Penrose tiles.jpg

The Music of Science: Oliver Morton takes one mathematician and some tiles—and brings Plato’s forms down to earth...

From INTELLIGENT LIFE magazine, November/December 2012 

Almost a quarter of a century ago, I was sitting outside the Turf Tavern in Oxford. It was a sunny late-summer day. I was a young reporter and with me, drinking a half-pint of cider, was Roger Penrose, the greatest British mathematician of his generation to apply himself to physics.

Until that drink, Plato's notion of a pure world of form to which the world perceived by humans was but a shadow was just something I had come across in books. It was familiar intellectual history, but hardly something I would expect anyone actually to believe. By the time our glasses were empty I knew that, at least to Penrose, it was an irreducible reality. The platonic world of pure form was not just as real to him as the cool gold of his drink or the sounds of my stammered questions. It was more real. For all the sunlight, I was a shadow.

The conversation in which I realised that Penrose, sincerely and without insanity, truly believed in a profoundly different and yet accessible world, was about one of the more intriguing of his many mathematical contributions: the tiles that bear his name. As I type, my wife is designing a pattern for square tiles in our kitchen. Because the tiles are square, she could choose a pattern that has fourfold symmetry—you could turn it round and every quarter turn it would look the same. With triangular tiles it is possible to create similar symmetries that are threefold. Twofold and sixfold are also possible. But there is no single shape of tile with which you can create a fivefold symmetry.

In the 1970s, Penrose found that you could, however, produce a set of tiles—now called Penrose tiles—which, used together, would cover a surface perfectly and show a fivefold symmetry. It was an odd symmetry (you couldn't get the pattern to repeat simply by moving a step or two in a given direction, as you can with square or triangular tiles) with intriguing new mathematical properties, and if it seemed more like a divertissement than a discovery, it's worth remembering that in maths the two can be quite close. But in the early 1980s an Israeli chemist, Dan Schechtman, found that some materials he had made for study seemed to be showing just such a peculiar symmetry, behaving like "quasicrystals". A few years later a researcher in Japan, An-Pan Tsai, produced further evidence that such things were real.

Hundreds of quasicrystalline solids have now been made in the lab, and they have been incorporated into some products. But they have seemed vanishingly rare in nature. Regular crystals, the equivalent of patterns in square or triangular tiles, abound, from sea-salt to snowflakes; the laws of thermodynamics encourage such patterns of growth. Natural quasicrystals were unknown. Did that mean that they only formed under precisely tuned conditions? Or did it mean people were not looking hard enough?

The latest evidence suggests that the problem was in part the latter. Paul Steinhardt, who had been interested in the possibility of quasicrystals even before Schechtman's work, inspired a systematic attempt to look at all plausible candidates in the world's mineralogy collections—and a few years ago a sample emerged from a museum in Florence that turned out to be almost identical to a creation of Dr Tsai's. But the sample, originally collected in Siberia, had various other attributes that were hard to explain on the basis of normal mineralogy. Did it come from peculiarly deep in the earth? From a lightning strike? From an aluminium smelter's slag pile?

The answer is no, every time. It now appears that, like kryptonite, the quasicrystalline mineral came from outer space. Hints to this effect could be found in the sample itself, and an expedition back to Siberia led by Dr Steinhardt has provided further evidence that the sample came from a meteorite. The expedition also furnished its participants with new samples both of the original quasicrystalline mineral and of some other stuff not yet described in the scientific literature.

In an odd way, this unearthly origin makes quasicrystals more normal. It now seems they probably can grow on their own, through well-established thermodynamic principles, in something like the way everyday crystals do. They just don't happen to do so under the conditions that are typically available for making minerals in the crust of the Earth. If we expand our purview to take in the conditions of the cloud of dust and gas in which the sun first took fire—the scene of the birth of many meteorites—the right conditions turn up and so do quasicrystals.

The difference between the conditions of a planet and a condensing pre-stellar dust cloud may seem to make them different worlds. But those differences are, at heart, parochial, compared with the fundamental rules of nature's order and symmetries. Space and Earth are just different places.

To find something from a truly different world, you have to think like Roger Penrose. If you do, then you will find the evidence for that other world not at the end of long treks across Siberia nor down a mineralogist's microscope, but in the recesses of your own thought. And from that world of form, all of this one—minerals, meteorites, the warmth of sun on sandstone and the cold of space before the sun—is of a piece. 


Illustration Gary Taxali