The Chaos Causing Solids

6 min readJun 23, 2021

**A Ho-Mg-Zn icosahedral synthetic quasicrystal. Image Credit: US Department of Energy**

I vividly remember my crystallography professor repeating the same mantra every other laboratory session during my first-year bachelor’s classes, “Crystals can only exhibit 2, 3, 4, and 6 fold rotational symmetry.” He’d say. Intuitively, it made sense. We couldn’t imagine stacking the 14 bravais lattices in any way to make a solid with, say, five symmetrical faces, without leaving a considerable amount of gaps in between. Scientists had studied the structure of crystalline solids for ages, and it was in consensus on how many kinds of basic crystal structure may exist. As a result, we all considered this declaration to be the absolute, unquestionable truth.

It was about a few days back I stumbled upon an article by nature, ‘First nuclear detonation created ‘impossible’ quasicrystals’. On a quick google search for the definition of quasicrystal occurred my first moment of disbelief. A material that could exhibit 5 fold symmetry? It would be an understatement to say ‘my lecturer dumbed it down for us’ about something that seemed so forbidden in its existence.

With a combination of wonder and inquisitiveness about the occurrence of these solids, I immediately started digging a little deeper. Based on my gasp-inducing discovery of quasicrystal facts, I immediately decided to put them all in one nice article for the benefit of other crystal and mineral geeks like myself.

Underdog of the crystallography world

A normal crystal has two types of symmetry: translational symmetry, and rotational symmetry. It means the pattern repeats itself as we transit along with the 3 crystallographic directions, as well as when we rotate the crystal at a certain degree (180°, 120°, 90°, or 60°). Hence crystals are periodic and symmetrical. On the other extremity lie amorphous material with neither periodicity nor any observable symmetry in their structure.

Quasicrystals (QC) are ordered structures but aperiodic, or rather, quasiperiodic. Their structures lie intermediary between crystalline and amorphous materials. We can think of them as crystals in a higher dimension. Hence, they will exhibit crystal-like periodicity and symmetry when observed in a higher dimension (say 6-D), but their projection in a 3-D space makes them lose their periodicity. Sounds straight out of science fiction, doesn’t it? Hold your horses, we’re just getting started!

To understand how a 3-D space can be filled with a material with 5-fold symmetry, first, we need to understand how we can fill a 2D space with the same. It was assumed that a tile with a non-repeating pattern exhibiting such symmetry could never be created for generations. Robert Berger was the first to construct a non-repeating pattern using 20,426 distinct shapes. Other scientists worked out how to make the pattern using smaller and smaller shapes over time until Roger Penrose in the 1970s was able to reduce it to just two shapes. The pattern famously came to be known as the ‘Penrose Tile’.

**Penrose Tiling designed by Roger Penrose. Image Credit: (Flicker et al., 2020)**

More than what meets the eye…and the microscope

When Penrose tiling was established, it was believed to be purely theoretical. Nobody expected them to occur in nature. It was in 1982 that an Israeli scientist Daniel Shechtman was doing his routine lab work. On his day’s task list was analyzing a few samples on an electron microscope, one of them being a unique aluminium-manganese alloy…the sample that would unknowingly change his life. The resultant diffraction pattern of that alloy was the very first observed pattern of five-fold symmetry. The result surprised and confused him as he verified it repeatedly. When he took to sharing the same with colleagues, he was mocked by many and even asked by his boss to leave his research group. Double Nobel laureate Linus Pauling went on to famously say, “There is no such thing as quasicrystals, only quasi-scientists.” For several years following that there was dramatic conflict and debate on the existence of these crystals.

“There is no such thing as Quasi Crystals, only quasi-scientists” — Linus Pauling

**Transmission electron microscopy diffraction pattern of a quasicrystal, exhibiting a 5-fold or an apparent 10-fold rotational symmetry. Image Credit: (Macia, 2011)**

But as they say, the truth always prevails; eventually, many scientists started looking into the findings of Dr. Schechtman with healthy scepticism. Other crystals with seemingly inconceivable patterns, such as eight and twelvefold symmetry, soon surfaced. One of the biggest contributors to the initial observation of 5-fold symmetry was Dr. Mackay’s experiment, where he used a Penrose mosaic as a diffraction grating. He obtained 10 bright dots in a circle, a pattern replicating what Dr. Schechtman had first observed. Putting two and two together, physicist Paul Steinhardt and Dov Levine realized the quasicrystals were the 3D equivalent of the Penrose tiles. The discovery ultimately led to Dr. Daniel Schechtman winning the 2011 Nobel Prize in Chemistry.

Over the years, we have not only developed perfect crystals exhibiting quasiperiodicity but also to everybody’s surprise, found an extremely rare naturally occurring mineral with a quasiperiodicity from a meteorite found in the Koryak Mountains of Far Eastern Russia.

At the heart of nature’s mystery lies another mystery

While the existence of 5 fold symmetry had started to gain some acceptance, the mystery behind the structure of such material still loomed. How exactly were the atoms arranged?

While Doughlas Adams would have you believe that the answer to everything is 42, other scientists will wager on something else altogether. A number continues reappearing in different forms in nature, be it in the human skull or in the ammonite shells. Yes, of course, I’m referring to the Golden Ratio (τ).

In an attempt to study the internal structure of quasicrystals, the scientists observed the golden mean popping up over and over again. For instance, the ratio between the prolate and oblate rhombuses in the Penrose tiling approaches τ as the pattern goes on increasing to infinity. Additionally, when you draw the ‘Ammann’ lines on the Penrose tile, the parallel lines are observed to occur at two different repeating distances, short (S), and long (L). The pattern in which these distances repeat follow Fibonacci’s sequence, whose ratio of two consecutive numbers is known to converge to the golden ratio.

**Ammann Lines on the Penrose Mosiac. Image Credit: (Walter, 2006)**

While these facts sound jaw-dropping, the meaning and importance of phi have been conceptualized by humans themselves. But, whether they actually play a role in the existence of these forbidden structures, or humans simply like to look at the world through phi-tinted glasses, the quasicrystal discovery has nonetheless opened up the possibility of a whole new class of solids, and with them, their exciting new properties and applications!

References and Further Reading

The Chaos Causing Solids

Written by Urja Zaveri

No responses yet