Bismillah. With thanks to Dr. Sabbir Rahman & Abu Ammar Mangoranca for these.
I was delighted to read the following short article, which clearly demonstrates how far ahead of their time Muslim geometers were:
Aperiodic Penrose Tilings are a very modern development yet,”The Darb-i Imam shrine was particularly remarkable because it showed girih tile patterns at two different scales, so that large girih tiles were broken up into smaller girih tiles. In principle, by repeatedly scaling up the tiling in this way, they could have covered an arbitrarily large wall with a Penrose tiling.”
Indeed, as the Wikipedia article on Penrose tilings should make clear, their development and discovery is highly non-trivial, and not something which is likely to be come upon purely by chance:
The connection of Penrose tilings with the golden ratio is also quite fascinating. Note that the golden ratio has featured in Islamic architecture from the earliest times:
This is a link to the technical paper associated with the first article:
The research of Dr Peter J. Lu on the Islamic art patterns is also described and with more illustrations at
Although this decorative art is found in medieval Muslim structures, especially mosques, the mathematical foundation of these patterns, or girih tiles, is only discovered in the West after 5 centuries.
Abu Ammar Mangorangca