Quasi-Crystal Conundrum Opens a Tiling Can of Worms
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NEWS OF THE WEEK MATHEMATICS Quasi-Crystal Conundrum Opens a Tiling Can of Worms The mosques and palaces of the medieval crystal pattern is “Penrose tiling,” named then he found a photo of the Darb-i Imam Islamic world are wonders of design. after Oxford University mathematician and shrine in Isfahan, Iran, built in 1453. Its Because tradition forbids any pictorial deco- cosmologist Roger Penrose. decagonally symmetric motifs on two dif- rations, they are covered with complex and In 2005, Lu, a doctoral student at Harvard, ferent length scales are a telltale sign of a intricate mosaics. These geometric patterns, noticed a geometric pattern on the wall of an quasi-crystal. Working with Steinhardt, his called girih in Arabic, may be even more Islamic school in Uzbekistan with surpris- former undergraduate adviser and a quasi- sophisticated than has been appreciated. ingly complex decagonally symmetric crystal expert, Lu found that the Darb-i On page 1106, physicists Peter Lu of motifs. “It got me thinking that maybe quasi- Imam girih pattern can map onto a Penrose Harvard University and Paul Steinhardt of crystals had been discovered by Islamic tiling. There were a few defects, but these Princeton University propose that architects architects long ago,” he says. Islamic archi- are superficial, says Lu, and were likely made a conceptual breakthrough sometime tects began to explore motifs with fivefold introduced by workers during construction or repair. “We realized that by the 15th cen- Middle Age masters. tury, these architects had the makings of The medieval architects quasi-crystals,” says Lu. who created complex The paper has had a mixed reception. tiling patterns, such as these on a madrasa in Crystal expert Emil Makovicky of the Uni- Bukhara, Uzbekistan, versity of Copenhagen, Denmark, studied may have been more girih patterns for 2 decades. His analysis of sophisticated than has the patterns on a tomb in Maragha, Iran, been appreciated. built in 1197, concluded that they map onto Penrose tiles and was published in a 1992 book about fivefold symmetry. Lu and on March 10, 2007 Steinhardt cite his work, he says, but “with- out proper quoting and … in a way that [the ideas] look like their own.” Physicist Dov Levine of the Israel Institute of Technology in Haifa agrees that Makovicky deserves more credit than he is given in the paper. “His analysis of [the Maragha tomb] patterns anticipates some of the ideas in the www.sciencemag.org Lu and Steinhardt paper,” he says. Joshua Socolar, a physicist at Duke University in Durham, North Carolina, agrees that between the 13th and 15th centuries. By first and 10-fold rotational symmetry during a Makovicky deserves credit for discovering visualizing a surface as a tiling of polygons, flourishing of geometric artistry between the “an interesting relation between the Maragha these unknown scholars created girih 11th and 16th centuries. pattern and a Penrose tiling with a few patterns that conform almost exactly to a Back at Harvard, Lu began to study archi- defects.” Both Levine and Socolar doubt that pattern called a quasi-crystal. If Lu and tectural scrolls from that period. On many the architects truly understood quasi-crystals Downloaded from Steinhardt are right, then the Islamic world scrolls, faintly sketched beneath the intricate but say Lu and Steinhardt have generated discovered a piece of mathematics 500 years lines of the girih design, was a polygonal interesting and testable hypotheses. before it was formally described in the West. tiling pattern. “I found the outlines of the Lu and Steinhardt say they were aware of But the paper has also sparked a rancorous same tile shapes appearing over and over,” Makovicky’s published work on the subject, dispute over who first made this insight, and he says. Lu realized that Islamic architects but “we have found serious problems with whether it is true at all. could have used a pattern of polygonal both his technical reconstruction and gen- Starting in the 1960s, mathematicians shapes—which he calls girih tiles—as the eral conclusions.” They say that they decided studying the geometry of tiling came up with starting point for their designs, creating a to limit their references to Makovicky “to the concept of the quasi-crystal. Tiling is wonderfully complex girih pattern by trac- avoid having to address the serious technical crystalline if it is made up of an infinitely ing lines from tile to tile following local problems with his work.” Makovicky dis- repeating pattern of some finite set of units. rules. And if the right shape of girih tiles agrees that his work is flawed. Quasi-crystals are also made up of a finite were laid together just so, the resulting pat- Beyond the question of credit, just set of interlocking units, but their pattern tern could be extended forever without how mathematically sophisticated these never repeats even if tiled infinitely in all repeating—a quasi-crystal. medieval architects really were remains UNIVERSTIY directions. Researchers also found that Lu examined “a few thousand” photos of open. “We haven’t done an exhaustive search ARD although pentagons and decagons don’t fit real mosques and found that although of Islamic architecture by any means,” says LU/HARV easily into normal tiling, in a quasi-crystal decagonal girih patterns became increas- Lu. “There could be a perfect quasi-crystal . P such fivefold and 10-fold rotational symme- ingly common from 1200, nearly all are pattern waiting to be found.” : tries are integral. The most famous quasi- periodic and so are not quasi-crystals. But –JOHN BOHANNON CREDIT 1066 23 FEBRUARY 2007 VOL 315 SCIENCE www.sciencemag.org Published by AAAS.