Shortly after von Laue’s discovery, the father and son team of British physicists, William Henry and William Lawrence Bragg, took the next big step. By carefully controlling the X-ray wavelength and direction, they showed that the pinpoint diffraction pattern could not only be used to reconstruct the symmetry, but also the detailed atomic arrangement throughout the crystal. The pinpoints became known as “Bragg peaks.”
The two breakthroughs immediately became indispensable in the exploration of matter. Over subsequent decades, tens of thousands of diffraction patterns were obtained from various natural and synthetic materials from all over the world. In later years, scientists obtained even more accurate information by replacing X-rays with electrons, neutrons, or the high-energy radiation produced when a beam of charged particles moving at relativistic speeds is bent at an angle by the magnets in a powerful particle accelerator called a synchrotron. But no matter the method, the original rules of symmetry derived from Haüy’s and Bravais’s work were always obeyed.
Based on the combination of mathematical reasoning and accumulated experimental experience, the rules became firmly fixed in the minds of scientists. It seemed certain, or at least as certain as any scientific principle could ever be, that matter could only have one of the long-prescribed symmetries. Nothing else. Five-fold symmetry remained verboten for more than two hundred years.
* * *
PASADENA, 1985: But now, here I was, standing in front of Richard Feynman explaining that these long-standing rules were wrong.
Crystals were not the only possible forms of matter with orderly arrangements of atoms and pinpoint diffraction patterns. There was now a vast new world of possibilities with its own set of rules, which we named quasicrystals.
We chose the name to make clear how the new materials differ from ordinary crystals. Both materials consist of groups of atoms that repeat throughout the entire structure.
The groups of atoms in crystals repeat at regular intervals, just like the five known patterns. In quasicrystals, however, different groups repeat at distinct intervals. Our inspiration was a two-dimensional pattern known as a Penrose tiling, which is an unusual pattern that contains two different types of tiles that repeat at two incommensurate intervals. Mathematicians call such a pattern quasiperiodic. Hence, we dubbed our theoretical discovery “quasiperiodic crystals” or “quasicrystals,” for short.
My little demonstration for Feynman was designed to prove my case using a laser and a slide with a photograph of a quasiperiodic pattern. I flipped on the laser, as Feynman had directed, and aimed the beam so that it passed through the slide onto the distant wall. The laser light produced the same effect as X-rays passing through the channels between atoms: It created a diffraction pattern, like the one pictured in the photo below.
I turned off the overhead lights so that Feynman could get a good look at the signature snowflake pattern of pinpoints on the wall. It was unlike any other diffraction pattern that Feynman had ever seen.
I pointed out to him, as I had done during the lecture, that the brightest spots formed rings of ten that were concentric. That was unheard of. One could also see groups of pinpoints that formed pentagons, revealing a symmetry that was thought to be absolutely forbidden in the natural world. A closer look revealed yet more spots between the pinpoints. And spots between those spots. And yet more spots still.
Feynman asked to look more closely at the slide. I switched the lights back on and removed it from the holder and gave it to him. The image on the slide was so reduced that it was hard to appreciate the detail, so I also handed him an enlargement of the tiling pattern, which he put down on the table in front of the laser.
The next few moments passed in silence. I began to feel like a student again, waiting for Feynman to react to the latest cockamamie idea I had come up with. He stared at the enlargement on the table, reinserted the slide in the holder, and switched on the laser himself. His eyes went back and forth between the printed enlargement on the table, up to the laser pattern on the wall, then back down again to the enlargement.
“Impossible!” Feynman finally said. I nodded in agreement and smiled, because I knew that to be one of his greatest compliments.
He looked back up at the wall, shaking his head. “Absolutely impossible! That is one of the most amazing things I have ever seen.”
And then, without saying another word, Dick Feynman looked at me with delight and gave me a huge, devilish smile.
TWO
* * *
THE PENROSE PUZZLE
PHILADELPHIA, PENNSYLVANIA, OCTOBER 1981: Four years before my encounter with Feynman, no one had ever heard of quasicrystals. Including me.
I had just joined the Department of Physics at the University of Pennsylvania and was invited to give the Physics Colloquium, a weekly lecture attended by the entire department. Penn had recruited me to the faculty based on my work at Harvard University in elementary particle physics, which was related to understanding the fundamental constituents of matter and the forces through which they interact. There was also great interest in my most recent research. My first graduate student, Andy Albrecht, and I were working feverishly on developing novel ideas about the creation of the universe, ideas that would eventually help set the foundation for what is now known as the inflationary theory of the universe.
But I decided not to talk about any of that. Instead, I chose to talk about a project that almost no one knew I had been working on and whose significance was not yet clear. I did not know the lecture would resonate with a young graduate student who was sitting in the audience, or that it would soon lead to a fruitful partnership and the discovery of a new form of matter.
Most of my presentation described a project that I had been exploring for the last year and a half with David Nelson, a theoretical physicist at Harvard, and Marco Ronchetti, a postdoctoral fellow working at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York.
Our project was to study how atoms in a liquid rearrange themselves if the liquid is rapidly cooled and solidified. It was well known among scientists that when a liquid is frozen very slowly, its atoms tend to reconfigure from the random arrangement of a liquid into the orderly, periodic arrangement of a crystal (as when water freezes into ice).
For the simplest case, in which all the atoms are identical and interacting under simple interatomic forces, the atomic arrangement would be one in which the atoms stacked together like oranges on display at a grocery store. The structure—technically called face-centered cubic—has the same symmetry as a cube, consistent with all the known rules of crystallography.
The three of us wanted to study what would happen if the liquid was cooled so rapidly that it solidified before the atoms had a chance to rearrange themselves into a perfect crystal. The common scientific assumption at the time was that the atomic arrangement would be like a snapshot of the liquid state. In other words, it would be entirely random, with no discernible order.
David Nelson and one of his students, John Toner, had conjectured that something more subtle might happen. Rapid solidification could result in a mixture of randomness and order. The atoms would be randomly placed in space, but the bonds between those atoms could align, on average, along the edges of a cube, they theorized. The atomic order would then be somewhere between order and disorder. They called the special phase “cubatic.”
To understand the significance of that idea, one must first understand a few basics. The physical properties of matter and how it can be used depend critically on the configuration of its atoms and molecules. For example, consider the crystals graphite and diamond. Based on their physical properties, it is hard to imagine that the two have anything in common. Graphite is soft, slippery, and opaque with a dark metallic appearance. Diamond is ultra-hard, transparent, and shiny. However, both are composed of exactly the same types of atoms and are 100 percent carbon. The only difference between the two substances is how the carbon atoms are arranged, as illustrated below.
In diamond, each carbon atom is bonded to four other carbon atoms in an interconnected three-dimensional network. In graphite, each carbon atom is bonded to only three other carbon atoms in a two-dimensional sheet. The sheets of carbon stack together, one atop the other, like sheets of paper.
The diamond’s network is sturdy and difficult to break apart. Sheets of carbon, on the other hand, easily slip past one another like pieces of paper. That is the basic reason why diamond is so much stronger than graphite. And that difference has a direct effect on their practical applications. Diamond, one of the hardest materials known, is used for drill bits. Graphite, on the other hand, is so soft it is used for pencils. Sheets of carbon peel off as the pencil moves across the page.
This example illustrates how knowing the symmetry of an atomic arrangement in a material makes it possible to understand and predict its properties and to figure the most effective uses for it. The same applies to rapidly cooled solids, which scientists call glassy or amorphous. They are a valuable alternative to slowly cooled crystals because they are observed to have different electronic, thermal, elastic, and vibrational properties. Slowly cooled crystal silicon, for example, is used extensively throughout the electronics industry. But amorphous silicon is less rigid than the slowly cooled material, which makes it advantageous in certain types of solar cells.
The issue that Nelson, Ronchetti, and I wanted to investigate was whether some rapidly cooled solids have a subtle kind of order that had not been detected before and that might suggest additional advantages and applications.
I had already been working for several years on developing ways to simulate the rapid cooling of liquids. I had been invited to spend my summers, first as an undergraduate and then again as postdoctoral fellow, to work on theoretical computer models at Yale University and the IBM Thomas J. Watson Research Center. My main scientific interests were pointing elsewhere at the time. But I took advantage of the summer research opportunities because I was intrigued by the fact that the atomic arrangement of something as rudimentary as amorphous matter was not yet known. In that, I was intentionally following one of the most important lessons learned from my mentor, Richard Feynman: It is wise to follow your heart and seek out good problems wherever they may lead, even if it is not in the direction you thought you were supposed to be headed.
I developed the first computer-generated continuous random network (CRN) model of glass and amorphous silicon in 1973, the summer before my senior year at Caltech. The model was widely used to predict structural and electronic properties of these materials. In later years, while working with Ronchetti, I developed more sophisticated programs to simulate the rapid cooling and solidification process.
In 1980, a chance conversation at Harvard with David Nelson gave new purpose to all my efforts on amorphous material. My computational models could be adapted to test Nelson and Toner’s speculation about cubatic matter.
After explaining all of the background history to the audience at Penn, I moved toward the climax of the talk: If the conjecture about a cubatic phase was right, the atomic bonds in my new computer simulations should not be oriented randomly. On average, the bonds should tend toward a “cubic orientation,” preferentially aligned along the edges of a cube.
We developed a sophisticated mathematical test for the experiment to check whether the average orientations of the bonds displayed the expected cubic symmetry, assigning a numerical score according to how strong the cubic alignment was.
And the result was . . . utter failure. We found no sign whatsoever of a preferential alignment of bonds along the edges of a cube, which Nelson and John Toner had predicted.
But by accident, we discovered something even more interesting. While devising a quantitative mathematical test to check for the orientation of the atomic bonds with the symmetry of a cube, we found that it would be easy to adapt the test to scan for every other possible rotational symmetry. So, as an afterthought, we used the test to give each symmetry a score, based on the degree of alignment among the atomic bonds along different directions.
To our great surprise, a forbidden symmetry scored much higher than the rest—the impossible symmetry of an icosahedron, shown in the left figure on the facing page.
I knew that some in the audience would already be familiar with an icosahedron because the three-dimensional shape was being used as a die in the popular game Dungeons & Dragons, seen below. Others would recognize it from biology, where the shape appears in certain human viruses. The more geometrically inclined would recognize it as one of the five Platonic solids, three-dimensional shapes in which every face is identical to every other, every edge is identical to every other, and every corner is identical to every other.
The significant feature of a three-dimensional icosahedron is that, staring directly down at any of the corners, one observes a pentagonal shape with five-fold symmetry. The same five-fold symmetry that is forbidden for two-dimensional tilings or three-dimensional crystals.
Of course, there is nothing wrong with having a single tile in the shape of a regular pentagon. You can make a single tile any shape you wish. But it is impossible to cover a floor with regular pentagons without leaving gaps. The same applies to an icosahedron. It is possible to make a single three-dimensional die in the shape of an icosahedron. But you cannot fill space with icosahedrons without leaving gaps and holes, as illustrated by the image on the next page.
With so many corners, each with forbidden five-fold symmetry, the icosahedron was well known among scientists studying the structure of matter to be the most forbidden symmetry for an atomic arrangement. This fact was considered so basic that it often appeared in chapter 1 in textbooks. Yet somehow, icosahedral symmetry had received the highest score for the alignment of atomic bonds in our computer experiment.
Strictly speaking, our results had not directly violated the laws of crystallography. Those rules only apply to macroscopic chunks of matter containing tens of thousands of atoms or more. For much smaller groups of atoms, as studied in our simulation, there was no absolute restriction.
In the extreme case of a small cluster containing only thirteen identical gold atoms, for example, the interatomic forces naturally move the atoms into an icosahedral arrangement. One atom is at the center and twelve surrounding atoms are positioned at the corners of an icosahedron. That occurs because interatomic forces are springlike and tend to draw atoms together into tightly packed symmetrical arrangements. The icosahedron occurs for thirteen atoms because it is the most symmetrical tightly-packed configuration that can be achieved. However, as more and more atoms are added, icosahedral symmetry becomes less favored. As shown on the opposite page with the Dungeons & Dragons die, icosahedrons cannot fit neatly together face-to-face, edge-to-edge, or in any other way that does not leave large gaps between them.
What was surprising about our calculation was that the icosahedral symmetry of the bond orientations extended nearly all the way across a simulation containing thousands of atoms. If you had asked most experts at the time, they would have guessed that icosahedral symmetry could not possibly extend over more than fifty atoms or so. But our simulations showed that there remained a high degree of icosahedral symmetry among the bond orientations even when averaged over many atoms. The laws of crystallography, however, demanded that icosahedral symmetry not extend indefinitely. And, sure enough, when we continued to average over yet more and more atoms, the symmetry score began to drop and eventually reached a level that was no longer statistically significant. Even so, the discovery of a high degree of bond alignment along the edges of an icosahedron for thousands of atoms was something truly remarkable.
I reminded the audience that the icosahedral ordering had arisen spontaneously from simulations containing only one type of atom. Most materials contain a combination of different elements with different sizes and different bonding forces. With an increasing number of different elements, I hypothesized, it might become easier to violate the known rules of crystallo
graphy so that icosahedral symmetry could extend over an increasing number of atoms.
Perhaps there could even be circumstances in which the symmetry extended without limit, I suggested. That would be nothing short of revolutionary, a direct violation of the laws Haüy and Bravais established more than a century ago. It was the first time I had expressed such an impossible idea in public, and I ended my lecture with that provocative thought.
There was an enthusiastic round of applause. Several faculty members asked me questions about this detail or that. And I received a lot of nice compliments afterward. But no one commented on my wild speculation about violating the laws of crystallography. Perhaps they all assumed it was merely a rhetorical flourish.
There was one person in the audience, though, who took me seriously. And he was about to gamble his entire future on the idea. The day after my talk, a twenty-four-year-old physics graduate student named Dov Levine showed up at my office and asked if I would be his new PhD advisor. Dov was specifically interested in working with me on the crazy idea he had heard at the end of my talk.
My initial reaction was not very encouraging. That idea is nuts, I told him. I would never recommend that kind of problem to a graduate student, I warned. I was not even sure I would recommend it to an untenured professor like myself. I had only a vague notion about where to start, and the chance of success was laughably small. I went on and on making an endless series of discouraging remarks, but nothing I said seemed to faze him. Dov was emphatic that he wanted to give it a go, no matter the odds.
When I asked Dov to tell me more about himself, he began by explaining that he was born and raised in New York City. That much was already obvious to me, based on his rapid cadence, brash attitude, and wry sense of humor. Dov could never go three sentences without making a joke or an irreverent remark, always accompanied by a mischievous smile.
The Second Kind of Impossible Page 3