The small Darb-i Imam shrine (1453), about 300 meters west of the Great Mosque, may in fact be one of the gems of Timurid architecture in Esfahan. The site is rather hidden in the labyrinthine lanes of the northern part of the old city of Esfahan .
The shrine consists of a funerary complex  with courtyards, shrine structures, and a small cemetery. During the centuries it had been steadily reconstructed and repaired, especially in the early and late 17th century. Characteristic are the two closely spaced domes, one bulbous with beautiful arabesques and one more slender with floral decoration, on high drums with highly stylized calligraphy.
Its pishtaq, or porch, contains several exquisite mosaics made of glazed tiles. Some of them are said to be created by Sayyid Mahmud-I Naqash, who has also decorated the southern iwan and the celebrated Timurid gate on Esfahan’s Masjed-e Jomeh.
What has recently attracted more interest are the geometric patterns made of black glazed and unglazed terracotta pieces in several spandrels and a porch next to the mentioned main pishtaq.
It had been suggested that they represent the so far only discovered example of an almost perfect Penrose tiling which had been created 500 years before their description in the West . In their meticulous reconstruction using the famous “kite-and-dart” type of Penrose tiling, Peter J. Lu and Paul J. Steinhardt very much focus on a spandrel which in fact matches almost perfectly with a Penrose tiling .
What makes this tiling so unique may be the subdivision rule painstakingly elaborated by Lu and Steinhardt:
“Perhaps the most striking innovation arising from the application of girih tiles was the use of self-similarity transformation (the subdivision of large girih tiles into smaller ones) to create overlapping patterns at two different length scales, in which each pattern is generated by the same girih tile shapes.”
It has been questioned whether the pattern on the spandrel is really self-similar. The difference between the large and small scales is very big. In their analysis, Lu and Steinhardt create the spandrel itself by four large length scale decagons and two bowties, while the small scale consists of three girih tiles, the decagon, the bowtie and the elongated hexagon. So, where is the large-scale elongated hexagon ? Can it be that it has been overlooked?
The pattern is in fact aperiodic. There is only one small-scale area in the whole spandrel which resembles the large-scale pattern: in the upper right corner. The area with the corresponding (yellow) borders of the small-scale spandrel is shown in the picture below. Here, a part of the (green) elongated hexagon shows in the lower corner.
Thus, the large-scale spandrel may be reconstructed in a different way, shown below. Although the bold blue lines do not exactly fit, the reconstruction here seems to support the concept of self-similarity and aperiodicity of the tiling on this particular spandrel of the Darb-i Imam .
 The historical city with its huge bazaar had been cut by Kh. Abdorrazaqq into two halves some 40 years ago in an attempt of urbanization.
 Penrose himself had been inspired by Johannes Kepler’s Harmonices Mundi (1619), where he constructed tilings around regular pentagons which can be extended into Penrose tilings. Pentilings, i.e. arrangements of regular pentagons in the plane in which each pentagon makes edge-to-edge contact with two, three, four, or five neighbors, thereby sharing vertices in such a way that no gaps large enough to contain another pentagon are left in the array, have even been described by Albrecht Dürer in 1525.
 In the supporting online material for their article in Science, Lu and Steinhardt (2007) have suggested, based on a more than 40-year-old photograph that the tiling on the western iwan of Esfahan’s Friday mosque can be subdivided in the same way as that on the Darb-i Imam. Meanwhile, it has been shown that the patterns are different and that the one on the Friday mosque contains, in addition to a decagon, an elongated hexagon and a bowtie, a fourth girih tile, a rhomb (see an illustration of the girih tiles here). The pattern is, in addition, periodic, similar as the pattern on the Gonbad-e Qabud in Maraghah, which had been constructed in fact 250 years earlier.
 While Lu and Steinhardt had elaborated only a subdivision of a decagon and a bowtie by smaller-scale decagons, elongated hexagons and bowties (see below), P. R. Cromwell has recently presented a corresponding subdivision of the hexagon.
See also this summary which has been inspired by an email exchange with Prof. Jost-Hinrich Eschenburg, University of Augsburg.