Over the next year, Domokos and his colleagues used geometric thinking to unpack the rules of molecular self-assembly — devising a new way to constrain the mosaics that molecules can form, using only the simple geometry of tessellation.

“At first, they did not believe that you can do it,” Domokos said. “They were doing artificial intelligence, supercomputing and all this kind of jazz. And now they are just looking at formulas. And this is very relaxing.”

From Planets to Atoms

 After Decurtins got in touch, Domokos tried to sell the idea to Krisztina Regős, his graduate student. Decurtins had sent a handful of images depicting patterns at an atomic scale — tilings of a molecule that had been designed and synthesized by his colleague Shi-Xia Liu — viewed through the eye of a powerful microscope. Domokos wanted to see if Regős could use the geometry that he had originally developed to describe geological fractures to characterize the patterns in Decurtins’ images.

To get started, Regős treated the 2D materials as simple polygonal tessellations — patterns that fit together with no gaps and infinitely repeat. Then, following Domokos’ approach, she calculated two numbers for each pattern. The first was the average number of vertices, or corners, per polygon. The second was the average number of polygons surrounding each vertex.

Together, those two average values are like a pattern’s GPS coordinates. They give its location within a landscape of all possible tessellations.

This landscape is called the symbolic plane. It is a simple 2D grid with the average number of shapes per vertex on the x-axis and the average number of vertices per shape on the y-axis. Each tessellation should plot to exactly one point within the plane. A perfect honeycomb pattern, for instance, is a tessellation of six-pointed hexagons that meet in trios at each vertex — a point at (3, 6) in the symbolic plane.

But most natural mosaics, from rock cracks to molecular monolayers, are not perfectly periodic tessellations.

For example, the cells of a real wax honeycomb are not all perfect hexagons. Bees make mistakes. But messy as it may be, a honeycomb is still, on average, a honeycomb. And on average, it still plots to a point at (3, 6) in the symbolic plane. Rather than being an oversimplification, Domokos’ method of calculating averages is insightful, said the mathematician Marjorie Senechal of Smith College, who reviewed the new study. By throwing out the mistakes and treating patterns as averages, it reveals a sort of ideal reality that’s normally buried beneath heaps of happenstance.

But when Regős tried to apply this method to Decurtins’ molecular pictures, she quickly ran into trouble. “I started to put them on the symbolic plane,” she said, “and then I realized that I can’t.”

The problem was scale. Unlike the geological patterns Domokos had worked with before, the molecular mosaics are really patterns within patterns. Viewed at different magnifications, they have different geometries. Regős couldn’t describe the molecular mosaics with a single pair of values because the patterns plotted different points on the symbolic plane, depending on an image’s magnification. It was a bit like zooming in on a hexagonal tiling and finding that its basic building blocks are really triangles.