# Ragged Edges

## The story of the quasicrystal

The story of quasicrystals is a strange one spanning six hundred years. It is a story marked by monsters, medieval mathematicians, higher dimensions, and extraterrestrial life. An impossible mathematical and material substance that denies our understanding of nature and yet recently found to exist within it. It is the story of the impossible becoming possible.

A quasicrystal is a material structure that hovers on the edge of falling apart. It is ordered but aperiodic. Unlike an ordinary crystal, whose molecular pattern is repetitive in all directions, the structural pattern of a quasicrystal never repeats itself. It is endless and uneven, described by the arrangement of modular parts. Small units aggregate to form larger figures that combine into even larger movements that are always a little bit different from the rest.

Quasicrystalline patterns have an infinite capacity to create and carry information. There is no end to the stories they can tell. The efficiency of modularity alongside the endless variety of pattern constitutes a step forward, or maybe a step sideways, for the discipline of architecture and design.

The infinite is immanent in every tile or, in terms of contemporary scientific thinking, the entire universe is contained within every piece of it. What is the importance of a pattern infinitely vast, small or large? Why has it been so difficult to accept the existence of quasicrystals in the West? Why were they always deemed so impossible? In the story of quasicrystals, several significant moments and figures stand out.

1453

A quasicrystalline motif appears in the tilework of a medieval Islamic mosque in Iran, five centuries before the pattern’s underlying mathematics are understood in the West. Workers applied a special set of tiles across the surface of a building, one after the other, to create aperiodic girih (geometric star-and-polygon) patterns. *1* Edge to edge, ad infinitum, these decagonal tilings form a continuous set of lines. They intuit forbidden, fivefold and tenfold, symmetries. *2*

*1In 2007, the quasicrystalline origin of girih tilings was discovered by Peter Lu and Paul Steinhardt in their article “Decagonal and Quasicrystalline Tilings in Medieval Islamic Architecture,” Science 315 (2007): 1106–1110.*

*2“Forbidden” symmetries are high degree symmetries, such as five, eight, ten, or twelve-fold. Because they disallow translational periodicity, crystals that displayed these types of symmetry were, until recently, thought incapable of packing space without leaving gaps, and therefore unable to form solids.*

1619

The mathematician,astronomer, and astrologer Johannes Kepler takes up the challenge of tiling with forbidden symmetries in his book Harmonice Mundi. 3 When asked if he succeeded in tiling the plane with five-sided shapes, Kepler ominously affirms that he has but that he has also found “monsters” in the resulting order. Monsters: pairs of fused decagons and stars. The unexpected shapes that repeat aperiodically. Kepler was the first to witness this strange behavior, the first recorded instance of monsters in the West.

*3 Kepler’s seventeenthcentury treatise Harmonice Mundi (Harmony of the World), published in 1619, explores the concept of congruence across geometric form, musical harmony, and the cosmos. In Book II, Kepler experiments with the symmetry of polyhedra and the orderly arrangement of pentagon tilings.*

1974

Roger Penrose decomposes Kepler’s four tilings — pentagons, pentagrams, decagons, and fused decagons — into a single pair of shapes that tile the plane aperiodically: stars, boats, diamonds, kites, darts, and rhombs. The Penrose pattern proves mathematically that aperiodic tilings exist, at least in theory — always changing. He effectively proves that a quasicrystal may exist ten years before it is actually found in nature.

1980

Nicolaas Govert de Bruijn creates an algorithm that turns Penrose’s tilings into a rule. *4* The projection algorithm is a nearperfect description of aperiodic order; its only problem (and not just for architecture) is an existential one. While it explains quite reliably how a crystal like this might form, the algorithm uses projection from a higher dimension down to more familiar three-dimensional space. What we see then is actually a hyperdimensional object. Only its shadow exists here in our reality. How de Bruijn made the conceptual leap from higher dimensions to our own to find the algorithm of projection has never been fully explained, but his method remains a bridge to another reality and back.

*4 Nicolaas Govert de Bruijn, “Algebraic Theory of Penrose’s Non-Periodic Tilings of the Plane,” Proceedings, vol. 84 (Eindhoven, NL: Eindhoven University of Technology, 1981).*

1982

The material scientist Dan Shechtman discovers a new type of synthetic solid-state matter: the icosahedral phase. Looking at a rapidly cooled alloy of aluminum and manganese (Al6Mn) through an electron microscope and using X-ray diffraction studies, Shechtman observes an atomic structure with fivefold rotational symmetry. He proves that aperiodic and non-repeating crystals do exist, however fleeting. But Shechtman’s discovery of this crystal structure is met with skepticism by the science community. It is not until 2011 that he is awarded the Nobel Prize in Chemistry for “the discovery of quasicrystals.”

1984 Two years later, Dov Levine and Paul Steinhardt coin the term “quasicrystal.”*5* By identifying the symmetries and parameters that define aperiodic arrangements in three dimensions, the scientists prove that a solid analogy exists for Roger Penrose’s two-dimensional aperiodic tiling — confirming Shectman’s observation from 1982. The search for the physical manifestation of this mathematically true concept ends. Skeptics will still contend that while the quasicrystal is a synthetic state of matter, it does not occur naturally.

*5 Dov Levine and Paul Steinhardt, “Quasicrystals: A New Class of Ordered Structures,” Physical Review Letters, vol. 53, no. 26 (December 1984).*

2009 According to the scientific community at large, quasicrystals still do not occur naturally. That is until scientist Luca Bindi reports the first natural occurrence in a fragment of Khatyrka. The decade-long search, beginning in 1999 by researchers at Princeton University, culminates in Bindi’s examination of the “khatyrkite” mineral found in the Koryak mountains in far-eastern Siberia, Russia. Steinhardt and Bindi establish that the mineral is dated to the formation of the solar system, delivered to earth by meteor. The latest episode in this winding story places the impossible crystal at the beginning of it all — not only possible but also extraterrestrial.

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This article is an excerpt from our book *Trace Elements* by Columbia Books on Architecture and the City.