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Patterns, Symmetry, and Solids
Joung Real Ahn, Sung Joon Ahn
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Patterns, Symmetry, and Solids


[Reproduced from the crossroads webzine]


Illustration by Kim Min-jeong

"Oddly satisfying photos to soothe your soul." "Give your eyeballs a massage with these perfect images."

No sooner have you forgotten one than another of these titles pops up on social media sites or internet community bulletin boards. Typically, a look at these posts shows a collection of pictures of objects in neat and orderly arrangements. They tend to get a lot of "likes"—a reflection, perhaps, of the resonance they hold for many. Well-organized patterns bring peace to your mind and boast beauty in themselves. It is this comforting and aesthetic quality that results in an appeal that transcends generations and cultures. Obviously, there are always a few impish replies offering something quite different from what the original poster intends—and including "infuriating" photographs to boot ...


Fig. 1: Photographs that relax our minds, and others that cause discomfort.

We can apply the concept of symmetry to explain the sense of comfort and beauty we feel when looking at neatly arranged objects. Simply put, symmetry refers to a situation where we can swap parts of an image and not see any difference from how it was before. Most of the things in nature that human beings perceive as "beautiful" have their own unique form of symmetry. For instance, the patterns on a butterfly's wings and the facial features of attractive actors exhibit mirror-like symmetry. Leaves neatly growing on a tree branch show parallel symmetry, and many flowers have their own particular rotational symmetry; snow crystals possess both rotational symmetry and expanding symmetry. Noting the beauty that such symmetry conveys, people since the dawn of civilization have introduced different patterns into their earthenware decorations, tomb paintings, and fabric weaving. Those ancient traditions have been carried on to today, with innumerable forms of symmetry to be found throughout our daily lives in places like the wallpaper in our homes, the tiles on building exteriors and floors, the pavement blocks on the roadside, and the stained glass of cathedrals, to name a few.


Fig. 2: Symmetry found in nature and created artificially.

Symmetry and patterns aren't just beautiful—they're also highly efficient. Perhaps they might not be used so widely today if they were merely pretty. Pavement blocks are a good illustration of the efficiency of patterns: one shape of block is enough to evenly and thoroughly fill even a very broad space. Another superb example of efficiency can be found in the desktop background images used in the days of Windows 95. Back then, computers fell far short of their counterparts today in terms of performance and storage capacity. Recalling the performance properties of the computer I used back in the day (a top-of-the-line model that I paid quite a bit of money for), I'd say it had about 32 megabytes of RAM and a 3.2-gigabyte hard drive; in comparison, computers today have RAM capacities roughly 500 to 1,000 times larger, and hard drive capacities about 1,000 to 2,000 greater. As an inexpensive way to fill the background with beautiful images without wasting precious capacity and performance, the use of regularly repeated patterns of small images may have been the best choice available to developers at the time.


Fig. 3: The efficiency of patterns: pavement blocks and Windows 95 desktop background patterns.

The efficiency of symmetry and patterns is a matter closely tied to research in solid state physics. The chief focus of solid state physics research is the crystal. A crystal is a substance in a solid state with regularly arranged atoms or ions. A crystal is produced through the combination of a lattice and a basis, where the lattice is the crystal's cycle (its parallel symmetry) and the basis is some repeated basic unit. By way of analogy, the lattice could be said to indicate a tile's size and shape, while the basis refers to the identical image depicted on each tile; arrange these tiles close together, and you obtain a large wall (crystal) with a repeating pattern. Once you've understood the larger wall as a combination of the repeated tiles (lattice) and the images within the tiles (basis), you can understand the entire wall (the material properties of the crystal as a whole) simply by analyzing a single tile (lattice unit)—even when the wall is very large indeed. This is the basis approach to researching materials in modern solid state physics.


Fig. 4: Salt as seen by the eye and in atomic form; a lattice and basis as the two constituent elements of a crystal (lattice + basis = crystal).

As mentioned above, a crystal is defined by the parallel symmetry of its lattice, but it can also possess its own rotational symmetry depending on the lattice's form. If the lattice is an ordinary parallelogram, for example, the crystal may possess symmetry with regard to a 180-degree rotation. (In other words, the crystal can be rotated by 180 degrees without any visible difference from the original crystal.) If the lattice is square in form, it may exhibit symmetry with regard to a 90-degree rotation; if it is a hexagon, it could have symmetry with regard to a 60-degree rotation. (As the figure below shows, even when the lattice itself exhibits 180-degree, 90-degree, or 60-degree rotational symmetry, it may not always exhibit rotational symmetry with regard to the angle in question, depending on the structure of the base.) There are five forms of rotational symmetry that a crystal can possess, with possible rotational angles of 360 degrees, 180 degrees, 120 degrees, 90 degrees, and 60 degrees (also referred to as 1-, 2-, 3-, 4-, and 6-fold symmetry). Crystals with other rotational symmetries besides these cannot exist; if we attempt to form a lattice with 72-degree rotational (5-fold) symmetry, we see that it does not achieve the parallel symmetry that is necessary for a crystal, as shown with the regular pentagons in the fourth image below.


Fig. 5: Examples of the possible rotational symmetry of lattices, and the reason a lattice with 72-degree rotational symmetry is not possible.

In 1982, an electron diffraction image observed by Israeli physicist Dan Shechtman (below left, showing the measurement of rapidly cooled sample of a metal alloy) sent shock waves through the fields of solid state physics and crystal studies. Alongside x-ray diffraction devices, electronic diffraction devices are some of the most important equipment in the study of crystals. As electron rays are fired at a crystal, the electrons diffracted as they collide with the crystal specimen form specific diffraction patterns on the screen, which indirectly show the crystal structure of solids. The two middle images below are diffraction patterns observed in specimens with 90-degree and 60-degree rotational symmetry, respectively. We can see that the diffraction patterns also exhibit 90-degree and 60-degree rotational symmetry. But a closer look at Shechtman's experimental findings shows a diffraction pattern with 36-degree (actually 72-degree) rotational symmetry. As noted in the preceding paragraph, it was more or less common wisdom among scholars that solid crystals cannot have rotational symmetries beyond the five types mentioned (360-degree, 180-degree, 120-degree, 90-degree, and 60-degree), yet here was a finding that seemed to utterly refute that. After closely analyzing the results, Shechtman dubbed this a "quasicrystal"—a new form of solid—and published his results in an academic journal. The findings were too shocking for the crystal community to readily accept, however. Shechtman was kicked out of his laboratory and lambasted by Professor Linus Pauling, one of the top authorities in the field at the time, who said there were "no quasicrystals, just quasi-scientists." But in a paper published later, Paul Steinhardt and Dov Levine were able to support Shechtman's findings by applying the concept of the Penrose tile (the far right image below). As more and more similar experimental results were published, Shechtman's quasicrystal findings were finally recognized by the academic community, and in 2011 he was awarded the Nobel Prize in Chemistry. Penrose tiles, the structures that account for the quasicrystal's structure and diffraction pattern, have a quite distinctive shape. Whereas the lattice structures that form crystals are shaped in such as a way that they occupy space as a single figure based on parallel symmetry, Penrose tiles fill space with two different forms of rhombus, producing 72-degree rotational symmetry. It's also common to find shapes that appear similar but are not completely identical; this quality (rotational symmetry and quasiperiodicity) may be seen as a unique structural property of the quasicrystal.


Fig. 6: Dan Shechtman's distinctive diffraction patterns and Penrose tiles.

Active research into quasicrystals has continued since then, with quasicrystal states reported in a variety of alloys; our research group recently published on the discovery of a new paradigm of graphene-based two-dimensional quasicrystals. A two-dimensional material consisting of a single layer of carbon, graphene has a hexagonal lattice structure with 60-degree rotational symmetry. Our team was the first in the world to discover a two-dimensional graphene quasicrystal with 30-degree rotational symmetry, which was the result of twisting two different sheets of graphene by 30 degrees and growing the layered structure. Like the Penrose tiles that supported Shechtman's experimental findings, the two-dimensional graphene quasicrystal with 30-degree rotational symmetry has a Stampfli-Inflation tile structure. The electron diffraction pattern confirmed the 30-degree rotational symmetry, while transmission electron microscopy (TEM) image analysis allowed us to directly verify the quasicrystal's Stamfli-Inflation tile structure. We were also able to confirm that the graphene quasicrystal possesses the expanding symmetry (self-similarity) of a fractal structure.


Fig. 7: Discovery of the graphene quasicrystal. Electron diffraction pattern, TEM analysis, expanding symmetry of the Stampfli-Inflation tile's fractal structure

Symmetry is a beautiful thing—but to physicists, it is truly beautiful, for it allows us to understand new natural phenomena and offers us new approaches and methods.

The flowering of spring is almost behind us as we head now into the summer. We hope that all of you reading this essay can take time from your busy schedules and find room in your hearts to enjoy the beauty of the different forms of symmetry that exist in nature or have been created by human beings. Thank you for taking the time to read this lengthy piece!




Joung Real Ahn is a professor at the Department of Physics of Sungkyunkwan University. He received his BA degree from the Department of Physics of Sungkyunkwan University and his MD and Ph.D. from POSTECH. His research interests include two dimensional materials.



Sung Joon Ahn is a post-doc at the Department of Physics of Sungkyunkwan University. He received his BA degree and Ph.D from the Department of Physics of Sungkyunkwan University. His research interests include two dimensional materials.

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