Remember those graph papers from school, covered in tiny squares?
They’re a perfect example of what mathematicians call periodic tiling—a pattern where shapes cover an entire area without gaps or overlaps.
If you slide or rotate the pattern by a specific amount (such as 90 degrees), the result looks exactly the same. This symmetry is the hallmark of periodic tiling.
But what happens when we step outside the neat, predictable world of squares? Imagine tiling your bathroom with pentagons instead.
It doesn’t work—the shapes leave gaps or overlap. This inability to fit together perfectly is why patterns like this don’t occur naturally in periodic systems.
Yet, in the 1970s, mathematician Roger Penrose found a way to create something extraordinary: a tiling pattern using two shapes that had rotational symmetry but no translational symmetry.
These patterns, now known as Penrose tilings, opened the door to an entirely new world of mathematics and materials science.
Fast forward to the 1980s, when physicist Dan Shechtman observed a similarly astonishing phenomenon in a material.
It was a pattern that seemed to defy the very definition of a crystal. What he discovered—and the ripple effects it caused—redefined our understanding of the material world.
Quasicrystals and a New Definition of Crystals
For centuries, crystals were thought to be periodic, like the graph paper pattern. But Shechtman’s discovery of a non-periodic yet rotationally symmetric structure in an aluminum-manganese alloy challenged this assumption.
The material exhibited symmetry when rotated by 72 degrees but lacked the repeating translational symmetry that defines traditional crystals.
This revolutionary structure was named a quasicrystal.
Initially, Shechtman’s findings were met with skepticism. Many scientists dismissed the idea as impossible, asserting that crystals without translational symmetry could not exist.
However, further studies confirmed the phenomenon, forcing the scientific community to redefine what constitutes a crystal.
Today, quasicrystals are recognized as a legitimate and fascinating category of materials. Their unique properties stem from an irrational number at the heart of their construction: the golden ratio (approximately 1.618).
This number, also known as φ (phi), governs the proportions of the shapes that make up these patterns.
The Role of the Golden Ratio in Quasicrystals
In Penrose tiling, the ratio of the side length of a pentagon to the side length of a star inscribed within it is the golden ratio.
This irrational number is key to understanding why quasicrystals lack translational symmetry but maintain rotational symmetry.
The structure’s beauty lies in its mathematical precision and its ability to never repeat.
Consider a three-dimensional quasicrystal. It exhibits five-fold rotational symmetry, which you can observe in its lattice structure.
In one example, the ten radial lines around a central point highlight the symmetry.
When you construct a model of this quasicrystal, using tools like Zometool, you can visualize the intricate arrangement of atoms or particles.
The white nodes represent atomic positions, while the connecting rods symbolize the bonds, revealing a pattern of unparalleled complexity.
How Quasicrystals Defy Nature’s Preferences
Nature typically favors periodic arrangements because they require minimal energy to assemble.
This is why crystals—from table salt to diamonds—have periodic structures. Quasicrystals, by contrast, are energetically less favorable and, therefore, rare in nature.
But their discovery has profound implications for both science and technology.
Researchers at the University of Leeds and other institutions have developed models to better understand how quasicrystals form.
These models identify two critical traits that enable their creation:
- Patterns at Two Length Scales: These scales must have an irrational ratio, such as the golden ratio.
- Strong Interactions Between Scales: These interactions stabilize the structure, allowing quasicrystals to form.
This research not only provides insight into the formation of quasicrystals but also helps scientists identify the conditions under which these patterns might arise naturally or be engineered.
From Lasers to Reflective Paint
Beyond their aesthetic and mathematical appeal, quasicrystals have exciting practical applications.
One promising area is in the development of high-efficiency lasers. Traditional lasers use periodic crystal patterns, which can limit their output power due to symmetry constraints.
By incorporating quasicrystal patterns, researchers can create lasers with fewer defects and higher peak output power.
This innovation could revolutionize industries ranging from telecommunications to medical devices.
Another intriguing application lies in surface coatings. Some researchers are exploring the use of quasicrystal patterns to create reflective finishes for household paint.
These coatings could offer unique optical properties, such as enhanced reflectivity and durability.
The Future of Quasicrystals
The study of quasicrystals is still in its infancy, but its potential is immense.
The mathematics behind these patterns offers a deeper understanding of symmetry and order in complex systems.
Moreover, the ability to design quasicrystals with specific properties could lead to breakthroughs in materials science, nanotechnology, and beyond.
As researchers continue to unravel the mysteries of quasicrystals, one thing is certain: these patterns, with their never-repeating yet harmonious structures, remind us that nature still has secrets waiting to be discovered.
From the elegance of Penrose tilings to the cutting-edge applications in modern technology, quasicrystals are a testament to the beauty and utility of mathematical innovation.
