects. Nature would have to be really lucky to make a quasicrystal as large as mine.

With a decagon in the middle, though, Nature couldn't go wrong. Every tile that fit on an edge would continue the pattern perfectly. Thus, a point defect at the center, the decagon seed, would allow a random growth algorithm to build a perfect quasicrystal except for the defect. The beauty of the author's work is that he showed you only need a point defect --- not an infinite line of them --- to do the same thing in three dimensions. He came up with a simple set of rules that make it impossible to mess up the tiling if you start with two special decagons stacked on top of each other. It's foolproof!