How a Scientific Heresy Rewrote the Rules of Matter
From a Nobel Prize-winning discovery to the patterns gracing ancient mosques, quasicrystals are a dazzling fusion of art, mathematics, and materials science.
Imagine a mosaic, perfectly fitted together, that never, ever repeats its pattern. No matter how far you extend it, you will never find a single tile that serves as the starting point for an infinite, repeating wallpaper. For centuries, scientists believed this was impossible for a crystal. The very definition of a crystal was an atomic structure with a repeating, periodic pattern. Then, in 1982, Dan Shechtman saw the impossible.
This is the story of quasicrystals—solids that break all the old rules. They are ordered but not periodic, a scientific paradox that forced textbooks to be rewritten and earned Shechtman the 2011 Nobel Prize in Chemistry. The field he spawned is beautifully detailed in Walter Steurer and Sofia Deloudi's seminal work, Crystallography of Quasicrystals, a map to this strange new world of matter.
Atoms in a crystal are arranged in a repeating 3D lattice, much like oranges stacked in a supermarket display. This periodic structure allows for only certain types of symmetry. You can rotate a classic crystal by 180°, 120°, 90°, or 60° and it will look the same. These are 2-fold, 3-fold, 4-fold, and 6-fold symmetries.
One rotation was considered impossible: 5-fold symmetry (72°). Try tiling your bathroom floor with pentagons; you'll always end up with gaps. Mathematicians had proven that a periodic pattern could not have 5-fold symmetry. It was simply incompatible with filling space.
Quasicrystals defied this fundamental law. They exhibit sharp, clear 5-fold symmetry in their diffraction patterns—the "fingerprint" of atomic arrangement—proving they are ordered. Yet, they lack a repeating unit cell, proving they are not periodic. They are aperiodic, living in a fascinating grey area between crystals and glass.
The discovery of quasicrystals is a masterclass in scientific rigor and the courage to challenge dogma.
Material: A rapidly cooled alloy of aluminum and manganese (Al₆Mn).
Technique: Melt-spinning. The molten metal is ejected onto a rapidly spinning copper wheel, cooling it at a rate of about a million degrees per second. This ultra-fast cooling prevents atoms from arranging into a stable, periodic crystal, instead "freezing" them in a novel, metastable structure.
Shechtman used a Transmission Electron Microscope (TEM). He fired a beam of electrons through a thin slice of his Al₆Mn sample.
As the electrons passed through the atomic lattice, they scattered and interfered with each other, creating a diffraction pattern on a fluorescent screen.
This pattern is a direct map of the symmetry and spacing of the atoms.
What Shechtman saw on that screen in April 1982 should not have existed. Instead of the fuzzy ring pattern of a glass or the symmetrical array of spots from a crystal with allowed symmetry, he saw a pattern of sharp, bright dots arranged in concentric circles.
Most stunningly, the inner circle had ten bright dots—exhibiting perfect 5-fold symmetry (10-fold is simply 5-fold symmetry viewed in a different way).
"10 Fold???" followed by "There is no such animal."
| Feature Observed | What It Means | Why It Was Revolutionary |
|---|---|---|
| Sharp Bragg Diffraction Spots | Indicates a highly ordered atomic structure with long-range order. | Proved it was not a disordered glass or amorphous material. |
| 5-Fold / 10-Fold Symmetry | The pattern remains identical when rotated by 36° or 72°. | Directly violated the crystallographic restriction theorem for periodic crystals. |
| Non-Periodic Spot Spacing | The ratios of the distances between spots were based on the irrational number φ (phi, the golden ratio ~1.618). | Proved the structure was not based on a repeating integer lattice, but on a quasiperiodic mathematical sequence. |
How can something be ordered but not repeat? The answer lies in a famous mathematical sequence and an irrational number.
0 1 1 2 3 5 8 13 21 ...
As the sequence grows, the ratio of successive numbers (e.g., 8/5, 13/8, 21/13...) gets closer and closer to the golden ratio, φ (~1.6180339887...).
Quasicrystals are essentially 3D projections of a perfectly periodic crystal that exists in a higher-dimensional space (e.g., 6D!).
Imagine the shadow of a 6D hypercube: the shadow can have 5-fold symmetry and a Fibonacci-like spacing, even though the original object is periodic.
| Feature | Classical Crystal | Quasicrystal |
|---|---|---|
| Atomic Order | Periodic (repeating unit cell) | Aperiodic / Quasiperiodic (no repeating unit cell) |
| Allowed Symmetry | 2, 3, 4, 6-fold | 5, 8, 10, 12-fold (and others) |
| Mathematical Basis | Integer mathematics | Irrational numbers (e.g., Golden Ratio φ) |
| Diffraction Pattern | Sharp spots with periodic spacing | Sharp spots with non-periodic, φ-based spacing |
| Dimensionality | Described in 3D | Often described as a projection from a higher (e.g., 6D) space |
Today, quasicrystals have moved from a controversial curiosity to a material with fascinating applications.
Used for frying pans and surgical tools due to their low friction properties.
Added to aluminum alloys to make strong, lightweight components.
Applied to engines due to their poor heat conductivity.
| Research Reagent / Tool | Function in Research |
|---|---|
| Aluminum-Transition Metal Alloys (e.g., Al-Mn, Al-Pd-Mn, Al-Cu-Fe) | The most common and stable base for forming quasicrystalline phases. Their specific atomic sizes and bonding characteristics favor aperiodic packing. |
| Melting Spinner / Rapid Quencher | A crucial device for synthesizing metastable quasicrystals by cooling molten metal at ultra-high speeds (~10⁶ K/s), freezing atoms in their non-equilibrium, quasicrystalline arrangement. |
| Transmission Electron Microscope (TEM) | The workhorse for discovery and analysis. It reveals the atomic-scale structure through electron diffraction patterns, directly showing the forbidden symmetries. |
| High-Dimensional Crystallography Software | Specialized computer programs used to analyze diffraction data by modeling the quasicrystal as a projection from a 4D, 5D, or 6D periodic lattice. |
The story of quasicrystals is a powerful reminder that scientific discovery often lies just beyond the edge of established knowledge. It teaches us to question definitions and to trust observation over dogma. Thanks to the pioneering work of Dan Shechtman and the detailed scientific mapping by researchers like Steurer and Deloudi, what was once "impossible" is now a vibrant and fruitful field of science, blending the beauty of mathematics with the utility of engineering and forever expanding our definition of order in the universe.