Roger Penrose, only 550 years behind Islamic artisans

It’s very easy to cover a flat surface with a regular pattern of geometric shapes – in mathematics, this set of problems is referred to as “tiling”. It’s much harder to cover a surface with a pattern that doesn’t repeat. In 1961, mathematician Hao Wang tried to prove that any set of tiles that cover a plane do so periodically – i.e., with a regular pattern. A number of mathematicians proved him wrong, first with a set of 20,000 tiles, then 104, then 92.

The big breakthrough in tiling came in 1973, when British mathematician Roger Penrose demonstrated sets of six, four and two tiles that could cover a surface non-periodically. His tiles have spawned a great deal of interest and research, and help an interesting problem in physics: quasicrystals, which are materials which difract light like crystals, but aren’t regular, periodic shapes like most crystals. Instead, they share much of the symmetry that Penrose’s tiles display.

While Penrose’s breakthrough is a fascinating chapter in mathematics, it turns out that he may be over 500 years late in solving the problem of non-periodic tiling. According to Peter J. Lu of Harvard University and Paul Steinhardt of Princeton University, the artists who decorated the Darb-i-Imam shrine in Isfahan, Iran, in 1453 used a non-periodic pattern that’s virtually identical to Penrose’s tiles.

The patterns, called girih, are composed from a small regularly shaped tiles, tesselated into complex patterns. The researchers believe that the technique was developed in the 13th century and by the 15th century had developed to a level of sophistication where non-repeating patterns were possible.

Peter Lu was inspired to pursue this research topic based on a pattern he’d seen on a building in Uzbekistan. I can think of no better argument for researchers to get out of the lab and onto airplanes more often.

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