The Second Kind of Impossible

by Paul Steinhardt

  PHILADELPHIA, 1983–84: Dov and I were meeting almost every day to flesh out our theory. Our immediate focus was to find a way to exploit the loophole we had discovered in the rules of crystallography: quasiperiodic ordering. Our aim was to use that loophole to create a three-dimensional structure with the forbidden symmetry of an icosahedron. It was an ambitious goal, but if we could show that such a geometrical structure existed, we could then begin to imagine how real-life atoms and molecules might be arranged the same way.

  It sounded crazy. But it was the same idea that had motivated me from the beginning, first as a teenager inspired by Vonnegut’s fictional ice-nine, and then, years later as a researcher, when a tantalizing hint of forbidden symmetry popped up in my work with David Nelson on rapidly cooled liquids.

  Roger Penrose’s discovery of how to design shapes with special interlocks to create his intricate patterns was a major achievement. For us to repeat that feat in three dimensions would be, in many ways, much more challenging.

  The icosahedron, like every other three-dimensional object, has different rotational symmetries in different directions. The forbidden five-fold symmetry appears along six different directions. Two- and three-fold symmetry can be seen if the icosahedron is viewed along other directions.

  Dov and I began by working with rhombohedrons, the three-dimensional equivalent of the rhombuses Penrose had used for his flat designs. We knew that rhombohedrons can be packed together in a periodic arrangement, as Haüy first discovered more than 200 years ago in his explorations of calcareous spar. But Penrose had found matching interlocks for his rhombuses that could prevent any such periodic patterns. The interlocks forced his fat and skinny rhombuses into a quasiperiodic arrangement. We needed to prove whether the same sort of mechanism existed for fat and skinny rhombohedrons. Dov and I discovered that we needed to use twice as many units as Penrose—two fat rhombohedrons and two skinny rhombohedrons, each with unique interlocks. More shapes, more interlocks, more complications.

  As always, we found it useful to construct physical models of the abstract theoretical objects we were studying in order to visualize the structure. So once again, we turned my office into a comical-looking arts-and-crafts studio.

  The easiest part was creating the two types of building blocks. We designed cardboard cut-outs of the fat and skinny rhombohedrons that could be folded to form the four different units—two fat and two skinny shapes. We tried taping them together according to our proposed interlock rules, but the process turned into a sticky nightmare. So we rolled up our sleeves and glued magnets into the corners of all the cardboard cut-outs. The magnets were precisely placed so they could play the same role as interlocks. The blocks would only stick together if the three-dimensional interlock rules were satisfied. It was highly organized chaos, or so I kept telling befuddled visitors to my office.

  On the next page are photographic examples of some of our constructions. On the top left there is one cluster of ten fat and ten skinny rhombohedrons joined together in a nearly spherical shape.

  The outside surface of this cluster has a daunting name. It is a “rhombic triacontahedron,” which is Greek for “thirty faces on the surface, each with the shape of a rhombus.”

  In the middle image, a skinny rhombohedron has been removed, revealing a bit of the interior. On the right, a fat rhombohedron has also been removed to reveal even more.

  The rhombic triacontahedron was the first step, we showed, in packing our fat and skinny rhombohedrons together in a quasiperiodic pattern of arbitrarily large size, while maintaining the symmetry of an icosahedron. Equally important, there were no gaps between our building blocks, the rhombohedrons, and our new interlocks specifically prohibited them from forming any other kind of structure, including the regular periodic arrangements of a crystal.

  Now that we knew a three-dimensional quasicrystal was theoretically possible, we needed to identify groups of atoms that could join together in analogous ways, with analogous matching rules, so that a quasicrystal would be the only possible result.

  We began considering what other previously forbidden rotational symmetries were possible with quasiperiodic order. The answer, incredibly enough, was all of them. Seven-fold, eight-fold, nine-fold . . . literally an infinite number of new possibilities once thought to be forbidden were now allowed. A beautiful example of a quasiperiodic tiling with seven-fold symmetry is shown below.

  Dov and I were now rapidly making so many discoveries and had so many new directions to explore that it was hard to decide when to stop researching and start writing a scientific paper. Not believing that there were any other competitors on the field, which might have given me a reason to rush, I made the fateful decision to continue working and delay publishing our results until we had made further progress.

  The early 1980s was one of the most fertile times of my career. Dov was not the only talented graduate student I was working with. Andy Albrecht and I were focused on an exciting new idea in cosmology, the inflationary theory of the universe, which had just been introduced by a physicist at the Massachusetts Institute of Technology named Alan Guth.

  Few scientific theories are ever complete when first introduced, and the inflationary theory was no different. Alan had proposed that inflation, a hypothetical period of rapid expansion a few instants after the big bang, could potentially explain, in part, why the distribution of matter and energy in the universe is so uniform today. In order to do that, however, he had to assume that inflation would stop after a short period of time. But there was the rub. Alan could not find any way to make inflation stop. Andy and I, along with Andrei Linde, a theorist working independently in the Soviet Union, solved that critical problem.

  Our “new inflationary theory” took hold rapidly. It had an explosive effect that triggered a period of prolific innovation in cosmology, astrophysics, and particle physics that continues today. Unlike my work with Dov on new forms of matter, the exploration of new inflationary cosmology was a crowded field with many sharp-elbowed competitors. There were many important follow-up projects that simply could not be ignored.

  Throughout this time, though, I was also quietly testing reactions to our new quasicrystal theory. I had begun discussing it informally with well-known condensed matter physicists and materials scientists. But to my surprise, the response was uniformly discouraging:

  You and Dov have an imaginative idea for a new form of matter that might be mathematically possible, but it seems far too complicated, compared to the simple case of a periodic crystal, to exist in the real world.

  I could understand their attitude. After all, Dov and I were challenging centuries of scientific wisdom by proposing a new state of matter based solely on the study of abstract tilings. What was needed was experimental proof that there existed combinations of atoms that would arrange themselves into a true quasicrystal. Without that, our idea was just another theoretical fantasy with no connection to reality.

  I was more sensitive to the criticism than Dov, who wanted to publish our basic idea right away. I wanted to wait until we had developed a more concrete proposal. I also wanted to be able to make a testable prediction, a necessary component of any scientific theory, to explain how to identify the new form of matter through experiment. Without that, I concluded, our work would probably be discounted. So there was no point in publishing.

  In 1983, Dov and I reached a compromise. We agreed to protect our intellectual investment by submitting a patent disclosure, which we filed with the University of Pennsylvania’s Technology Licensing office. The submission would present our concept and formally establish priority. But we would not release our idea to the general scientific community until we had made more progress.

  The disclosure, partially reprinted below, described our building blocks, the rhombohedrons, and the matching interlocks. It explained that the connections were designed to force the building blocks into a noncrystalline pattern with the symmetry of an icosahedron. It also explained how the idea could
potentially lead to a new phase of matter with properties different from either liquids or crystals. Dov and I called our theoretical invention “crystalloids” in the 1983 disclosure, but later renamed them “quasicrystals.”

  Was this just an abstract theory, as critics claimed, or was it actually a valid scientific theory and somehow testable? And how would we ever recognize a quasicrystal if we were lucky enough to find one? Dov and I spent months grinding away at calculations only to find that the answer was relatively simple. An ordinary X-ray or electron diffraction pattern would reveal the quasiperiodicity and forbidden symmetry of its atomic arrangement.

  Compared to a crystal, a quasicrystal’s diffraction pattern has a much richer, more complex structure partly because it is composed of atoms that repeat with different frequencies related by an irrational number, such as the golden ratio.

  If X-rays or electrons magically diffracted from only one type of atom in the quasicrystal, it would produce true pinpoint diffraction, known as Bragg peaks, with equal spacing between the peaks. But in reality, X-rays and electrons diffract from all of the atoms in a quasicrystal. Different subgroups have different pinpoint diffraction patterns as well as different spacings between the atoms. An icosahedron has multiple symmetries, which also adds to the complexity.

  Our predicted diffraction pattern took different forms depending on whether the X-ray or electron beam was shone down an axis of five-fold, three-fold, or two-fold rotational symmetry. The image on the opposite page shows our computed prediction when the beam is aimed along the “impossible” five-fold symmetry.

  We had deciphered the mathematical formula behind the secret symmetry and were able to make a bold quantitative prediction that could be tested by experiment: The diffraction pattern of a quasicrystal would be composed of absolutely perfect pinpoints arranged in a snowflake pattern.

  The archival figure shown above is the first such pattern ever computed. Our computer code drew a circle centered at each predicted pinpoint. The radius of each circle was chosen in proportion to the predicted intensity of diffracted X-rays. The figure we created was the first visual representation of the bright and dim spots that we would expect to see in the diffraction pattern of an actual quasicrystal.

  If it were possible to see the ever-dimmer points, one would find that between every pair of spots there were yet more spots. And between every pair of those spots, there would be even dimmer spots, and so on. If Dov and I had created a circle for each predicted spot, the pattern would have been so crowded that the circles would have merged into a white featureless cloud. We knew that experiments would only be able to detect the brightest spots. So we figured that our image would approximate the signature diffraction pattern for a quasicrystal.

  By creating the figure, Dov and I had made a prediction that could be used to test and potentially disprove our theory. So now, we had arrived at another milestone. Time to publish? Once again, I held back. I knew there was something else we needed to accomplish if we wanted the radical new theory to be taken seriously. We had to show that it would be possible to replace the rhombohedron blocks we used for our theoretical model with real matter.

  By the summer of 1984, the demanding commitments from my work on the new inflationary theory had finally subsided. So I was able to carve out the large block of time needed to focus on the final stage of our quasicrystal research. I took a sabbatical from the University of Pennsylvania and headed back to the IBM Thomas J. Watson Research Center, where I had done much of my earlier work on the atomic structure of amorphous metals.

  My plan was to work with crystal experts to try to create the world’s first synthetic quasicrystal. But unbeknownst to me, someone else had already done that. It was much easier to do than I had imagined. In fact, the discovery was a complete accident.

  * * *

  GAITHERSBURG, MARYLAND, 1982–84: “No such animal!” Dan Shechtman reportedly thought, as he looked at the strange sample under his electron microscope. The forty-one-year-old Israeli scientist had accidentally come across a material with all of the impossible properties Dov and I had anticipated, although he had no inkling about any of our ideas or any understanding of what his discovery actually signified. But Shechtman recognized that he was looking at something remarkable. And it would ultimately earn him the 2011 Nobel Prize in Chemistry.

  Shechtman was working as a visiting microscopist at the National Bureau of Standards with John Cahn, whom he had met as a graduate student at the Technion, Israel’s premier technological institute. Cahn was considered a major figure in condensed matter physics and was especially famous for his work on the processes that occur when hot metallic liquids are cooled and solidified.

  Cahn had invited Shechtman to take a two-year leave from the Technion to work on a large project being funded by the National Science Foundation and the Defense Advanced Research Projects Agency (DARPA). The goal of the project was to synthesize and categorize as many different aluminum alloys as possible by rapidly cooling liquid mixtures of aluminum and other metals. Other scientists would create the alloys. Shechtman would use the electron microscope to study, identify, and classify the samples. It was an important service to the materials community because aluminum alloys are useful in many applications. But it was also a relatively dull and tedious assignment.

  Robert Schaefer, one of the metallurgists at the lab, was especially interested in creating alloys composed of aluminum and manganese because of its superior strength as compared to pure aluminum. He and his colleague Frank Biancaniello made a series of samples composed of aluminum combined with varying amounts of manganese, and each sample was dutifully sent to Shechtman for analysis.

  On April 8, 1982, Shechtman studied a sample of rapidly quenched Al6Mn (scientific shorthand for an alloy with six atoms of aluminum for every atom of manganese), which had tiny feathery grains with roughly pentagonal shapes. A larger sample with beautiful flowerlets and clearly evident five-fold symmetries was later synthesized by An-Pang Tsai and his team at Tohoku University, and is shown on the next page.

  When Shechtman fired a beam of electrons through the grains to obtain its diffraction pattern, he found something shocking. The pattern had what initially appeared to be rather sharp spots, as expected for crystal. But to Shechtman’s surprise, the spots revealed an apparent ten-fold symmetry, which he, as well as every other scientist in the world, knew was impossible.

  Shechtman sketched the pattern on one side of a page in his notebook. On the other side he noted a partial catalog of the diffraction peaks in which he wrote “10-fold???”

  When Shechtman showed his results to his colleagues they were not particularly impressed. They, too, had been taught that true ten-fold symmetry was impossible. Everyone assumed the strange diffraction pattern could be explained by something called “multiple twinning.”

  A crystal twin is commonly formed when two crystal grains oriented at different angles grow together. A multiple twin is when three or more grains oriented at different angles combine. Two examples are shown in the images on the opposite page. The one on the left is an example of “triple twinning.” It is easy to see with the naked eye that the combined crystals are oriented at three different angles.

  The image on the right is much more subtle. It is an example of multiple-twinned gold. The sample is composed of five distinct wedges, which have been made more obvious with added lines. The atoms are blurry white spots within each wedge. At first glance, the overall shape suggests a quasicrystal with five-fold symmetry. But that would be a false conclusion. This is not a quasicrystal.

  Under the microscope, it becomes clear that each of the five wedges is made up of a regularly repeating hexagonal pattern of atoms. Therefore, each individual wedge is a crystal conforming to all the rules of crystallography. Taken as a whole, this is an example of a multiple-twinned crystal. It is a collection of crystals that just happened to come together in five wedge-shaped pieces forming the shape of a pentagon. Any solid composed of a combinat
ion of crystal wedges is always defined as a crystal no matter the number of wedges or how they are arranged.

  Multiple-twinning is an everyday occurrence. So it was natural for Shechtman’s colleagues, including John Cahn, to be convinced that the Al6Mn sample was merely another example of the same phenomenon. No one was expecting to find anything the least bit unconventional in the midst of a dull survey of aluminum alloys. The lab dismissed the notion that Shechtman had discovered anything remarkable.

  Shechtman, however, did not agree. He refused to relent and continued to press his case with the senior scientists. It was something novel, he argued. Unconvinced, John Cahn told him that there was a test that could settle the issue. Cahn told Shechtman to focus an electron beam on a very narrow region of the sample. If the sample was a multiple-twinned crystal, as the rest of the lab suspected, many of the spots in the ten-fold pattern would disappear and the remaining spots would form a pattern with one of the well-known crystal symmetries. On the other hand, if the sample truly violated the long-established principles and was uniformly ten-fold symmetric, all of the spots marking ten-fold symmetry would continue to appear no matter where the beam was focused.

  Shechtman went back to his microscope and performed the crucial test. Wherever he looked in the Al6Mn sample he found the same impossible ten-fold symmetry. It was an astonishing result that eliminated the routine explanation of multiple-twinning. History does not make clear, however, whether he showed the results to Cahn or anyone else in the lab before he completed his two-year term in America and returned to Israel.

  What is known, however, is that Shechtman never gave up. He realized that his discovery was so outrageous that he would never be taken seriously without offering some plausible explanation. But he was an electron microscopist, not a mathematically-trained theorist. So he later teamed up with an Israeli materials scientist named Ilan Blech, whom he hoped could provide a possible theory.