One clue was that the Penrose tilings obey something called a “deflation rule.” Namely, each fat and skinny rhombus in a Penrose tiling can be subdivided into smaller pieces that create another Penrose tiling. In the figures seen below, the original tiles have solid lines. The subdivision, or deflation, rule for each fat and skinny tile is indicated by the dotted lines. As shown on the right, the dotted lines join to form a new Penrose tiling with more pieces than the original.
Beginning with a small cluster of tiles, repeated deflation can produce a Penrose tiling with as many pieces as you wish. The inverse process, replacing a group of smaller tiles with larger tiles, is called an inflation rule. The deflation and inflation rules proved to Dov and me that the Penrose tiling has some kind of predictable hierarchical structure.
Dov and I were convinced that the combination of five-fold symmetry, matching rules, and deflation-inflation rules were unmistakable evidence that Penrose’s arrangement of tiles was ordered in some novel, nonintuitive way, but exactly what kind of ordering was it?
It was all incredibly frustrating. Dov and I knew if we could answer that question, we would discover a loophole in the long-accepted rules about symmetry, which dictate what types of matter are possible. And that would be the key to a major paradigm shift and to discovering a range of materials unlike any seen before.
But what in the world could that loophole possibly be? We were stuck.
THREE
* * *
FINDING THE LOOPHOLE
PHILADELPHIA, 1982–83: Dov and I discovered an important clue to unlocking the secret symmetry of the Penrose tilings in the unpublished work of a brilliant amateur mathematician named Robert Ammann.
He was an unusual, reclusive man. Ammann was talented enough to be accepted at Brandeis University in the mid-1960s. But he only attended college for three years, during which time he rarely left his room. The administration finally dismissed him, and he never completed a formal degree.
Next Ammann studied computer programming on his own and obtained a job as a low-level computer programmer. Unfortunately, his job was eliminated during a company-wide cutback. So he began sorting mail at the post office, which was the kind of job that did not require much human interaction. His co-workers considered him uncommunicative and extremely introverted.
What the other postal workers probably never knew was that Ammann was a mathematical genius. Privately, he was engaged in the same world of recreational mathematics as the academic superstars Roger Penrose and John Conway. With typical modesty, Ammann simply described himself as an “amateur doodler with math background.”
Dov and I stumbled across Ammann’s ideas in two short papers in lesser-known journals, written by Alan Mackay, a crystallographer and professor of materials science at the University of London. Mackay shared our fascination with the icosahedron, Penrose tilings, and the fantasy of materials with forbidden five-fold symmetry. His two papers, which were more like speculative essays than research papers, presented some of his notional thoughts on the issue. They included two illustrations that piqued our interest.
In the first, Mackay showed a pair of fat and skinny rhombohedrons, illustrated below. The three-dimensional shapes were already very familiar to Dov and me. We knew they were the obvious three-dimensional analogs of the fat and skinny rhombuses that can be used to create two-dimensional Penrose tilings. So Mackay appeared to be on the same track that we were.
But we were disappointed to find that his paper did not present any matching rules that would prevent the three-dimensional building blocks from forming periodic crystal structures. It was essential for Dov and me to find those particular matching rules. Without them, the atoms would still be able to arrange themselves into any one of a number of ordinary crystal structures, instead of being forced into the impossible structure we were hoping to discover.
Fat Rhombohedron
Skinny Rhombohedron
The second figure Mackay included (not pictured here) also intrigued us. It was a photo of a diffraction pattern created by shining a laser through the image of a Penrose tiling. From Mackay’s image, it was clear that the complex diffraction pattern included some fairly sharp spots, including some at the corners of a decagon and some at the corners of a pentagon. But we could not determine if the spots were truly sharp or somewhat fuzzy pinpoints, or whether they were arranged along perfectly straight lines.
For physicists like Dov and me, those details were critically important. Truly sharp pinpoints aligning in perfectly straight rows, along with arrays of spots in the shapes of perfect decagons and pentagons, would be a diffraction pattern no one had ever seen before. That, of course, would indicate an atomic order no one had ever seen before.
Fuzzy spots with imperfect alignment would be much less exciting. That would indicate a combination of both atomic order and disorder, similar to the arrangements David Nelson and I had already studied, and not a new form of matter.
Clearly, the first possibility, which would indicate something truly novel, was what Dov and I were hoping for. But once we contacted Mackay to ask about matching rules and the precise mathematical nature of the diffraction pattern in his photo, he had no answers to our questions. Mathematics was not his forte, Mackay explained. So he did not know how to prove whether the diffraction spots for a Penrose tiling were perfectly sharp or somewhat fuzzy. He also confessed that he only had one photograph, which was unfortunate because a photograph always introduces a little distortion. So he could not be sure about the diffraction properties.
Mackay also informed us that the fat and skinny rhombohedrons he discussed in his paper were not his own creation. They were taken directly from the work of an unknown amateur named Robert Ammann. That was the first time we had ever heard mention of the mysterious genius who communicated with very few people other than Martin Gardner, the Scientific American guru of recreational mathematicians. Mackay suggested we contact Gardner for help.
Dov immediately wrote to Gardner, who in turn referred us to Branko Grunbaum and Geoffrey Shephard, who were writing an upcoming book about tilings that included some of Ammann’s ingenious inventions. From them, we discovered that Ammann had independently invented rhombus tiles with matching rules that force five-fold symmetry similar to Penrose’s discovery. Incredibly enough, he had also invented another set of tiles with matching rules that force the equally impossible eight-fold symmetry.
Ammann was not a trained mathematician, so he did not provide any proof that his matching rules worked and he never wrote a scientific paper to that effect. He just intuitively knew it to be so.
Gardner also provided us with some of Ammann’s notes that expounded on his ideas for building blocks with icosahedral symmetry. But once again, there were no rigorous proofs nor any attempts at plausible arguments.
Several years later, Dov and I managed to track down the elusive genius in the Boston area and were able to entice him to visit us in Philadelphia. Ammann was every bit as brilliant as I envisioned. He was full of creative geometric ideas and intriguing conjectures that were never published, but which often turned out to be correct. Some, like his ideas about the rhombohedrons that first appeared in Mackay’s illustration, were things that Dov and I had discovered independently as a result of hard work and painstaking proofs. For Ammann, everything was just intuitively obvious. Sadly, Ammann died several years later and Dov and I were never able to meet with him again.
Ammann’s most impactful invention, as far as Dov and I were concerned, was his introduction of the eponymous Ammann bars, which were a powerful matching rule. Using rhombuses with perfectly straight edges, Ammann drew a set of bars across each fat and skinny rhombus according to the precise prescription shown below as dash-line segments.
Ammann’s matching rule is that two tiles can only join together if a bar from each continues straight across any edge where two tiles meet. That produces the same kind of constraints as we had seen with Penrose’s ribbons or interlocks. So at first b
lush, it was nothing remarkable.
But upon closer inspection, the Ammann bars changed everything. Dov and I discovered that the bars revealed something about Penrose tilings that even Penrose himself had not recognized. And that was the discovery that launched Dov and me into the strange new world of impossible symmetries.
Dov and I observed that when the tiles are joined together according to the matching rules, the individual Ammann bars connect to form Ammann lines that stretch across the entire tiling in straight lines. The image opposite shows the tiles and, superimposed, the crisscrossing array of straight Ammann lines. The array consists of five sets of parallel lines oriented along different directions.
Dov and I found that each of the five sets of parallel lines are identical and that the angles between pairs of crisscrossing sets of lines were precisely the same angles as between the edges of a pentagon. This was the simplest proof we could ever imagine that the tiling had perfect five-fold symmetry.
For Dov and me, it was an absolutely thrilling moment. Now we knew for sure that we were heading for a discovery that would be in direct conflict with the centuries-old theorems of Haüy and Bravais. We were confident the Ammann lines held the clue to evading those established theorems, and to explaining the secret symmetry of Penrose tilings. But we still had to decipher their meaning.
The key was to focus on just one of the five sets of parallel lines, such as the set shown with thick lines in the image on the next page. Here we see that the channels between parallel Ammann lines are limited to one of two possible widths, wide (W) and narrow (N). For us, the two most important things would be the ratios between the two channel widths and the frequency with which they repeated in the pattern. We were about to discover that those two features, the ratio and the sequence, were related to two very famous mathematical concepts called the “golden ratio” and the “Fibonacci sequence.”
The golden ratio is often discovered in nature, and has been incorporated in artistic works since ancient times. The Egyptians are thought to have used it to design the Great Pyramids. In the fifth century BCE, the Greek sculptor and mathematician Phidias is purported to have used the golden ratio to create the Parthenon in Athens, which is considered a monument to Greek civilization. The ratio is sometimes signified by the Greek letter Φ, pronounced “phi,” in honor of Phidias.
Euclid, the Greek mathematician who is considered the father of geometry, provided the earliest recorded definition of the golden ratio using a simple object. He considered how to break a stick into two pieces such that the ratio of the shorter length to the longer length was the same as the ratio of the longer length to the sum of the shorter and longer lengths. The solution he found was that the longer piece must be exactly Φ times the length of the shorter piece, where Φ is
. . . a never-ending, never-repeating decimal.
Numbers with nonrepeating decimal forms are called irrational because they cannot be expressed as a ratio of integers. This is to be contrasted with rational numbers, like 2/3 or 143/548, which are ratios of integers and whose decimal forms, 0.333 and 0.26094890510948905109, are seen to regularly repeat if you follow the digits out far enough.
It was not all that surprising to Dov and me that the golden ratio is found in the five-fold symmetry of a Penrose tiling, because the ratio itself is directly related to the geometry of a pentagon. For example, in the left image below, the ratio of the length of the upper line that connects opposite corners of the pentagon to the length of one of the edges of the pentagon is golden. The icosahedron (pictured on the previous page, to the right) also incorporates the golden ratio; the twelve corners form three perpendicular rectangles, and each rectangle has a length to width ratio equal to the golden ratio.
What was surprising to Dov and me, though, was to discover that the golden ratio was incorporated in the sequence of (W)ide and (N)arrow channels, as well.
Consider the channel sequence of Ws and Ns in the figure below. It never settles into a regularly repeating rhythm. If you were to count the Ws and Ns, and then compute the ratio at certain points along the way, you would find that the ratio after the first three channels is 2 to 1; after the first five channels, 3 to 2; after the first eight channels, 5 to 3; etc.
There is a bit of simple arithmetic that can generate this sequence. Consider the first ratio, 2 to 1. Add the two numbers (2 + 1 = 3), and then compare the sum (3) to the larger of the original two numbers (2). This new ratio is 3 to 2, which is also the next ratio in the sequence of channels. Add the next two numbers (3 + 2 = 5), and once again, compare that number to the larger of the previous two: 5 to 3.
You could continue the process indefinitely to obtain 8 to 5, 13 to 8, 21 to 13, 34 to 21, 55 to 34, and so on. The ratios would precisely predict the sequence of Ammann channels.
Dov and I immediately recognized that sequence of integers: 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . They are known as Fibonacci numbers, named after the Italian mathematician Leonardo Fibonacci, who lived in Pisa during the thirteenth century.
The ratios of consecutive Fibonacci numbers—2:1, 3:2, 5:3, . . . —are ratios of integers and, therefore, all rational. But a famous property of the Fibonacci numbers is that, as the integers get larger, the ratios get closer and closer to the golden ratio. That is how Fibonacci numbers and the golden ratio are interrelated.
And it turns out the only way to obtain a pattern of Ws and Ns that reproduces the Fibonacci numbers is to have the Ws repeat with a greater frequency than the Ns, as the Penrose tiling extends out in all directions, by a factor precisely equal to the golden ratio, an irrational number. And that is the secret of the Penrose tiling in a nutshell.
A sequence composed of two elements that repeat at different frequencies, the ratio of which is an irrational number, is called quasiperiodic. A quasiperiodic sequence never repeats exactly.
For example, no two channels in the Fibonacci sequence are surrounded by the same exact pattern of Ws and Ns, though one has to go out quite far in some cases to find the differences. The same applies to Penrose tiles. Check far enough out and you find that no two tiles have the exact same surrounding configuration.
At last, Dov and I could point precisely to the loophole in the centuries-old rules of Haüy and Bravais. The fundamental theorem of crystallography states: If a pattern of tiles or atoms is periodic, occurring with a single repeating frequency, then only certain symmetries are possible. In particular, five-fold symmetry along any direction is truly impossible for an arrangement of atoms that is periodic. We might call this the first kind of impossible, meaning absolutely inviolable, just like 1 + 1 can never equal 3.
However, when scientists asserted to generations of students that five-fold symmetry was impossible for matter of any type, it was a case of the second kind of impossible—a claim resting on an assumption that may not always be valid. In this case, physicists and materials scientists were assuming without proof that all orderly arrangement of atoms are periodic.
The Penrose tiling, Dov and I now understood, is a geometric example of an orderly arrangement that is not periodic. It is quasiperiodic, precisely described in terms of tiles or atoms with two different repeating frequencies with an irrational ratio. That was the loophole we had been seeking. Scientists had been assuming that atoms were always arranged either periodically or randomly in matter. They had not considered quasiperiodic arrangements before.
If real atoms could somehow be arranged into a pattern in which they repeated with two different frequencies whose ratio is irrational, the result would be a whole new form of matter that shattered the rules established by Haüy and Bravais.
It all seemed so simple, and yet so profound. It was as if a new window had magically appeared in front of us, a window that only Dov and I could peer through.
In the distance, I knew, was an entire field of potential breakthroughs. For now, that field was ours and ours alone to explore.
FOUR
* * *
A TALE OF TWO LABORATORIES
Dov and I did not realize it, but we had just entered a race against time. Ever since we had discovered that quasiperiodic order was the secret to creating matter with forbidden symmetries, we had been developing our theory about a new form of matter on our own timetable.
We had no concern that another theoretical physicist would duplicate our work. The peculiar approach we had taken, using recreational mathematics and tilings for inspiration, was far too unconventional to be mimicked. We had not published our ideas yet, so no one else could run ahead with them. And how could an experimentalist who never heard about our quasicrystal theory ever compete? That seemed impossible.
The one thing we did not anticipate was serendipity. Sometimes a simple experiment can produce an unintended outcome. And if the right person is paying attention, there is always the chance of a scientific breakthrough. As it turned out, during the same period of time that Dov and I were systematically developing our radical theory, an unknown scientist named Dan Shechtman, who was working in a lab less than 150 miles from us, had stumbled upon a seemingly nonsensical experimental result.
It was a bizarre coincidence that would become an unusual footnote in the history of science. Two teams with no knowledge of the other were independently challenging the same rigid principles accepted for centuries. It would be two years before either team learned of the other. And once they did, it would quickly become apparent that each team needed the other to reach their goal.
* * *
The Second Kind of Impossible Page 5