The Second Kind of Impossible
ONE IMPOSSIBLE!
PASADENA, CALIFORNIA, 1985: Impossible!
The word resonated throughout the large lecture hall. I had just finished describing a revolutionary concept for a new type of matter that my graduate student, Dov Levine, and I had invented.
The Caltech lecture room was packed with scientists from every discipline across campus. The discussion had gone remarkably well. But just as the last of the crowd was filing out, there arose a familiar, booming voice and that word: “Impossible!”
I could have recognized that distinctive, gravelly voice with the unmistakable New York accent with my eyes closed. Standing before me was my scientific idol, the legendary physicist Richard Feynman, with his shock of graying, shoulder-length hair, wearing his characteristic white shirt, along with a disarming, devilish smile.
Feynman had won a Nobel Prize for his groundbreaking work developing the first quantum theory of electromagnetism. Within the scientific community, he was already considered one of the greatest theoretical physicists of the twentieth century. He would eventually achieve iconic status with the general public, as well, because of his pivotal role identifying the cause of the Challenger space shuttle disaster and his two bestselling books “Surely You’re
Joking, Mr. Feynman!” and “What Do You Care What Other People Think?”
He had a wonderfully playful sense of humor, and was notorious for his elaborate practical jokes. But when it came to science, Feynman was always uncompromisingly honest and brutally critical, which made him an especially terrifying presence during scientific seminars. One could anticipate that he would interrupt and publicly challenge a speaker the moment he heard something that was, in his mind, imprecise or inaccurate.
So I had been keenly aware of Feynman’s presence when he entered the lecture hall just before my presentation began and took his usual seat in the front row. I kept a careful watch on him out of the corner of my eye throughout the presentation, awaiting any potential outburst. But Feynman never interrupted and never raised an objection.
The fact that he came forward to confront me after the talk was something that probably would have petrified many scientists. But this was not our first encounter. I had been lucky enough to work closely with Feynman when I was an undergraduate at Caltech about a decade earlier and had nothing but admiration and affection for him. Feynman changed my life through his writings, lectures, and personal mentoring.
When I first arrived on campus as a freshman in 1970, my intention was to major in biology or mathematics. I had never been particularly interested in physics in high school. But I knew that every Caltech undergraduate was required to take two years of the subject.
I quickly discovered that freshman physics was wickedly hard, thanks in large part to the textbook, The Feynman Lectures on Physics, Volume 1. The book was less of a traditional textbook than a collection of brilliant essays based on a famous series of freshman physics lectures that Feynman delivered in the 1960s.
Unlike any other physics textbook that I have ever encountered,
The Feynman Lectures on Physics never bothers to explain how to solve any problems, which made trying to complete the daunting homework assignments challenging and time-consuming. What the essays did provide, however, was something much more valuable—deep insights into Feynman’s original way of thinking about science. Generations have benefited from the Feynman Lectures. For me, the experience was an absolute revelation.
After a few weeks, I felt like my skull had been pried open and my brain rewired. I began to think like a physicist, and loved it. Like many other scientists of my generation, I was proud to adopt Feynman as my hero. I scuttled my original academic plans about biology and mathematics and decided to pursue physics with a vengeance.
I can remember a few times during my freshman year when I screwed up enough courage to say hello to Feynman before a seminar. Anything more would have been unimaginable at the time. But in my junior year, my roommate and I somehow summoned the nerve to knock on his office door to ask if he might consider teaching an unofficial course in which he would meet once a week with undergraduates like us to answer questions about anything we might ask. The whole thing would be informal, we told him. No homework, no tests, no grades, and no course credit. We knew he was an iconoclast with no patience for bureaucracy, and were hoping the lack of structure would appeal to him.
A decade or so earlier, Feynman had given a similar class, but solely for freshmen and only for one quarter per year. Now we were asking him to do the same thing for a full year and to make it available for all undergraduates, especially third- and fourth-year students like ourselves who were likely to ask more advanced questions. We suggested the new course be called “Physics X,” the same as his earlier one, to make it clear to everyone that it was completely off the books.
Feynman thought a moment and, much to our surprise, replied
“Yes!” So every week for the next two years, my roommate and I joined dozens of other lucky students for a riveting and unforgettable afternoon with Dick Feynman.
Physics X always began with him entering the lecture hall and asking if anyone had any questions. Occasionally, someone wanted to ask about a topic on which Feynman was expert. Naturally, his answers to those questions were masterful. In other cases, though, it was clear that Feynman had never thought about the question before. I always found those moments especially fascinating because I had the chance to watch how he engaged and struggled with a topic for the first time.
I vividly recall asking him something I considered intriguing, even though I was afraid he might think it trivial. “What color is a shadow?” I wanted to know.
After walking back and forth in front of the lecture room for a minute, Feynman grabbed on to the question with gusto. He launched into a discussion of the subtle gradations and variations in a shadow, then the nature of light, then the perception of color, then shadows on the moon, then earthshine on the moon, then the formation of the moon, and so on, and so on, and so on. I was spellbound.
During my senior year, Dick agreed to be my mentor on a series of research projects. Now I was able to witness his method of attacking problems even more closely. I also experienced his sharp, critical tongue whenever his high expectations were not met. He called out my mistakes using words like “crazy,” “nuts,” “ridiculous,” and “stupid.”
The harsh words stung at first, and caused me to question whether I belonged in theoretical physics. But I couldn’t help noticing that Dick did not seem to take the critical comments as seriously as I did. In the next breath, he would always be encouraging me to try a different approach and inviting me to return when I made progress.
One of the most important things Feynman ever taught me was that some of the most exciting scientific surprises can be discovered
in everyday phenomena. All you need do is take the time to observe things carefully and ask yourself good questions. He also influenced my belief that there is no reason to succumb to external pressures that try to force you to specialize in a single area of science, as many scientists do. Feynman showed me by example that it is acceptable to explore a diversity of fields if that is where your curiosity leads.
One of our exchanges during my final term at Caltech was particularly memorable. I was explaining a mathematical scheme that I had developed to predict the behavior of a Super Ball, the rubbery, super-elastic ball that was especially popular at the time.
It was a challenging problem because a Super Ball changes direction with every bounce. I wanted to add another layer of complexity by trying to predict how the Super Ball would bounce along a sequence of surfaces set at different angles. For example, I calculated the trajectory as it bounced from the floor to the underside of a table to a slanted plane and then off the wall. The seemingly random movements were entirely predictable, according to the laws of physics.
I showed Feynman one of my calculations. It predicted that I could throw the Super Ball and that, after a complicated set of bounces, it would return right back to my hand. I handed him the paper and he took a glance at my equations.
“That’s impossible!” he said.
Impossible? I was taken aback by the word. It was something new from him. Not the “crazy” or “stupid” that I had come to occasionally expect.
“Why do you think it’s impossible?” I asked nervously.
Feynman pointed out his concern. According to my formula, if someone were to release the Super Ball from a height with a certain spin, the ball would bounce and careen off nearly sideways at a low angle to the floor.
“And that’s clearly impossible, Paul,” he said.
I glanced down to my equations and saw that, indeed, my prediction did imply that the ball would bounce and take off at a low angle. But I wasn’t so sure that was impossible, even if it seemed counterintuitive.
I was now experienced enough to push back. “Okay, then,” I said. “I have never tried this experiment before, but let’s give it a shot right here in your office.”
I pulled a Super Ball out of my pocket and Feynman watched me drop it with the prescribed spin. Sure enough, the ball took off in precisely the direction that my equations predicted, scooting sideways at a low angle off the floor, exactly the way Feynman had thought was impossible.
In a flash, he deduced his mistake. He had not accounted for the extreme stickiness of the Super Ball surface, which affected how the spin influenced the ball’s trajectory.
“How stupid!” Feynman said out loud, using the same exact tone of voice he sometimes used to criticize me.
After two years of working together, I finally knew for sure what I had long suspected: “Stupid” was just an expression Feynman applied to everyone, including himself, as a way to focus attention on an error so it was never made again.
I also learned that “impossible,” when used by Feynman, did not necessarily mean “unachievable” or “ridiculous.” Sometimes it meant, “Wow! Here is something amazing that contradicts what we would normally expect to be true. This is worth understanding!”
So eleven years later, when Feynman approached me after my lecture with a playful smile and jokingly pronounced my theory “Impossible!” I was pretty sure I knew what he meant. The subject of my talk, a radically new form of matter known as “quasicrystals,” conflicted with principles he thought were true. It was therefore interesting and worth understanding.
Feynman walked up to the table where I had set up an experiment to demonstrate the idea. He pointed to it and demanded, “Show me again!”
I flipped the switch to start the demonstration and Feynman stood motionless. With his own eyes, he was witnessing a clear violation of one of the most well-known principles in science. It was something so basic that he had described it in the Feynman Lectures. In fact, the principles had been taught to every young scientist for nearly two hundred years . . . ever since a clumsy French priest made a fortuitous discovery.
PARIS, FRANCE, 1781: René-Just Haüy’s face turned ashen, as the small sample of calcareous spar slipped out of his hands and fell to the floor with a crash. As he bent to collect the pieces, though, his sense of embarrassment melted away, replaced by curiosity. Haüy noticed that the surfaces where the sample had split apart were smooth and neatly angled, not rough and chaotic, as the outer surface of the original sample had been. He also noticed that the smaller pieces had facets that met at the same precise angles.
It was certainly not the first time someone had cracked open a rock. But this was one of those rare moments in history when an everyday occurrence leads to a scientific breakthrough because the person involved has both the instincts and the acumen to recognize the significance of what has just occurred.
Haüy had been born to humble beginnings in a French village. Early on, priests at a local monastery recognized his intellectual abilities and helped him achieve an advanced education. He eventually joined them in the Catholic priesthood and accepted a position teaching Latin at a Parisian college.
It was only after his theological career was under way that Haüy
discovered his passion for the natural sciences. The turning point came when one of his colleagues introduced him to botany. Haüy was fascinated by the symmetry and the specificity of plants. Despite their tremendous variety, plants could be precisely classified on the basis of their color, shape, and texture. The thirty-eight-year-old priest soon became an expert in the subject, frequently visiting the Jardin du Roi in Paris to test his identification skills.
Then, during one of his many visits to the Jardin, Haüy was exposed to another field of science that was to become his true calling. The great naturalist Louis-Jean-Marie Daubenton had been invited to give a public lecture about minerals. During the presentation, Haüy learned that minerals, like plants, come in many different colors, shapes, and textures. But at that point in history, the study of minerals was a much more primitive discipline than botany. There was no scientific classification of the various types of minerals nor any understanding about how they might be related to one another.
Scientists knew that minerals, like quartz, salt, diamond, and gold, are solely composed of one pure substance. If you were to smash them to bits, each bit would consist of exactly the same material. They also knew that many minerals form faceted crystals.
But unlike plants, two minerals of the same type can have very different colors, shapes, and textures. Everything depends on the conditions under which they are formed and what happens to the mineral afterward. In other words, minerals seemed to defy the neat and tidy classification that Haüy had come to appreciate about botany.
The lecture inspired him to ask an acquaintance, the wealthy financier Jacques de France de Croisset, if he could examine his private mineral collection. The visit was a joy for Haüy, up until the fateful moment when he dropped the sample of calcareous spar.
The financier graciously accepted Haüy’s apologies for the damage
he had caused. But he also noticed his guest’s absolute fixation on the shattered remains and generously offered to let him take some of the pieces home for further study.
Back in his room, Haüy took a small fragment of an irregular shape and carefully cleaved its surfaces, chipping away, bit by bit, until the exterior consisted entirely of smooth, flat facets. He noticed that the facets formed a small rhombohedron, the relatively simple shape of a cube pushed on an angle.
Rhombohedron
Haüy then took another calcareous spar fragment with a rough outer shape and repeated the same operation. Once again a rhombohedron emerged. This time the size of the rhombohedron was somewhat larger, but it had the exact same angles as the one he had tested before. Haüy repeated the experiment many times, utilizing all of the different fragments he had been given. Later he did the same for many other samples of calcareous spar found in different regions of the world. Each time he found the same result: a rhombohedron with the same angles between facets.
The simplest explanation Haüy could think of was that the calcareous spar was composed of a basic building block that was, for some unknown reason, shaped like a rhombohedron.
Haüy then expanded his experiments to include other types of minerals. In each case, he found that the mineral could be cleaved and reduced to a building block with a certain precise geometrical shape. Sometimes it was a rhombohedron, just like the calcareous spar. Sometimes it was a rhombohedron with different angles between facets. Sometimes it was a different shape altogether. He shared some of his findings with French naturalists and won broad acceptance from the scientific community, which enabled him to continue
his methodical study of minerals for the next two decades, including throughout the French Revolution.
Haüy finally published his masterpiece, the Traité de Minéralogie, in 1801. It was a superbly illustrated atlas compiling his results and presenting the “laws of crystal forms” that he had discovered while gathering his data.
The publication was an instant classic. It earned him an academic scientific position, the admiration of his peers, and a place in history as the “Father of Modern Crystallography.” Haüy’s scientific contributions were considered so important that Gustav Eiffel chose to include him on the list of seventy-two French scientists, engineers, and mathematicians whose names are engraved on the first floor of the Eiffel Tower.
A profound implication of Haüy’s work was that minerals are composed of some kind of primitive building block, which he called la molécule intégrante, that repeats over and over throughout the material. Minerals of the same type are constructed from the same building block no matter where in the world they may originate.
Several years later, Haüy’s discovery helped inspire an even bolder idea. British scientist John Dalton proposed that all matter, not just minerals, is made of indivisible and indestructible units called atoms. According to this idea, Haüy’s primitive building blocks corresponded to a cluster of one or more atoms whose type and spatial arrangement determined the type of mineral.
The ancient Greek philosophers Leucippus and Democritus are often credited with introducing the concept of atoms in the fifth century BCE. But their ideas were strictly philosophical. It was Dalton who transformed the atomic hypothesis into a testable scientific theory.
From his experience studying gases, Dalton concluded that atoms are spherical in shape. He also proposed that different types of atoms have different sizes. They were far too small to be seen by cleaving
minerals or using any of the other technologies available in the nineteenth century. So it would take more than a century of fierce debate and the development of new technologies and new types of experiments before the atomic theory was fully accepted.
Despite their accomplishments, neither Haüy nor Dalton could explain one of Haüy’s most important discoveries. No matter which mineral he studied, the primitive building block, la molécule intégrante, was either a tetrahedron, a triangular prism, or a parallelepiped, which is a broader category that includes the rhombohedron that Haüy originally observed. Why should that be so?
The search for an explanation continued for many decades, ultimately leading to the creation of a new and pivotal field of science known as “crystallography.” Based on rigorous mathematical principles, crystallography would eventually make an enormous impact on other scientific disciplines, including physics, chemistry, biology, and engineering.
The laws of crystallography would turn out to be powerful enough to explain all of the possible forms of matter known at the time and to predict many of their physical properties, such as hardness, response to heating and cooling, conduction of electricity, and elasticity. Crystallography’s success in explaining so many different properties of matter relevant to so many different disciplines has long been considered one of the great triumphs of nineteenth-century science.
Yet, by the early 1980s, it was precisely these celebrated laws of crystallography that my student Dov Levine and I were challenging.
We had figured out how to construct novel building blocks that could be packed into arrangements that were supposedly impossible. The fact that we had discovered something new in what was thought to be a well-understood, fundamental principle of science was what had grabbed Feynman’s attention during my lecture.
To fully appreciate his surprise warrants a brief introduction to the three simple principles that are at the foundation of crystallography:
The first principle is that all pure substances, such as minerals, form crystals, as long as there is enough time for the atoms and molecules to move into an orderly arrangement.
The second states that all crystals are periodic arrangements of atoms, meaning their structure is entirely composed of one of Haüy’s primitive building blocks, a single cluster of atoms that repeats over and over in any direction with equal spacing.
The third principle is that every periodic atomic arrangement can be categorized according to its symmetries, and there is a finite number of possible symmetries.
This third principle is the least obvious of the three, but can be easily illustrated with everyday floor tiles. Imagine that you want to cover a floor with regularly spaced tiles that have identical shapes, as seen in the examples opposite. Mathematicians call the resultant pattern a periodic tiling. The tiles are two-dimensional analogs of Haüy’s three-dimensional primitive building blocks because the entire pattern is composed of repeated elements of the same unit. Periodic tilings are frequently used in kitchens, patios, bathrooms, and entryways. And those patterns often include one of five basic shapes: rectangles, parallelograms, triangles, squares, or hexagons.
But what other simple shapes are possible? Stop and think about this for a moment. What other basic shape could you use to tile your floor? How about a regular pentagon, a five-sided shape whose edges have equal length and whose angles have equal measure?
The answer may surprise you. According to the third principle of crystallography, the answer is no. Absolutely not. A pentagon won’t work. In fact, nothing else works. Every two-dimensional periodic pattern corresponds to one of the five patterns shown above.
You might find a floor’s tiling pattern that seems to be an exception to the rule. But that is a bit of trickery. If you take a closer look, the tiling will always turn out to be one of the same five patterns in disguise. For example, you could make more complicated-looking patterns by replacing each of the straight edges with identical curvy ones. You could also cut or divide each tile—for example, a square along the diagonal—and then fit them back into a pattern using other geometric shapes. Or you could choose a picture or design and insert it in the center of each tile. But, from a crystallographer’s point of view, none of that changes the fact that the framework is equivalent to one of the five possibilities above. No other fundamental patterns exist.
If you asked your contractor to cover your shower floor with regular pentagons, you would actually be asking for a lot of water damage. No matter how the tiler tried to jam the pentagons together, there would always be gaps (see the next page). Lots of them! The same would be true if you tried a regular seven-sided heptagon, eight-sided octagon, or nine-sided nonagon. The list of forbidden shapes goes on and on forever.
The five periodic patterns are key to understanding the basic structure of matter. Scientists also classify them according to their “rotational symmetry,” which is a complicated-sounding name for a straightforward concept. Rotational symmetry is defined as the number of times you can rotate an object within 360 degrees so that it always looks unchanged, as compared to the original.
For example, consider the leftmost square tiling shown on the opposite page. Let’s say you turn your back and your friend rotates the square tiling 45 degrees, as illustrated by the middle figure. When you return to face the tiling, you are able to see it does not look the same as it did originally, being obviously oriented in a different direction. So this rotation of 45 degrees is not considered a “symmetry” of the square.
But, starting over again, if your friend rotates the tiling 90 degrees (the right-hand figure on the facing page) then you will not be able to detect that anything has changed. The tiling looks exactly the same as
it did originally. This rotation by 90 degrees is considered a rotational “symmetry.” In fact, 90 degrees is the minimal angle of rotation that creates a symmetry of the square pattern. Any rotation of the square less than 90 degrees alters the apparent orientation.
It then becomes clear that two rotations by 90 degrees, for a total of 180 degrees, is also a symmetry. So are three rotations (270 degrees) and four (360 degrees). Since it takes a total of four rotations to complete 360 degrees, the square tiling is said to have four-fold symmetry.
Let’s give your friend a tiling composed of equal rows of rectangles whose long sides are oriented horizontally. Rotating the tiling by 90 degrees makes it look different because the long sides are now oriented vertically. But rotating it 180 degrees makes it looks the same as it did originally. So in the case of a rectangle, 180 degrees is the smallest rotation that is a symmetry. Twice that rotation is 360 degrees. So a tiling of rectangles has a two-fold symmetry.
Similarly, for a tiling of parallelograms, the only rotation that leaves the tiling looking unchanged is 180 degrees. Therefore, a parallelogram tiling also has two-fold rotational symmetry.
Using the same method, an equilateral triangle can be seen to have three-fold symmetry. A hexagon has six-fold.
Finally, there is one other possible rotational symmetry that can
be made from any of the five patterns. If we make the edges of any one of the shapes irregularly jagged, for example, the only rotation that leaves the pattern looking unchanged would be a complete turn of 360 degrees, or one-fold symmetry.
And that completes the list of possibilities. One-, two-, three-, four-, and six-fold are the only rotational symmetries allowed for two-dimensional periodic patterns, a fact that has been known for millennia. Ancient Egyptian artisans, for example, used rotational symmetries to create beautiful mosaics. But it was not until the nineteenth century that those trial-and-error methods were fully explained by rigorous mathematics.
But let’s get back to your shower floor. The fact that your contractor cannot make a periodic pattern using only regular five-sided pentagon tilings without leaving substantial gaps and creating water damage is a vivid demonstration that five-fold symmetry is impossible according to the laws of crystallography. But it is not the only forbidden symmetry. The same is true for seven-, eight-, and any higher-fold symmetry.
Remember that Haüy discovered that crystals are periodic, just like the tiles on your floor, with regularly repeating patterns. So by extension, the same types of restrictions that apply to tilings also apply to three-dimensional crystals. Only certain patterns can fit together without creating gaps.
But despite that similarity, three-dimensional crystals are much more complicated than floor tiles because crystals can have different rotational symmetries along different viewing directions. The symmetries vary depending on your vantage point. However, no matter what vantage point one chooses, the only possible symmetries for regularly repeating three-dimensional structures and periodic crystals are one-, two-, three-, four-, and six-fold, the same restriction that applies to two-dimensional tiles. And no matter what vantage point one
chooses, five-fold rotational symmetry is always forbidden, along with seven-, eight-, and any higher-fold symmetry.
How many distinct combinations of symmetries as viewed along all possible vantage points are possible for periodic crystals? Finding the answer was a great mathematical challenge.
The problem was finally solved in 1848 by French physicist Auguste Bravais, who showed that there are exactly fourteen distinct possibilities. Today, these are known as the Bravais lattices.
But the challenge to understanding crystal symmetries did not end there. A more complete mathematical classification was later developed combining rotational symmetries with even more complicated symmetries, known as “reflections,” “inversions,” and “glides.” When all of these additional possibilities are added to the mix, the number of possible symmetries altogether grows from 14 to a total of 230. But among all of those possibilities, five-fold symmetry remains forbidden along any direction.
These discoveries brought the beauty of mathematics together with the beauty of the natural world in a most remarkable way. The identification of all of the 230 possible three-dimensional crystal patterns was accomplished using pure mathematics. And each of those patterns could also be found in nature by cleaving minerals.
The remarkable correspondence between the abstract mathematical crystal patterns and the real crystals found in nature was indirect but compelling evidence that matter is composed of atoms. But exactly how were those atoms arranged? Cleaving could reveal the shape of the building blocks, but it was far too crude of a tool to determine how atoms are arranged within.
A precise tool capable of obtaining this information was invented by German physicist Max von Laue at the University of Munich in 1912. He discovered that he could precisely determine the hidden symmetry of a chunk of matter simply by shining a beam of X-rays through a small sample of the material.
X-rays are a type of light wave whose wavelength is so small that they can easily pass through the channels of empty space between the regularly spaced rows of atoms in crystals. When the X-rays passing through the crystal are then projected onto a piece of photographic paper, von Laue showed, they interfere with one another to produce a pinpoint pattern of sharp spots known as an “X-ray diffraction pattern.”
If the X-rays are pointed along a line of rotational symmetry through the crystal, the pinpoint diffraction pattern has precisely the same symmetry. By shining the X-rays through a crystal along various directions, the full set of symmetries of its atomic structure can be revealed. And from that information, the crystal Bravais lattice and the shape of its building block can also be identified.
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.