ROBERT SIEGEL, host:

Historic buildings in the Islamic world are often covered with breathtakingly intricate geometric designs. Both artists and mathematicians have long puzzled over them, wondering how the designs were created. Well, now a Harvard physicist has some new ideas.

As NPR's Nell Boyce reports, she says that some of the designs show remarkably advanced mathematics

NELL BOYCE: Peter Lu is a physics graduate students at Harvard University. A couple of years ago, he was traveling with his cousin through central Asia.

Mr. PETER LU (Physics Graduate Student, Harvard University): When we were in Bukhara, Uzbekistan, we were walking around looking at the buildings, and one tiling we walked by was this gigantic wall on a madrassa.

BOYCE: It was a blue wall with a stunning design.

Mr. LU: And it had a bunch of 10-pointed stars.

BOYCE: The stars sort of reminded Lu of something he'd studied, a strange geometric pattern that mathematicians first described just 30 years ago. Lu wondered - could Islamic artists have known about this weird geometry 500 years earlier? So when he got back home, he went to see an expert on Islamic art.

Mr. LU: And I said what do you think? And she, like many Harvard professors, said well, start and read my book, and so I did.

BOYCE: The book taught Lu about a 15th-century scroll. It's a rare window into how Islamic artists did their work. They normally hid their trade secrets. This scroll is like an instruction manual. It has panels that outline any different geometric designs.

Mr. LU: Basically, I just sort of started looking through these panels. I saw a bunch of five-fold pentagons, 10-fold stars.

BOYCE: Lu saw that in addition to the black outlines, there were some faint red marks in five shapes. He realized they were basic building blocks.

Mr. LU: The set, sort of, of universal puzzle pieces - that they may have been assembled in order to build a whole range of different kinds of tilings that are all sort of similar looking, but certainly not the same.

BOYCE: Lu says, once artists had these building blocks, they could start drawing increasingly sophisticated patterns. And in at least one case, they generated the kind of advanced mathematical pattern he'd been looking for. It's over an archway at a shrine in Iran, built back in 1453. There are silver stars surrounded by black and gold shapes.

In the current issue of the journal Science, Lu shows that the pattern has all the hallmarks of what mathematicians call a quasicrystal.

Mr. LU: So they were really conceiving of these tilings and thinking about how to make these patterns 500 years before people in the West were really seriously considering them. So I think that's really fascinating.

BOYCE: Paul Steinhardt is a physicist at Princeton University who worked with Lu. He helped coin the term quasicrystal two decades ago. He says when some people first look at a quasicrystal pattern, they immediately notice repeating motifs.

Mr. PAUL STEINHARDT (Physicist, Princeton University): And other people hone in on it and say no, I don't see - every time I try to follow the pattern, it doesn't quite - parts of it repeat, but they don't repeat regularly, so it must be random.

BOYCE: In fact it isn't random. Steinhardt says if you do enough math, you see that it fits together in predictable way, but intuitively, it's hard for some people to get their minds around it.

Mr. STEINHARDT: It is hard to picture, and it's hard for humans to process these patterns and interpret them.

BOYCE: Which raises the question of whether the medieval artists really understood the math behind their creation.

Mr. CRAIG KAPLAN (Computer Scientist, University of Waterloo): That is a much stronger claim and one that would be much more difficult to establish. I get very nervous when people talk like that.

BOYCE: Craig Kaplan is a computer scientist at the University of Waterloo, in Canada. He studies star patterns made by Islamic architects.

Mr. KAPLAN: We know that they were good mathematicians, we know that they studied Euclid, but we don't know exactly how they worked.

BOYCE: That hasn't stopped today's scientists from trying to figure out the secrets hidden in these geometric patterns. Kaplan says they are so appealing and mysterious that researchers will keep trying to unravel them, using all of the modern tools at their disposal. Nell Boyce, NPR News.

SIEGEL: And you can see examples of the complex designs Peter Lu is studying at npr.org.

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