Years later, when Peter had a chance to travel to Bukhara, Uzbekistan, on his summer vacation, he found many examples of periodic patterns that included ten-fold stars as part of the repeating motif. That experience inspired him to search through catalogs of Islamic tilings when he returned home. Many tilings were like the ones he saw in Bukhara—periodic patterns with regularly placed five- and ten-fold stars. But a pattern found on the Darb-i Imam shrine in Isfahan, Iran, a monument with an inscription dated 1453, defied simple description (see color insert, image 3).
* * *
Peter contacted me soon afterward to help analyze the complex tiling. We transposed the photograph into a precise geometrical pattern composed of three shapes known as “girih tiles,” as shown in image 4 in the color insert. From this, we discovered that the pattern was almost perfectly quasiperiodic, except for a small percentage of errors that may have been due to later repairs. Furthermore, we found that one could construct this pattern and an infinite extension of it using a kind of deflation, or subdivision, rule that was much more complex than the one for Penrose tiles.
Unfortunately, we have no record of how the artisans of Darb-i Imam designed the complex pattern. One can only hazard a guess, based on the fragments observed today on the shrine. Although the design suggests some knowledge of the deflation rule Peter and I identified, there was no evidence that the artisans applied any matching rules. At the present time, there is no other Islamic tiling with as many surviving tiles that is known to be as perfectly quasiperiodic.
The Islamic tiling project was a fascinating digression into art and archaeology, but I was not ready to give up on my quest for a natural quasicrystal. I was still hoping that someone would respond to the paper Peter and I had published describing our search through the ICDD catalog.
In our concluding paragraph, we had offered to share the remaining list of potential candidates that we had been unable to examine with anyone willing to join the search. “Those interested are encouraged to contact PJL and PJS [Peter and Paul].”
We had hoped the invitation would act as a scientific homing beacon. But no one answered our call for help . . . no one . . . not for six long years. And then . . .
EIGHT
* * *
LUCA
PRINCETON, BOSTON, AND FLORENCE, ITALY, 2007: On May 31, 2007, Peter Lu and I received an email from an Italian mineralogist named Luca Bindi. We were caught off guard. Neither of us had ever heard of Luca before. But he had obviously heard of us.
Luca had been studying a special class of minerals known as “incommensurate crystals” whose atoms are spaced quasiperiodically, similar to a quasicrystal, but in a way that maintains the long-standing rotational symmetry rules of Haüy and Bravais, which quasicrystals do not.
While researching the topic, Luca had unearthed our paper describing our methodical search for natural quasicrystals. He had taken note of the fact that we were inviting potential collaborators to contact us and was writing to take us up on the offer.
Luca identified himself as the head of the Department of Mineralogy at the Museo di Storia Naturale dell’Università di Firenze, the Natural History Museum at the University of Florence. He volunteered to study any potential quasicrystal mineral candidates that could be found in his museum’s collection.
In other words, I thought, here is an Italian scientist I have never heard of, who is volunteering to join a wild-goose chase designed by a couple of American scientists whom he has never met, and whose search for natural quasicrystals had, by the way, failed to produce any tangible results for the past eight years. Who is this guy? I wondered.
Peter was, by then, an advanced graduate student at Harvard working on projects unrelated to quasicrystals. He asked if I thought we should proceed to work with this unknown scientist. Why not? I figured.
Almost immediately, Luca became as obsessed about the search for natural quasicrystals as me. Although he was athletic and loved being outdoors, he also had the patience required to spend countless hours working alone in the laboratory, even if the chance of success was minuscule.
Peter and I began by sending Luca our list of prime quasicrystal suspects based on the ICDD catalog. Luca then proceeded to collect samples in his museum that were on our list and carefully analyze them. It did not go well. Over the next several months, he reported disappointing results to me on a regular basis. Failure after failure after failure.
At one point, I suggested to Luca that, rather than continue to search terrestrial minerals, “Meteorites are more promising because they include a variety of pure metal alloys. I would be interested to work on them with you.” That thought would later prove to be prophetic. But Luca did not pick up on my suggestion at the time, perhaps because he was a mineralogist, and meteorites were outside his area of expertise.
It was during this period of back-and-forth communication that Luca and I began to develop a deep friendship, which was tested from the very beginning by the ongoing frustrations he was experiencing in the lab. Our respect for each other was grounded in science and nurtured by daily emails and Skype chats.
* * *
FLORENCE AND GENOA, ITALY, NOVEMBER 3, 2008: After more than a year of disappointing results, Luca suddenly, and unilaterally, did what any good scientist would do. He cast aside the losing strategy and adopted a new one.
Although the ICDD file that Peter and I had used for our initial analysis contained diffraction patterns for thousands of minerals, there were some rare and recently discovered natural minerals that were not yet included. Luca decided to specifically target those minerals. He narrowed the search even more by focusing on the ones that included metallic aluminum and copper, which was a popular combination of elements being used at the time to create many synthetic quasicrystals.
In early November of 2008, I went to Italy to attend the annual Festival della Scienza in Genoa. I had been invited to the festival to speak about Endless Universe, a book written for the general public about the cyclic theory of the universe that I developed with physicist Neil Turok. The cyclic theory is the leading alternative to the inflationary model, which I also helped develop a few decades earlier but no longer consider viable.
I had not heard from Luca for some time. So I had not bothered to tell him that I would be in Italy that week. As I was walking through the Piazza San Lorenzo in front of my hotel in search of some good Italian coffee, I felt a vibration from my BlackBerry. It was Luca. I opened the message, expecting news of yet another failure. Instead, there was an email which began:
I studied a museum sample (belonging to the mineralogical collection of Museum) labelled khatyrkite (CuAl2). By means of a preliminary SEM [scanning electron microscope] study, I realized that such a sample actually consisted of four different phases, i.e., cupalite (CuAl), khatyrkite (CuAl2), an unknown phase with composition CuFeAl, and finally a phase with stoichiometry Al65Cu20Fe15 (normalized to 100 atoms).
The rest of the message focused on the last phase, the mineral Luca described with the chemical formula Al65Cu20Fe15, meaning 65 percent aluminum, 20 percent copper, and 15 percent iron.
The sample he was referring to is shown in the color insert (image 5) in its original plastic box next to a 5-cent euro coin included for scale. Most of what you see inside the box is a lump of putty holding the rock in place so it would not knock around and shatter when the box was moved. The entire sample, a mere three millimeters in diameter (shown enlarged in image 6 of the color insert), was being held in place at its uppermost tip by the wad of putty.
The image with the 5-cent euro coin was my first peek at the tiny grain that was about to launch a grand adventure.
The box described the sample as “Khatyrkite” (pronounced KAT-er-kite), a crystal mineral composed of CuAl2 (one atom of copper for every two atoms of aluminum). Khatyrkite was listed in the International Mineralogical Association (IMA) official catalog, which meant that its composition and periodic structure were already known, and that its properties had
been carefully measured and documented. The sample was registered in the official Florence Museum catalog as number 46407/G. The label on top of the plastic box included, for some unknown reason, the number 4061 along with the word “Khatyrka,” the name of a river in Far Eastern Russia, and “Koriak Russia,” an alternate spelling referring to the Koryak mountain range on the Kamchatka Peninsula.
The close-up (color insert, image 6) revealed that the grain contained a complex conglomeration of minerals. Luca found that the parts with lighter colors included common minerals like olivines, pyroxenes, and spinels. The darker material consisted primarily of alloys of copper and aluminum. The box was labeled “Khatyrkite” because whoever wrote the label considered the crystal CuAl2 to be the component that made the sample worthy of interest.
Luca had already sliced the mineral open to study its composition. He made six delicate sections, each the thickness of a human hair. In order to create the slices, though, Luca was forced to sacrifice the bulk of the sample. Ninety percent of what would eventually turn out to be an extremely precious mineral sample was destroyed in the process. Shown below the close-up of the sample in the color insert is image 7, the thin section that Luca excitedly recounted in his email.
The grayscale image shows a mix of different materials that look like they have been randomly kneaded together. Using an electron microprobe, which bombards a sample with a narrow beam of electrons to measure the chemical composition, Luca was able to identify most of the minerals in the section. Each dot in the image corresponds to a different measurement.
The yellow dots correspond to places where Luca found khatyrkite, CuAl2 (as mentioned, one atom of copper for every two atoms of aluminum). The red dots correspond to another rare crystal known as “cupalite,” CuAl, a 50-50 mixture of copper and aluminum atoms.
Then, he found some truly puzzling spots. The green dots denote areas with roughly an equal mix of aluminum, copper, and iron atoms, which is a combination that did not appear anywhere in the official IMA catalog of natural minerals. The blue dots are Al65Cu20Fe15, another composition that did not exist in the catalog.
Luca was eager to isolate the two mysterious minerals corresponding to the green and blue dots in order to obtain their powder-diffraction patterns and identify them. For this, he took a huge gamble and used a special tool to punch out the regions. The operation took remarkable hand-eye coordination since the regions were microscopic and the slice wafer-thin. Luca succeeded in capturing the material. But the rest of the fragile slice was destroyed in the process.
As a result, valuable information about how the different minerals connected to one another was lost. To be fair, Luca did not realize at the time how rare and important the specimen would turn out to be, or how essential the information would become. His only goal was to isolate the individual mineral grains as soon as possible so he could take an X-ray diffraction pattern to determine if they were promising quasicrystal candidates.
Once this process was complete, all that remained of the original sample were two tiny specks of material, which Luca glued onto the tips of a pair of slender glass fibers. Tiny as they were, the specks were large enough for Luca to obtain an X-ray powder-diffraction pattern.
Luca compared the results with the published patterns for synthetic quasicrystals and was excited by the possibilities. But he could not be sure if it was a true match. He did not have the computer program needed to perform the sophisticated tests that Peter and I had devised, and he could not detect the rotational symmetry of an atomic arrangement based solely on a powder-diffraction pattern. That was the same problem Peter and I had encountered with the ICDD database.
Within minutes of reading Luca’s email, I had forwarded his powder-diffraction data to Peter for immediate study. The test would perform a quantitatively precise comparison of the powder-diffraction pattern of Luca’s sample with the data we would expect to see from a natural quasicrystal. Until the results were known, there was no point getting excited by Luca’s find.
Two days later, I was back in the United States and received the initial results. Based on the test, there was a reasonably good chance that the grains with the blue dots contained a natural quasicrystal. But it was still too early to get our hopes up. As I explained to Luca, “a good chance” was not nearly the same thing as proof. Peter and I had experienced nothing but false positives during our previous studies. More testing was needed before we could determine if we had actually found a natural quasicrystal.
At that time, however, there were only two tiny specks remaining of the original rock. A quick search for other samples of khatyrkite in collections near Florence or Princeton came up blank. So we had no choice but to focus on the specks we already had in hand. Luca’s lab did not have the high-precision instruments needed to perform definitive tests on the remaining material. But I had access to the right equipment as well as the best person to perform the tests. I would call on Nan Yao, director of the Princeton Imaging Center.
On November 11, 2008, about a year and a half since I first started working with Luca, a plastic box arrived at my office from Florence, Italy. The box contained two small brass cylinders used to hold samples for a powder-diffraction experiment. Extending from each cylinder was a thin glass fiber. And glued to the top of each glass fiber was a nearly invisible speck of dark material.
I wanted to make sure the sample had arrived safely. I remember opening the package, taking out the plastic box, and squinting hard to see if I could make out the specks at the end of the fibers. I explained to the student who happened to be in my office at the time that I had spent more than ten years searching for a natural quasicrystal, and was hoping that if I ever found one, it would be at least as large as a pebble.
“It will be awfully frustrating,” I said, “if the first natural quasicrystal turns out to be something that I cannot even see!”
NINE
* * *
QUASI-HAPPY NEW YEAR
PRINCETON, NOVEMBER 21, 2008: I kept a firm grip on the small box as I trudged up the hill from my office to the Princeton Imaging Center. Inside were the two brass cylinders I had received from Luca. Each cylinder held a thin glass fiber about an inch long with a precious speck of material glued onto the end.
Nan Yao, the director of the Center, had his head down and was busy working at his desk when I arrived. I took a quick look around his office. Every available nook and cranny was stacked with books, journals, or boxes of samples related to one project or another.
The widespread clutter was tangible proof of the enormous amount of time Nan devoted to working with various faculty members and students from every corner of campus. I myself was already in his debt. He had been contributing some of his own time and discretionary funds to support our investigation.
Nan stood up from his desk and cheerfully maneuvered his tall, slender frame around all the stacks of material to greet me. We exchanged pleasantries, and he invited me to sit down. I looked around, unsure which way to move, because even the chairs and small coffee table in his office were covered with research and debris from earlier meetings. But Nan quickly collected everything and piled it on the floor in order to make room for me.
Nan, seen on the left in his laboratory, knew that I was bringing him Luca’s samples to examine. So I quickly handed him the box and sat back to watch his reaction. Nan is a highly regarded fellow of the Microscopy Society of America, and always manages to maintain a cool and collected professional demeanor. But he was visibly taken aback when he looked inside the box and saw that there were only two specks, each about a tenth of a millimeter in size, to work with. I was already concerned about the lack of material, and Nan’s reaction made me even more nervous.
Things are definitely as bad as I suspected, I thought.
Removing the specks from the glass fibers would be a risky endeavor, Nan told me. So before we attempted to do anything that might damage the samples, we decided to see how much we could learn by simply leaving them in plac
e. We would repeat the same X-ray diffraction measurements that Luca had performed but with a more precise instrument to see if the sample was truly as promising as the initial tests suggested.
After several weeks, though, our own results were also inconclusive. Even though Nan was using a better instrument, he could not improve on Luca’s results. The powder-diffraction peaks we measured were more or less the same as what Luca had obtained. The flimsy glass fiber mounts were likely the source of the problem, we thought. They wobbled around too much as Nan rotated the sample, smearing the X-ray diffraction pattern.
We considered removing the specks from the original glass fibers and regluing them onto newer, stiffer mounts. But as Nan and I had already discussed, detaching the tiny samples would be a risky operation. If we were going to take such a chance, I decided, we should make it worth our while and not just try to re-glue the samples and repeat the same test. We should jump ahead to the most decisive test: transmission electron diffraction from individual grains in the speck.
The advantage of transmission electron diffraction is that it uses a beam of electrons that can be finely focused. The beam can then be aimed to penetrate a tiny fraction of a single grain of material among the many different grains contained within a single speck. The result is a direct diffraction pattern which reveals the telltale symmetry of the atomic arrangement.
Preparing our sample for the test would prove to be a daunting challenge. It required removing a speck from the glass fiber, separating it into its many microscopic individual grains, and then sorting among all of those grains to find one that would be thin enough for the electron beam to pass through.
The Second Kind of Impossible Page 11