Did medieval tile-makers discover quasicrystal geometry before Penrose?

Probably only partly. Some medieval tile-makers made patterns that never repeat, and the physicists Peter J. Lu and Paul Steinhardt presented evidence that a Penrose tiling underlies girih patterns at the Darb-e Imam shrine in Isfahan. But other scholars point out that there is no evidence the medieval artisans used a Penrose set, and that they seem not to have realized what their tool could do.

Girih, Persian for "knot", are patterns of angled lines that form an interlaced strapwork. They can be drawn with a straightedge and compass, or with a set of five girih tiles with lines drawn on them. By the 15th century some patterns were no longer periodic, and the Topkapı Scroll, from the late 15th century, shows the lines drawn over the tiles that make them. The Darb-e Imam shrine of 1453 has a much more complex pattern than any seen before, aperiodic and drawn at two scales: a large pattern from a distance and a smaller one from close up.

Why do physicists care? A repeating pattern, like wallpaper, cannot use five-fold shapes such as a decagon, but quasi-periodic patterns can. Roger Penrose investigated such tilings in the 1970s. In 1982 Dan Shechtman observed unusual diffractograms in aluminium–manganese alloys, which are now seen as showing quasicrystals; fearing the scientific community's reaction, he took two years to publish. He won the 2011 Nobel Prize in Chemistry.

The medieval makers stopped at two levels of design, since a third would have been too large or too small to see, and it appears that they had a tool with the potential for highly complex patterns, but never realized it.