I tried not to reveal what I was thinking as I listened to Dov argue why we should pursue my wild idea. But I secretly approved that he was apparently a stubborn person who was not easily dissuaded. That is the kind of attitude one needs to take on a super-high-risk problem, I thought. A good sense of humor would come in handy, as well, since we were likely to encounter more than a few difficulties.
There was something else working in Dov’s favor—a dream of mine that stretched back to when I was thirteen and read the novel Cat’s Cradle, by Kurt Vonnegut. The book is about the potential misuse of science, which was admittedly a strange novel to inspire a budding scientist.
In the book, Vonnegut imagined a new form of frozen water called “ice-nine.” When a seed crystal of ice-nine makes contact with ordinary water, it causes all of the H2O molecules to rearrange themselves into a solid block. A single seed crystal, if thrown in the ocean, could trigger a chain reaction and solidify all of the water on the planet.
Ice-nine was, of course, a fictional creation. But the novel brought to my attention a scientific fact that I had never considered before, which was that the properties of matter can be radically changed by simply rearranging its atoms.
Maybe, just maybe, I thought, there were other forms of matter whose arrangements of atoms had not yet been observed by scientists. And maybe, I imagined, they did not even occur on this planet.
Dov had no way of knowing it, but he was about to help provide me with the opportunity to pursue my longtime scientific fantasy. I agreed to take him on as a student on a trial basis. If we made no progress after six months, we both understood that he might have to find a different topic and a different advisor.
We began by trying to determine the largest number of atoms we could place in a tightly packed arrangement with the symmetry of an icosahedron. In order to visualize what we were doing, Dov, as seen on the left, and I needed to construct some kind of tangible model. But there, we immediately ran into a roadblock. Chemists constructed such models using commercially available kits containing plastic spheres and rods. That was fine, as long as one was studying ordinary crystalline arrangements.
Dov and I were trying to do something different. We needed pieces that could produce bond angles and bond distances appropriate for the symmetry of an icosahedron. Since the symmetry was impossible for crystals, chemistry kits did not include such pieces. Everyone, including the model makers, knew that five-fold symmetry was forbidden. So we had to improvise, and finally resorted to experimenting with Styrofoam balls and pipe cleaners. Before too long, my office started looking like an arts-and-crafts project gone berserk.
We began by assembling a cluster of thirteen Styrofoam balls into the shape of an icosahedron, like the one I had described in my Penn lecture, with one ball in the middle and the other twelve lying at the corners of an icosahedron, as shown on the opposite page.
Then we tried to surround this first icosahedron with twelve more identical icosahedrons, constructing a larger, more complex structure—an “icosahedron of icosahedrons.” But that created an immediate problem. The icosahedrons did not fit together very well. There were large gaps in between them. So we tried to preserve the structure by adding more Styrofoam balls and more pipe cleaners to fill all of the open spaces between the individual icosahedrons. That method worked well enough for us to build a large cluster with the symmetry of an icosahedron containing more than two hundred atoms.
We then tried to repeat our success, this time using thirteen copies of this large cluster to build an even larger cluster. But the gaps that we wound up creating were much larger now, and the model kept falling apart.
Our simple arts-and-crafts project appeared to illustrate a fundamental limitation in creating atomic structures with icosahedral symmetry. Because individual icosahedrons do not fit together neatly, there would always be ever larger gaps to fill as more atoms were added to the structure. From this experience, we hypothesized that it would be impossible to extend icosahedral symmetry beyond a few hundred, or perhaps a few thousand, atoms.
Dov and I wrongly assumed that our strategy of building hierarchically, from one cluster to clusters of clusters, was the only way to maintain icosahedral symmetry. To this day, I keep one of the pipe cleaner models in my office as a reminder of how close we came to reaching the wrong conclusion.
The two of us were thinking about publishing a paper describing our conclusion about the impossibility of icosahedral symmetry. But Dov managed to save us from that embarrassment when he brought me a four-year-old Scientific American article about Penrose tilings. Penrose? I certainly knew the name. But not because of anything related to forms of matter or geometrical tilings.
Roger Penrose (now Sir Roger Penrose), a physicist at Oxford University, was already recognized worldwide for his many contributions to general relativity and its applications to understanding the evolution of the universe. In the 1960s, Penrose proved a set of influential “singularity” theorems showing that, under a wide range of conditions, a universe that is expanding today must have emerged from a big bang. More than four decades later, some cosmologists, including me, are considering ways of avoiding those initial conditions in order to avoid the big bang and replace it with a big bounce.
As luck would have it, the only reason Dov knew about Penrose tilings was because he had originally come to Penn to work on general relativity. In December 1980, a year before he attended my lecture, he had heard Penrose talking about his tiling patterns at an international conference.
* * *
BALTIMORE, MARYLAND, 1980: Dov was attending the “Tenth Texas Symposium on Relativistic Astrophysics.” It was an odd name for the convention, because Baltimore is about 1,300 miles from Dallas. The name follows an informal tradition. Texas was home to the first Symposium on Relativistic Astrophysics, so every meeting thereafter keeps the original name, even if it is held in Geneva, Switzerland.
Dov was strolling down the halls in between scientific talks when he happened to see Roger Penrose chatting with a group of students. Hoping to learn something about Penrose’s latest work in relativity, he moved closer to eavesdrop on the conversation.
To Dov’s surprise, Penrose was not talking about general relativity or cosmology. Instead, he was telling the students about a novel tiling pattern he had constructed a few years earlier for his own amusement. He basically discovered it by doodling. Penrose made sketches of tiles, and groups of tiles, in his notebook until he came up with a tiling that could solve a famous mathematical puzzle. In addition to being a creative genius with unbounded curiosity, Penrose was also an extraordinarily talented artist who could draw precise figures freehand. Throughout his career, Penrose has often used his intricate hand-drawn illustrations to clarify highly technical points in his seminars.
Inventing a new type of tiling may seem like an odd form of amusement. For Penrose, it was an exercise in “recreational mathematics,” a pastime that entails exploring certain well-known mathematical puzzles and challenges. Aficionados range from total amateurs to famous mathematicians, and from young to old.
The leading exponent of recreational mathematics at the time was Martin Gardner, who wrote a monthly column in Scientific American for twenty-five years entitled “Mathematical Games.”
The article Dov brought me was Martin Gardner’s Scientific American article about Penrose tilings published in 1977, about three years after Penrose had invented the tilings. The article explained how Penrose had found a neat solution to a challenge that recreational mathematicians had been discussing for many years: Is it possible to find a set of tiles that can cover a floor without leaving gaps and do so only nonperiodically?
Triangles can cover a floor nonperiodically if, for example, they are arranged in a spiral, as shown by the pattern in the left illustration below. However, triangles can also make patterns that are periodic, as shown in the right illustration below. So triangles would not be a valid solution to the challenge.
Mathematici
ans once thought it was impossible to find any shape or combination of shapes that could satisfy the challenge. But in 1964, mathematician Robert Berger constructed a valid example composed of 20,426 different tile shapes. Over the years, others managed to find examples using many fewer tile shapes.
In 1974, Penrose made a major breakthrough when he found a solution to the challenge using only two tile shapes, which he called “kites” and “darts” (see opposite page). Each of the tiles is marked with circular arcs or “ribbons.” Penrose imposed a rule that two tiles can only be joined together edge-to-edge if the ribbons on both sides of the joint edge match. Following this “matching rule” prevents the tiles from being put together in any regularly repeating pattern. The tiling on the opposite page shows the complex ribbon pattern that emerges when many kites and darts are put together following Penrose’s matching rules.
Nonperiodic
periodic
PHILADELPHIA, OCTOBER 1981: The Gardner article described many surprising features of the original tilings Penrose had discovered, as well as additional properties subsequently discovered by his friend, Cambridge University mathematician John Conway.
Conway has made countless contributions to number theory, group theory, knot theory, game theory, and other fundamental fields of mathematics. For example, Conway invented the Game of Life, a famous abstract mathematical model known as a cellular automaton, which mimics aspects of self-replicating machines and biological evolution.
When Penrose introduced Conway to the new tilings, he went wild with excitement. Conway immediately began cutting out pieces of paper and cardboard, piecing them together, and filling tables and surfaces all over his apartment with configurations of his cut-out shapes in order to study their properties. Gardner’s Scientific American article included many of Conway’s valuable insights that helped Dov and me flesh out certain properties of Penrose tilings that were not obvious at first sight.
In reading other articles, we learned that the precise shapes of the tiles were not important so long as the tiles fit together in ways that were equivalent to kites and darts. A version that was easier for Dov and me to analyze was constructed from a pair of fat and skinny rhombuses, the four-sided shapes used to construct the tiling pattern seen at the top of the opposite page.
It is possible to arrange just the fat rhombuses into a periodic pattern, or just the skinny shapes into a periodic pattern, or to arrange many different combinations of the two shapes together into various other periodic patterns.
But the rhombuses are not the whole story. To prevent all the periodic possibilities and force a nonperiodic arrangement, it is necessary to introduce some sort of matching rule. One approach is to apply ribbons analogous to ones Penrose invented for the kites and darts and impose the rule that two tiles can only be joined together if the ribbons match along the edge where they meet.
Another way to prevent an ordinary periodic pattern is to transform their straight edges into curves and notches analogous to puzzle interlocks, as shown in a beautiful example at the bottom of the facing page, which is made of individual wooden pieces. The tiling constructed of wooden tiles is equivalent to the tiling made of gray and white rhombuses in terms of the arrangement of units. The only difference is that interlocks have been added to the wooden tiles. With interlocks, the pieces fit together like a puzzle and there is no way to put them together in any regularly repeating pattern.
If this is the first time you are viewing a Penrose tiling, take a few moments to study it and see what your first impression is. How would you characterize it? Are you looking at an orderly or a disorderly pattern? If you think the tiles follow an orderly progression, how would you predict which tiles occur next?
Looking at the tiling of gray fat and white skinny rhombuses, Dov and I noticed certain frequently repeating motifs, such as a star-shaped cluster of five gray tiles surrounding a central point, which is not what one would expect from a random pattern. But we also noticed that those clusters did not repeat with equal spacing, as one would expect in a periodic pattern. Nor did the spacing between repeats appear to be arbitrary, as one would expect for a random pattern.
Comparing the configurations of tiles that immediately surround the star-shaped clusters, we observed that not all stars had the same surroundings. And when we considered the next layer of surrounding tiles, we observed even more differences. Study the figure on the previous page and you will note those differences. In fact, no two stars ever have exactly the same surroundings if one continues far enough out from the center of the star.
This was significant, because Dov and I knew it was the opposite of what one finds in a periodic pattern. Every tile in a square tiling always has exactly the same surrounding as any other tile no matter how far out from the center of the design you look.
From that simple observation, we confirmed that the Penrose pattern could not be periodic. Then again, a pattern composed of clusters that were nearly the same and repeated frequently in the tiling could not be considered random, either. That led to the question: What kind of pattern could be both non-periodic and non-random at the same time?
There was no ready answer, which really intrigued me. No one had ever seen anything quite like a Penrose pattern before he invented them in 1974. Even Penrose himself did not appear to fully appreciate what he had invented. In his original paper, Penrose described his pattern as “non-periodic,” which precisely defines what the tiling is not. It is not periodic. But it does not say what the tiling actually is. And that was the crucial issue for Dov and me.
The moment we began to study Penrose’s tiling, we imagined that we might be able to construct an analogous three-dimensional pattern using a pair of building blocks. Then, by replacing each building block shape with a certain type of atom or cluster of atoms, we hoped to construct an atomic structure that would achieve our dream of a new type of matter.
But first, in order to show that the new atomic structure was truly novel and to figure out its distinctive physical properties, we needed to identify its symmetries. Merely describing the matter as non-periodic or non-random was not going to be good enough. So we spent the next few months focusing on Penrose’s tiling to see if we could discover the mathematical secret to its symmetries.
The first remarkable property of Penrose tilings that Dov and I established was that they had a subtle kind of five-fold rotational symmetry, which was, of course, supposed to be impossible.
To see the five-fold symmetry of the Penrose tiling requires some effort. The image on the following page shows again an enlargement of the tiling composed of gray fat and white skinny rhombus tiles. Take a moment and study the tiles that immediately surround any one of the star clusters. The arrangement is highly complex. Imagine rotating it one-fifth of the way around a circle, or 72 degrees. Does the arrangement look the same as the original?
If you try the experiment, you will find that the answer is “it depends.” For some stars, the answer is “no.” So ignore them and choose another. Continue until you find a star cluster for which the answer is “yes.” You won’t have to look very far to find one.
Now consider the second layer of tiles surrounding the star cluster you have chosen. Repeat the rotation by 72 degrees, one-fifth of the way around a circle, and ask if this larger configuration of tiles, which extends to two layers of tiles around the original star cluster, looks identical to the original.
Once again, the answer for some stars will be “no.” So ignore those, too, and continue until you have found one of the rarer star clusters for which the answer is “yes.” Now, repeat the process again for this subset going out three layers. And so on.
As you check more and more layers, you will be discarding more and more star clusters, but you will also find that there will always be some clusters remaining that have five-fold symmetry. The procedure is much more tedious than what is needed in order to check the symmetry for a periodic tiling, but it is still enough to prove that Penrose tiling has fi
ve-fold rotational symmetry.
A more sophisticated mathematical analysis can be used to show that, technically speaking, the Penrose tiling has more than five-fold symmetry. It actually has ten-fold symmetry. But for Dov and me, it made little difference whether the tiling had five- or ten-fold symmetry. Either way, the symmetry was strictly forbidden according to the mathematics of tilings and the established laws of crystallography.
That could only mean there was a faulty assumption underlying those laws that everyone had been missing for more than two hundred years. There was some kind of loophole. Once we realized that, Dov and I were hooked. We just had to find the loophole.
We already knew about matching rules, the mysterious interlocks that prevented the tiles from being put together in any kind of periodic pattern. Matching rules meant the shapes would only be allowed to fit together in patterns with the forbidden five-fold symmetry.
Using our ball and stick models, Dov and I had already begun to construct analogous three-dimensional structures composed of building blocks, where each block represented one or more atoms. For our model, we translated the Penrose interlocks into atomic bonds, which connected the atoms represented by one of our three-dimensional building blocks with those of another. The atoms would then be naturally prevented from solidifying into any type of crystal with a regular periodic pattern. The atoms would, instead, be forced to make the new type of matter with the icosahedral symmetry we were seeking.
This line of thinking was personally intriguing for me, because it was remarkably reminiscent of Vonnegut’s imaginary ice-nine, in which a new arrangement of water molecules—ice-nine—was more stable than ordinary crystalline ice. The new form of matter we were pursuing, if we ever found it, might turn out to be a remarkably stable material that was harder than ordinary crystals. But what kind of regularity were the matching rules imposing?
The Second Kind of Impossible Page 4