Chemical Solution for the Einstein Project

The Einstein problem is a mathematical question that was solved in 2023 by the English amateur mathematician David Smith [1]. It asks: Can an infinite plane be seamlessly tiled using only a single shape in such a way that the resulting pattern never repeats? The term “Einstein” is a wordplay on the German phrase ein Stein (meaning “one stone”) and does not originate from the physicist Albert Einstein.

Karl-Heinz Ernst, Empa, Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland, Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic, and University of Zürich, Switzerland, Narcis Avarvari, University Angers, CNRS, France, and colleagues have discovered that during the crystallization of tris(tetrahelicenebenzene) or t[4]HB on a silver surface, instead of the expected regular structure, irregular patterns form that never seem to repeat. Moreover, each time the experiment is repeated, different patterns emerge.

The researchers originally aimed to better understand chirality and chose a molecule that easily switches its chirality at room temperature. They expected the molecules to arrange themselves according to their chirality—either alternating or forming groups with the same chirality. Instead, the molecules assembled seemingly at random into triangles of varying sizes, which in turn formed irregular spirals on the surface—creating a non-repeating or aperiodic structure.

The team found that the arrangement of the molecules was not entirely random. The molecules formed triangles with sides measuring between two and fifteen molecules. In each experiment, one triangle size dominated, while triangles of one size larger and one size smaller were also present, but no others.

The researchers explain this that under the experimental conditions, the molecules “prefer” to cover the silver surface as densely as possible, as this is energetically most favorable. However, due to chirality, the triangles they form do not fit together perfectly at the edges and must be slightly offset. To maximize surface coverage efficiency, both smaller and larger triangles are necessary. This arrangement also leads to defects in certain areas, which can become the center of a spiral.

Defects are usually energetically unfavorable, but in this case, they enable a denser arrangement of triangles. This balance also explains why the researchers never observed the same pattern twice: if all patterns have the same energy state, entropy determines the outcome.

Surfaces with atomic- or molecular-scale defects can show unique properties. Specifically, for an aperiodic surface, it has been predicted that electrons behave differently, potentially giving rise to a new kind of physics. To investigate this, the aperiodic molecule would need to be studied under the influence of magnetic fields on a different surface.