In 2018, on an afternoon at the Swiss Federal Laboratories for Materials Science and Technology (Empa), chemist Karl-Heinz Ernst stared at his microscope, frowning. His doctoral student Jan Voigt had just deposited an organic molecule called tris(tetrahelicenebenzene) onto a silver surface. According to the textbooks, these chiral molecules should have arranged themselves into a neat crystal lattice—left-handed molecules clustering together, right-handed ones grouping up, or left and right alternating. That is the standard playbook for chiral molecular crystallization.
But the microscope showed nothing orderly. No tidy rows, no repeating pattern. Instead, there was a jumble of triangles of different sizes, fit together crookedly, like building blocks knocked over by a mischievous child. Even stranger: Voigt repeated the experiment about 100 times, and every run produced a different pattern—none of which ever looked like it would repeat.
"The experiment must be wrong." That was Ernst's first reaction. He had Voigt do it again. And again. And again.
The result was always the same: the molecules stubbornly refused to form any repeating pattern.
A Math Problem with a Pun Hidden in Its Name
To understand Ernst's confusion, we have to go back to a problem that had troubled mathematicians for over half a century.
In the 1960s, logician Robert Berger constructed a set of 20,426 shapes that could tile the entire plane, but whose pattern would never repeat. A tile set that "can tile but not repeat" is called an aperiodic tile set by mathematicians.
Over the following decades, mathematicians kept whittling the number down. Roger Penrose reduced it to two in the 1970s—the famous Penrose tiles, which still decorate the grounds of Oxford University.
But the most fundamental question remained: can a single tile do it?
German geometer Ludwig Danzer gave this hypothetical single tile a name: "einstein"—not in homage to the famous physicist, but as a German pun: "ein Stein," meaning "one stone."
One stone to cover an infinite plane, with a pattern that never repeats. It sounded like an impossible task, and mathematicians had begun to doubt it existed.
A Retired Print Technician's Kitchen-Table Experiment
In November 2022, in the seaside town of Bridlington, England, retired print technician David Smith was sitting at his computer doing what he loved most—playing with shapes. Smith is a puzzle enthusiast who enjoys fiddling with geometric figures using software called PolyForm Puzzle Solver. That day, he casually assembled an unremarkable hat-shaped tile, then tried to tile the screen with copies of it.
Usually, the tiles he created either quickly fell into a repeating pattern or failed to extend very far. But the hat tile was different. 30 tiles with no repetition, then 60. Smith printed out 60 paper copies and cut them out to tile on his table—still no repetition.
Smith sent the discovery to Craig Kaplan, a computer scientist at the University of Waterloo in Canada. Kaplan tested it with his own program; typically, a tile reveals its repeating nature within 6 rings of expansion. The hat tile? Still going at 16 rings. Kaplan stopped the program—they already had enough data.
In March 2023, Smith, Kaplan, and two other collaborators officially announced: the hat tile is the einstein that mathematicians had sought for 60 years.
The news shook the mathematical world. Doris Schattschneider, professor emerita at Moravian University, used the word "flabbergasted." Marjorie Senechal, professor emerita at Smith College, called it "mind-boggling."
Even more remarkably, this was not the first time amateurs had made major breakthroughs in tiling problems. In the 1970s, postal clerk Robert Ammann independently discovered a variant of the Penrose tiles; in 1975, California housewife Marjorie Rice found new families of pentagonal tilings; amateur mathematician Joan Taylor from Tasmania discovered the Socolar-Taylor tiles.
As Professor Senechal put it: "Perhaps amateurs are not as constrained as mathematicians by knowing how difficult something is."
Chemists Puzzling at the Kitchen Table
Back to Ernst's lab. When the hat tile was published, Ernst and Voigt had already been puzzled by their inexplicable molecular patterns for five years.
They had originally set out to study how chiral molecules crystallize on metal surfaces. Chirality—a molecule's "handedness"—is crucial in the pharmaceutical industry: more than half of modern drugs are chiral molecules, and all the amino acids, sugars, and proteins in the human body share a single handedness. Taking the wrong handedness of a drug can be ineffective at best and lethal at worst. So controlling chirality is a major concern in the chemical industry, and surface crystallization is one of the cheapest and most effective methods of controlling it.
Ernst's chosen molecule has a special ability: it can easily flip between left- and right-handedness at room temperature, which most chiral molecules cannot do. The researchers expected to see neat chiral separation—left with left, right with right.
Instead, the molecules gave a completely unexpected answer.
They formed triangles of varying sizes—with edge lengths ranging from 2 to 15 molecules. In each experiment, one size of triangle dominated, with triangles one size larger and smaller also present, but all other sizes entirely absent. These triangles then fit together, but their edges could not meet perfectly due to chirality mismatches and had to shift slightly. The misalignment created defects, and the defects became the centers of spirals.
The entire surface formed a pattern that never repeats.
Like true puzzle enthusiasts, Ernst and Voigt not only ran computer simulations but also assembled cardboard pieces at their kitchen tables at home—just as David Smith had assembled paper hat tiles.
When the hat tile paper appeared in 2023, Ernst finally found the key to understanding his experiment: what his molecules were doing on the silver surface was, in essence, the same thing Smith's hat tile was doing on the plane.
Entropy: Order in the Most Disordered Way
But there is one key difference between molecules and tiles. The hat tile is a mathematically rigorous aperiodic monotile—it can only tile the plane non-periodically. The molecules' aperiodicity is more of a "statistical preference"—they tend not to repeat, but it is not their only option.
The driving force behind this preference is surprising: entropy.
Under Ernst's experimental conditions, the molecules "want" to cover the silver surface as densely as possible, because that is the lowest-energy state. But chirality prevents the triangle edges from meeting perfectly, forcing misalignments. Misalignment produces defects, and defects are energetically unfavorable—but the denser packing they enable compensates for that energy cost.
Here is the key: since almost all non-repeating arrangements carry roughly the same energy cost, entropy becomes the deciding factor. In statistical mechanics, when multiple states have the same energy, the system tends to occupy the state with the most possible configurations—the state of maximum entropy. Non-repeating arrangements vastly outnumber repeating ones, so the molecules naturally flock to the non-repeating ones.
This is what physicists call "order by disorder."
It sounds paradoxical, but consider shuffling cards: a deck has 52! possible arrangements, and only a tiny fraction are ordered (by suit and rank). If you shuffle randomly, you will almost never produce an ordered arrangement—not because some force prevents it, but because there are so many more disordered states.
The molecules are the same. They are not "choosing" aperiodicity; aperiodicity is simply the most natural outcome.
From Quasicrystals to New Physics
The story has an even deeper layer.
In 1982, Israeli materials scientist Dan Shechtman saw an impossible diffraction pattern in an aluminum-manganese alloy—tenfold rotational symmetry. This meant the crystal looked identical every 36 degrees of rotation, which was forbidden in classical crystallography. Shechtman wrote "(10 Fold ???)" in his lab notebook, then spent the whole afternoon trying to find a twin crystal that could explain the phenomenon—if it were just two ordinary crystals grown together, it would be nothing interesting.
He found no twins.
When he told his colleague John Cahn, Cahn's first reaction was: "Go away, Danny, those are just twins, nothing interesting."
But Shechtman persisted. He faced not only academic skepticism—two-time Nobel laureate and the most famous chemist of the 20th century, Linus Pauling, publicly opposed the existence of quasicrystals, reportedly saying: "There is no such thing as quasicrystals, only quasi-scientists."
Pauling kept trying to explain Shechtman's data with increasingly elaborate twinning models, failing again and again. He died in 1994 still not accepting quasicrystals. Shechtman won the Nobel Prize in Chemistry in 2011.
The discovery of quasicrystals rewrote the definition of a crystal. In 1991, the International Union of Crystallography redefined "crystal," no longer requiring periodic arrangement, only sharp diffraction spots. Quasicrystals also have unusual physical properties: anomalously low electrical conductivity, high hardness, and low friction, with practical applications in non-stick pan coatings and pen tips.
Ernst's molecularly aperiodic surfaces may push things further. Physicists have predicted that on aperiodic surfaces, electrons behave very differently than in ordinary crystals—potentially giving rise to an entirely new physics. Felix Flicker, a physicist at the University of Bristol, has even built computer simulations of quasicrystals using the hat tile, predicting it would exhibit "super-graphene"-like properties.
Nature Doing Math
Ernst said something in an interview worth savoring:
"This is nature doing math."
There are several layers to this statement.
First: the molecules never studied the einstein problem, never read David Smith's paper, never heard of aperiodic tiling. Yet they found the answer themselves—or rather, the answer was sitting in the laws of physics, waiting to be discovered.
Second: mathematicians and chemists arrived at the same endpoint from completely different directions. Smith used intuition and paper cutouts; Ernst used molecules and a microscope. One started from abstraction, the other from experiment, separated by 60 years of mathematical research and 5 years of confusion—but ultimately saying the same thing.
Third, and deepest: the boundary between order and disorder is far blurrier than we assume. We habitually equate "order" with repetition and "disorder" with chaos. But einstein tiles and molecular aperiodic surfaces tell us there is a third state—structured but non-repeating, rule-governed but non-cyclic. Quasicrystals are the physical embodiment of this state.
This suggests an analogy. In AI, when we train language models, we also wrestle with "order" and "disorder." A model that is too ordered produces repetitive boilerplate ("As an AI language model..."); too disordered, and it produces gibberish. The best output—insightful, creative, structured without mechanical repetition—lives precisely in that "aperiodic" middle ground.
Perhaps creativity itself is an einstein tile: patterned, but not repeating; rule-bound, but not cyclical. And the force driving this state is nothing mysterious—just simple statistics: the state with the most possibilities happens to be the most interesting one.
Ernst has now retired, leaving the study of electron behavior on aperiodic surfaces to others. "I have a bit of awe toward physics," he says with a smile.
But the molecules have no awe. They simply keep tiling the silver surface, over and over, with patterns no one has ever seen.
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*References:*
- *Voigt, J. et al. "An aperiodic chiral tiling by topological molecular self-assembly." Nature Communications (2025).*
- *Smith, D. et al. "An aperiodic monotile." arXiv:2303.10798 (2023).*
- *Shechtman, D. et al. "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry." Physical Review Letters (1984).*