Quasi-periodicity in 15th century Islamic Art and whether it actually has been developed as a concept is still a matter of a somewhat controversial debate. There are three sites in the city of Esfahan which have been studied in this regard in considerable detail, two rather small patterns on a spandrel and a portal on the Darb-i Imam shrine in the Dardasht quarter of the old city (1453) and a huge pattern on the western iwan of the Great Mosque. Several authors have, in the meantime, tried to overlie Penrose patterns of kites and darts or thick and thin rhombuses in order to prove that medieval artisans were able to apply what had been described by Roger Penrose only five centuries later. See, for example, Lu and Steinhardt (2007), their response to some additional work by Makovicky (2007), or Cromwell (2008) here, here, and here.
Certainly, it would have been possible to assemble the five-point star (purple in the left picture) by a bobbin and two bowties instead. How it would look then can be seen in the right picture above (by Benjamin R. Schleich from his dissertation, which can be found here). A nice animated gif, instantly explaining the concept of subdivision and self-similarity, can be found here.
The picture below shows the western iwan, or sofe-shāgird, the iwan of the student (sic!). Sofe-e ustadh (the iwan of the master) faces it, it is the eastern iwan.
That medieval artisans just fell short of creating something that would really have amazed us 500 years later may be regarded as indirect proof that they had not really penetrated the mathematical concept. But if they actually had, why would that have been more interesting? Medieval artisans were keen to produce, with the help of eminent mathematicians, interesting ornaments and designs on mosques and buildings, not mathematical breakthroughs. They achieved a ‘dazzling’ appearance anyway; I have pointed to that several times here on this blog, see here and here. By removing color as a common element, for instance in the case of the Gonbad-e Qabud in Maraghah (1196), they even created something what would absorb and dazzle scientists even eight centuries later.
That has made a lasting impact, hasn’t it?
First published at Freelance.