The thought got me very excited. Over the next few months, I made targeted inspections of the mineral collections at several museums, including the Franklin Institute in Philadelphia, the American Museum of Natural History in New York, and the Smithsonian National Museum of Natural History in Washington, D.C. I went from display case to display case hunting for a misidentified quasicrystal. It was such a wild hunch that I never tried to talk to anyone at the museums and eventually came up empty. Perhaps my insight about the possibility of natural quasicrystals was not so insightful after all.
* * *
The Shechtman team’s paper announcing their experimental results appeared in Physical Review Letters on November 12. Our theoretical explanation of those results appeared in the same journal on December 24, the penultimate issue of 1984.
Perfect timing and a perfect fit, I thought.
The two papers drew attention and strongly positive responses from scientists and journalists from around the world. There were articles in scientific journals as well as the consumer press, including Physics Today, Nature, New Scientist, and the New York Times. The Times article, headlined “Theory of New Matter Proposed,” described how we had “postulated a new quasicrystalline state of matter that apparently explains the bewildering results of a test recently conducted at the National Bureau of Standards.”
As news of the breakthrough spread across the globe, Dov and I were surprised to learn about scientists in other parts of the world who had been developing related ideas. Some were interested in the mathematics of Penrose tilings; some were interested in quasiperiodicity; some were even thinking about materials with icosahedral symmetry. In the pre-Internet days, it was much more difficult to share information. So Dov and I were previously unaware of these papers because they were not published in journals well known to physicists. But now, we were being contacted by the authors and, in turn, devouring everything they had written.
We were particularly struck by the work of Dutch mathematician Nicolaas de Bruijn, who had written a series of beautiful papers in 1981 with an ingenious “multigrid” method for generating two-dimensional Penrose tiling patterns without relying on any of the normal matching or subdivision rules. Dov and I teamed up with another talented young graduate student at Penn named Joshua Socolar to further develop those ideas. The three of us were able to generalize De Bruijn’s multigrid method to create quasiperiodic patterns with any symmetry in any number of dimensions including purely mathematical constructs beyond three dimensions.
Our generalized multigrid method demonstrated in a straightforward, direct way something that Dov and I had already proven in a more abstract, indirect mathematical way: Quasicrystal patterns could be made for an infinite number of different symmetries forbidden to crystal patterns. Now it was simple for anyone to see that the number of possible forms of matter had gone from being strictly limited to unlimited. This was a major paradigm shift.
The “projection method” was another important idea developed by several independent groups of theorists. According to this approach, the Penrose tiling and other quasiperiodic patterns are obtained by projections or “shadows” of a higher-dimensional periodic packing of “hypercubes,” which are the equivalent of three-dimensional cubes, but in imagined geometries with four or more dimensions of space. Most people cannot visualize how the method works without advanced training, but mathematicians and physicists find the concept very powerful for analyzing the atomic structure of quasicrystals and computing their diffraction properties.
The generalized multigrid and projection methods are powerful mathematical tools for generating patterns of rhombus tiles in two dimensions or rhombohedrons in three dimensions. But they have a major limitation: They provide no information about matching rules. For example, the patterns with eleven-fold symmetry (see the color insert, image 1) and with seventeen-fold symmetry (as shown below) were generated by the multigrid method.
These beautifully intricate patterns are composed of simple rhombus shapes: some fat, some medium, and some thin. But there are no notches or interlocks to prevent the tile shapes from being arranged in a crystal pattern.
So if you were given a pile containing these tile shapes and asked to cover a floor with them without using pictures of the complete pattern as a guide, you might wind up with an ordinary crystal pattern because they are so simple to construct. You might also make a random pattern. But the chance that you would be able to make a quasicrystal pattern is very small. You would need matching rules to guide you, and to help in recognizing if you made a mistake along the way.
Imagine replacing each different tile in the pattern opposite with a group of atoms. Even though the precisely ordered quasicrystal arrangement would be possible, it seems intuitively much more likely that a liquid would tend to solidify into either a crystal or a random arrangement if there were no interactions between atoms that could act as matching rules to prevent that from happening. There are many more of those arrangements and each of them requires less delicate coordination of the atoms than a quasicrystal.
That was why Dov and I had originally worked so hard to show that it is possible to construct interlocks for our fat and skinny rhombohedrons which act as matching rules that prevent both crystal and random arrangements and force quasicrystal ones.
But were matching rules enough to explain why quasicrystals form? I wondered. Maybe a quasicrystal would need additional properties in order for the atoms to organize naturally into the ideal quasiperiodic arrangement.
* * *
PRINCETON, JANUARY, 1985: Josh Socolar volunteered to work with me on this challenging question. He had already proven his talents in our previous work generalizing the multigrid approach to arbitrary symmetries, so I was delighted that he wanted to take on a bigger project. Josh was tall and lanky, and always managed to convey a sense of patience and thoughtfulness, which was unusual for someone so young. I felt like I was always the one apt to get overly excited and that Josh brought a sense of calm to the discussion. He also had a remarkable geometrical intuition that would prove to be invaluable in all of our collaborations, which have been fruitful and continue to the present day.
Josh and I decided to go back to the Penrose tiling for guidance. We noted that Penrose’s matching rules for two-dimensional patterns included two other properties that the fat and skinny three-dimensional rhombohedrons Dov and I studied did not share. The first missing element was Ammann lines, the wide and narrow channels that appear when each rhombus is decorated with bars and the tiles are put together in a Penrose pattern. Josh and I decided to incorporate the three-dimensional analog of Ammann lines, which we called “Ammann planes,” in our geometrical constructions. The second missing property was deflation-inflation rules, protocols for subdividing the two rhombuses in a Penrose tiling into smaller pieces.
Josh and I conjectured that an alternative set of building blocks with all three properties of matching rules (interlocks), Ammann planes, and deflation-inflation rules might be the secret to understanding how real atoms come together in a liquid to form a quasicrystal. The Ammann planes and deflation-inflation rules might be important in explaining how atoms beginning from some random arrangement organize into a precise quasiperiodic arrangement, and the interlock rules Dov and I developed might be important in explaining how they remain locked in that configuration.
The subtle reasoning was as follows: If the building blocks could be construed as lying along quasiperiodically spaced Ammann planes, then it would be possible to imagine a liquid solidifying into a quasicrystal beginning with some small seed cluster of atoms to which more atoms would attach one layer at a time. Each layer would correspond to an Ammann plane.
The layer-by-layer growth would be analogous to the way many periodic crystals form, so it was reasonable to imagine that something similar occurs for quasicrystals.
The three-dimensional deflation-inflation rules seemed to suggest another way that quasicrystals could grow. First, atoms in a liqu
id might form many small clusters; then those clusters could come together to make larger clusters; then larger clusters could come together to form yet larger clusters; and so on. This hierarchical clustering of smaller bits to make larger bits might correspond to the way small tiles combine together into larger tiles according to the deflation-inflation rules.
We also imagined that some quasicrystals might solidify by using a combination of layer-by-layer and hierarchical growth.
The fat and skinny rhombohedrons Dov and I had constructed with our cardboard cutouts had interlock rules, but nothing like Ammann planes or deflation-inflation rules. The challenge for Josh and me was to find another set of building blocks that had all three properties. To accomplish that for the complicated case of icosahedral symmetry in three dimensions would be a significant mathematical feat, comparable to what Penrose had accomplished with his two-dimensional designs. But if we succeeded, we could explain that growing quasicrystals in a liquid can be as simple and natural as growing ordinary crystals.
But did building blocks exist that possessed all three properties?
Josh and I set out to find the answer. Shortly after the first papers on quasicrystals appeared at the end of 1984, we began to work intensively on a new mathematical approach to generate quasicrystals based on lessons learned from Penrose tilings.
Our approach involved a weird combination of algebra using pencil and paper and physical geometrical constructions in three dimensions. Algebraic equations had to be solved to predict the precise positions of the Ammann planes in three dimensions, which was my job. Josh would then see where the Ammann planes intersected and use our generalized multigrid method to determine the shapes of the building blocks and how the Ammann planes would pass through them.
The fact that we were working in two different physical locations made the project even more challenging. Josh was at the University of Pennsylvania in Philadelphia, but I was still continuing my sabbatical leave and had become a visiting fellow at the Institute for Advanced Study in Princeton, New Jersey. Skype would not be invented for nearly two more decades. So Josh and I could only communicate over the phone and could not exchange any images.
I would telephone Josh and describe how my algebraic calculations dictated how the Ammann planes should be arranged. He would then describe to me the building blocks implied by my calculations. Josh was able to combine our separate ideas and construct some truly remarkable physical models out of transparent, colored plastic sheets that remain fixtures on my office shelf today. When I finally saw the models several weeks later, I was excited to see that our two calculations fit together hand in glove. We submitted our paper to Physical Review B in September of 1985. There was no question that we had solved the problem.
Now we knew for sure that there were building blocks for three-dimensional icosahedral symmetry with interlocking matching rules, Ammann planes, and deflation-inflation rules. They had all the same properties of two-dimensional Penrose tiles but with much more complex symmetry. The work was directly relevant to explaining real-life quasicrystals with icosahedral symmetry.
Josh and I ultimately found a manufacturing company that could fabricate the four types of building blocks we had invented to solve the problem. The plastic blocks have specially designed LEGO-like connections, which serve as substitutes to enforce all of our matching rules.
One shape was the same fat rhombohedron that Dov and I had used, represented by the white blocks seen in the color insert (image 2). The other three shapes were different from any of the ones Dov and I originally studied. They have complicated Greek names based on the number of facets, all of which are rhombuses of the exact same size and shape. The actual names are not so important, but, for those who enjoy practicing their Greek, they are, in increasing order of size: rhombic dodecahedron (twelve rhombus facets, blue), rhombic icosahedron (twenty rhombus facets, yellow), and rhombic triacontahedron (thirty rhombus facets, red).
I have to admit that I enjoy the fabricated units. They not only illustrate how the new building blocks fit together, they also represent a big improvement over the arts-and-crafts experiments Dov and I had performed with Styrofoam balls and pipe cleaners, and then with cardboard cutouts and magnets.
A few layers showing how the four three-dimensional shapes fit together are shown in the color insert (image 2).
This mathematical tour de force made me feel much more secure that there were no theoretical roadblocks that would prevent us from extending the concept of quasicrystals from the abstract world of two-dimensional Penrose tilings to the realistic world of three-dimensional matter.
Our construction was timely because, by the spring of 1985, the discovery of quasicrystals had launched a hot new field of research. News about new experiments, new potential quasicrystal alloys, and new theoretical ideas from diverse groups all around the world seemed to come out every week. The excitement led to a continuous stream of conferences, workshops, and invited lectures, including the lecture at Caltech that led to my deeply satisfying encounter with Richard Feynman.
It was during this same period of time that Dan Shechtman invited me to visit his laboratory at the Technion in Haifa, Israel. We had met previously at a conference, but only had time for a brief exchange. My visit to Haifa was our first opportunity to spend a substantial amount of time together exchanging ideas.
Dan was a gracious host. He was proud of his work and his nation. He showed me his lab and latest data, and then took me on a tour of the Haifa region all the way up to the Golan Heights.
I admired the courage and independence of mind that had led Dan to make his great discovery. I was not satisfied with our scientific discussion, though. Dan’s expertise was in electron microscopy and diffraction, and he had limited interest in theory. It soon became clear to me that he was still enamored with the idea that Ilan Blech had originally proposed to explain his Al6Mn alloy, which was that the material consisted of icosahedral clusters whose orientations are, for some inexplicable reason, all aligned in the same way despite the fact that their positions in space are random. For some reason, Dan seemed to consider Blech’s idea to be equivalent to our quasicrystal theory.
I tried to explain to him the key differences: The Blech model was incomplete since it had large gaps between clusters that were not accounted for; it was not a stable configuration; it would therefore not represent a new phase of matter; nor did it have a diffraction pattern consisting of sharp pinpoints aligned along straight lines.
But I could tell that Dan was not impressed by these distinctions. He apparently thought the Shechtman-Blech model composed of randomly placed icosahedral clusters was easier to imagine and did not seem to consider the important differences that I was pointing out. I felt badly that I had failed to convince him to change his opinion. In fact, he would continue to use the Shechtman-Blech picture rather than the quasicrystal model in his presentations for many years to come.
Shechtman was not the only one resisting the quasicrystal picture. Within a few months, other plausible alternative explanations for the strange aluminum-manganese alloys would begin to surface. Even more disturbingly, a serious problem with the quasicrystal concept was about to be exposed.
Alternate theories. Conceptual problems. Much to my dismay everything would soon lead to a growing consensus within the scientific community that quasicrystals were, as I had been told all along, impossible.
SIX
* * *
PERFECTLY IMPOSSIBLE
PHILADELPHIA, 1987: More than two years had passed since Dov and I had published our paper introducing the concept of quasicrystals. During this period, attitudes toward the concept had undergone a series of mood swings in the scientific community.
For the first year after our paper appeared, the quasicrystal theory was embraced as the only viable scientific explanation for the newly announced alloy with icosahedral symmetry. In fact, the idea seemed to catch the scientific world by storm and triggered a marvelous series of new disc
overies.
Scientists began combining aluminum with elements other than manganese, the components of the original experiment, and discovered even more quasicrystal alloys with the symmetry of an icosahedron. In the process, they found a material with eight-fold symmetry, another with ten-fold, and another with twelve-fold, firmly establishing the existence of matter with yet other symmetries previously thought to be impossible.
I admired what all the other scientists were accomplishing. And so far, everything was in agreement with what one would expect according to the quasicrystal theory we had presented. But the good news would not last much longer. The pendulum began to swing in the other direction as competing explanations surfaced, along with serious criticisms.
The first and most vociferous critic was two-time Nobel Laureate Linus Pauling. Pauling was a towering figure in the scientific community. As one of the founders of quantum chemistry and molecular biology, he was widely regarded as one of the most important chemists of the twentieth century.
“There is no such thing as quasicrystals,” Pauling liked to joke derisively. “Only quasi-scientists.”
Pauling proposed that all the peculiar alloys that had been discovered were complex examples of multiple-twinned crystals, similar to what senior scientists at the National Bureau of Standards had originally suggested. But Pauling had a very different and very explicit atomic arrangement in mind that he claimed could explain the diffraction pattern.
If Pauling was right, there would be nothing newsworthy about any of the new materials. All of our work would sink into obscurity as nothing but a historical curiosity. For those in the materials science and chemistry fields, like Dan Shechtman and his colleagues, Pauling’s objections were frightening and considered a serious threat. During the course of his scientific career, Pauling had consistently challenged and prevailed over conventional wisdom. He was not someone you wanted as an intellectual opponent.
The Second Kind of Impossible Page 8