An Unexpected Experiment
One afternoon in 2018, chemist Karl-Heinz Ernst at the Swiss Federal Laboratories for Materials Science (Empa) 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 organized into a tidy lattice—left-handed molecules clustering together, right-handed ones grouping, or the two alternating. That is the standard playbook for chiral molecular crystallization.
Instead, the microscope showed nothing orderly. No rows, no repeating units—just a jumble of triangles of different sizes, fitted together crookedly like dropped building blocks. Strangest of all, Voigt repeated the experiment roughly 100 times and every pattern was different, with no visible repeats.
"The experiment went wrong," was Ernst's first reaction. He had Voigt do it again. And again. The result never changed: the molecules stubbornly refused to form any repeating pattern.
A Math Problem Hidden in a Pun
To understand Ernst's confusion, you 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 cover the entire plane but never produce a repeating pattern—what mathematicians call an "aperiodic tiling set."
Over the following decades mathematicians kept shrinking the number. Roger Penrose reduced it to two in the 1970s—the famous Penrose tiles, still embedded in the ground at Oxford.
But the fundamental question remained: could a single tile do it?
German geometer Ludwig Danzer gave this hypothetical tile a name: "einstein"—not in honor of the famous physicist, but a German pun: "ein Stein," meaning "one stone." One stone, tiling an infinite plane, never repeating. It sounded impossible, and mathematicians had begun to doubt it existed.
A Retired Print Technician's Kitchen Table
In November 2022, in the English seaside town of Bridlington, retired print technician David Smith was doing what he loved—playing with shapes. Using software called PolyForm Puzzle Solver, he assembled an unremarkable-looking hat-shaped tile and tried to tile the screen with copies.
Usually his tiles either quickly fell into repetition or failed to extend far. But the hat tile went 30 tiles without repeating, then 60. Smith printed and cut out 60 paper copies and puzzled on his kitchen table—still no repetition.
He sent his discovery to Craig Kaplan, a computer scientist at the University of Waterloo. Kaplan's own program typically caught repetition within about six layers of ring-like expansion. The hat tile passed 16. Kaplan stopped the program—they had enough data.
In March 2023, Smith, Kaplan, and two coauthors formally announced: the hat tile is the einstein mathematicians had sought for 60 years.
The news stunned the mathematical world. Doris Schattschneider, professor emerita at Moravian University, said she was "flabbergasted." Marjorie Senechal of Smith College called it "mind-boggling."
Remarkably, this was not the first time amateurs had made breakthroughs in tiling: postal clerk Robert Ammann independently found a Penrose tile variant in the 1970s; California housewife Marjorie Rice discovered new pentagonal tiling families in 1975; Tasmanian amateur Joan Taylor co-created the Socolar–Taylor tile. As Senechal put it: "Perhaps amateurs are not as bound by 'how hard this is' as mathematicians are."
Chemists Puzzling at the Kitchen Table
Back in Ernst's lab, he and Voigt had spent five years baffled by their molecular patterns. They had originally set out to study how chiral molecules crystallize on metal surfaces. Chirality—molecular "handedness"—is critical in the pharmaceutical industry: more than half of modern drugs are chiral, and the amino acids, sugars, and proteins in the human body all share a single handedness. Taking the wrong handedness of a drug can be ineffective at best, lethal at worst. Surface crystallization is one of the cheapest and most effective ways to control chirality.
Ernst's molecule had a special ability: it can flip between left- and right-handed forms at room temperature, which most chiral molecules cannot. The researchers expected clean chiral segregation. Instead, the molecules gave a completely unexpected answer.
They formed triangles ranging from 2 to 15 molecules per side. In each experiment one size dominated, with neighboring sizes present but all others absent. Because the triangles' edges could not mate perfectly due to chirality mismatch, they had to shift slightly, producing defects—and the defects became the centers of spirals.
The entire surface formed a never-repeating pattern.
Like true puzzle enthusiasts, Ernst and Voigt not only ran computer simulations but assembled cardboard models at home—much as David Smith had cut out paper hat tiles.
When the einstein paper appeared in 2023, Ernst finally had the key: what his molecules were doing on silver 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 a key difference between molecules and tiles. The hat tile is rigorously proven to tile *only* aperiodically. The molecules' aperiodicity is more like a statistical preference—they *tend* not to repeat, but are not forced to. And the driving force is surprising: entropy.
Under Ernst's conditions, the molecules "want" to cover the silver surface as densely as possible, the lowest-energy state. But chirality prevents the triangles' edges from aligning perfectly, forcing misalignments. Misalignment creates defects, which are energetically unfavorable—but the denser packing compensates for the energy cost.
Here is the crucial point: since all the non-repeating arrangements cost roughly the same energy, entropy becomes the deciding factor. In statistical mechanics, when many states have equal energy, the system favors the state with the most possible configurations—the highest entropy. Non-repeating arrangements vastly outnumber repeating ones, so the molecules naturally flood into aperiodicity.
This is what physicists call "order by disorder."
It sounds paradoxical, but consider shuffling cards: a deck has 52! arrangements, and only a tiny fraction are ordered. Shuffle randomly and you will almost never get an ordered deck—not because some force prevents it, but because disordered states vastly outnumber ordered ones. The molecules are not "choosing" aperiodicity; aperiodicity is simply the most natural outcome.
From Quasicrystals to New Physics
The story has deeper layers. In 1982, Israeli materials scientist Dan Shechtman saw an impossible diffraction pattern in an aluminum–manganese alloy: tenfold rotational symmetry—crystals that look identical every 36 degrees, forbidden by classical crystallography. Shechtman wrote "(10-fold???" in his lab notebook and spent an afternoon trying to find a twin-crystal explanation. He found none. When he told colleague John Cahn, Cahn's first response was: "Go away, Danny, those are just twins, nothing interesting."
Shechtman persisted despite academic hostility—two-time Nobel laureate Linus Pauling publicly opposed quasicrystals, reportedly saying: "There are no quasicrystals, only quasiscientists." Pauling tried increasingly elaborate twinning models, failing each time, and died in 1994 still unconvinced. Shechtman won the 2011 Nobel Prize in Chemistry.
Quasicrystals forced the International Union of Crystallography to redefine "crystal" in 1991—no longer requiring periodic arrangement, only sharp diffraction spots. They also have unusual properties: anomalously low conductivity, high hardness, low friction, used in nonstick coatings and pen tips.
Ernst's aperiodic molecular surfaces may go further. Physicists predict electrons behave very differently on aperiodic surfaces than on ordinary crystals—possibly enabling an entirely new physics. Bristol physicist Felix Flicker has even built quasicrystal simulations from hat tiles, predicting "super-graphene"-like properties.
Nature Doing Math
Ernst said something in an interview worth savoring:
"This is nature doing math."
It has several layers. First: the molecules never studied the einstein problem or read David Smith's paper, yet they found the answer—or rather, the answer was sitting in the laws of physics, waiting to be discovered. Second: mathematicians and chemists arrived at the same destination from opposite directions—Smith through intuition and paper cutouts, Ernst through molecules and microscopes—separated by 60 years of mathematics and 5 years of confusion, yet saying the same thing.
Third, and deepest: the boundary between order and disorder is far blurrier than we assume. We equate "order" with repetition and "disorder" with chaos. But einstein tiles and aperiodic molecular surfaces reveal a third state—structured but non-repeating, rule-bound but non-cyclic. Quasicrystals are its physical realization.
There is even an analogy in AI: language models struggle with the same tension. Too orderly and the output is repetitive boilerplate; too disorderly and it becomes gibberish. The best outputs—insightful, creative, structured but never mechanically repetitive—live in that "aperiodic" middle ground.
Perhaps creativity itself is an einstein tile: patterned, but never repeating; ruled, but never cycling. And what drives it is no mysterious force, 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 before.
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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).*