Update below.

Readers may have noticed that I have attributed considerable recreational time on this blog to conceive and convey quasi-periodic patterns on ancient Islamic buildings. Following a first visit of Esfahan’s Darb-i Imam shrine in the old city in late 2007 and comparing the patterns with those on the celebrated western iwan of Esfahan’s Great Mosque, a more common tourist attraction which I had opportunities to admire numerous times before, I had always asked myself: How have they done that?

Lu and Steinhardt (2007), in their famous paper in Science magazine, had frankly rejected any possibility of creating complicated geometric designs with tesserae by compass and straightedge. They had offered instead a solution using so-called girih tiles (or, by other authors, prototiles) which can, with some attention, be identified on all kinds of tesselation in Iran and elsewhere: a decagon, a rhombus, a so-called bowtie, and an elogated hexagon.

When in Esfahan, I had noticed frequently reconstruction work on either building. There were no signs of girih tiles used for that purpose. Unfortunately, I had no time to ask for respective workshops but got the impression that something must be wrong with Lu and Steinhart’s solution.

That construction of complicated, even quasi-periodic, tilings is in fact possible with compass and straightedge, even more than 900 years ago, has recently been shown by Rima A. al-Ajlouni (2012), see [pdf], an assistant professor at the College of Architecture of Texas Tech University, in Lubbock. A decagonal, quasi-periodic tiling (not exactly a Penrose tiling) which can be found at two different portals of the Darb-i Imam shrine and the Great Mosque in Esfahan (and, in essence, also on the somewhat disputed Gunbad-i Kabud in Maragha, Iran) is constructed from a framework of nested decagrams originating from a seed unit, a decagon. It grows based on the Fibonacci sequence and serves as the underlying basic grid for the quasi-periodic pattern. The positions of star units are entirely determined by the intersections of the network of nested decagrams. Connecting formations are formed by overlapping main units. The pattern can be grown infinitely by using the resulting cartwheel pattern as new seed unit.

See a presentation by Dr. Ajlouni on her discovery here.

Update January 23, 2014. A reader had pointed to a public lecture which was given at the Middle East Association on 27 April 2007 by R. Henry of the British Museum in London, where a workshop in Esfahan is pictured and the creation of tenfold symmetrical patterns with compass and unmarked straightedge demonstrated. There are no girih tiles involved.