The secret of Islamic patterns
This is the pre-edited version of my latest piece for news@nature. The online version acquired some small errors that may or may not be put right. But what a great paper!

Muslim artists may have used a sophisticated tiling scheme to design their geometric decorations

The complex geometrical designs used for decoration by Islamic artists in the Middle Ages, as seen in buildings such as the Alhambra palace in southern Spain, were planned using a sophisticated tiling system that enabled them to make patterns not known in the West until 20 years ago, two physicists have claimed.

By studying many Islamic designs, Peter Lu of Harvard University in Cambridge, Massachusetts, and Paul Steinhardt of Princeton University in New Jersey have decided they were put together not using a compass and ruler, as previously assumed, but by tessellating a small number of different tiles with complex shapes.

The researchers think that this technique was developed around the start of the thirteenth century, and that by the fifteenth century it had become advanced enough to generate complex patterns now known as quasiperiodic. These were 'discovered' in the 1970s by the British mathematical physicist Roger Penrose, and were later found to account for puzzling materials called quasicrystals. Discovered in 1984 in metal alloys, quasicrystals initially foxed scientists because they seemed to break the geometric rules that govern regular (crystalline) packing of atoms.

The findings provide a further illustration of how advanced Islamic mathematics was in comparison with the medieval West. From around the eleventh century, much of the understanding of science and maths in the Christian West came from Islamic sources. Arabic and Persian scholars preserved the learning of the ancient Greeks, such as Aristotle, Ptolemy and Euclid, in translations and commentaries.

The Muslim writers also made original contributions to these fields. Western scholars learnt Arabic and travelled to the East to make Latin translations of the Islamic books. Among the mathematical innovations of the Islamic world were the use of algebra, algorithms (both of which are words derived from Arabic) and the use of numerals now known as 'Arabic' (although derived in turn from Indian notation).

The mathematical complexity of Islamic decoration has long been admired. The artists used such motifs because representational art was discouraged by the Koran. “The buildings decorated this way were among the most monumental structures in the society, combining both political and religious functions”, says Lu. “There was a great interest, then, in using these structures to broadcast the power and sophistication of the controlling elite, and therefore to make the ornament and decoration equally monumental.”

Lu and Steinhardt now propose that these designs were created in a previously unsuspected way. They say that the patterns known as girih, consisting of geometric polygon and star shapes interlaced with zigzagging lines, were produced from a set of just a handful of tiling shapes ranging from pentagons and decagons (regular ten-sided polygons) to bow-ties, which can be pieced together in many different ways. The two physicists show how these tiles could themselves be drawn using geometric constructions with compasses that were known by medieval Islamic mathematicians.

Some scrolls written by Islamic artists to explain their design methods show tiles with these shapes explicitly, confirming that they were used as 'conceptual building blocks' in making the design. Lu says that they’ve found no evidence that the tiles were actually made as physical objects. “But we speculate they were”, he adds, “so as to be used as templates in laying out the actual tiling on the side of a building.”

Lu and Steinhardt say that designing this way was simpler and faster than starting with the zigzag lines themselves: packing them together in different regular arrays automatically generates the complex patterns. “Once you have the tiles, you can make complicated patterns, even quasicrystalline ones, by following a few simple rules”, says Lu.

The researchers have shown that many patterns on Islamic buildings can be built up from the girih tiles. The resulting patterns are usually periodic – they repeat again and again, and so can be perfectly superimposed on themselves when shifted by a particular distance – but this regularity can be hard to spot, compared say with that of a hexagonal honeycomb pattern.

The patterns also contain many shapes, such as polygons with 5, 10 and 12 sides, that cannot themselves be packed together periodically without leaving gaps. This property of the polygons means that scientists long believed that it was impossible for crystals to show five- ten- and twelvefold symmetries, such that rotating them by a fifth, tenth or twelfth of a full circle would allow them to be superimposed on themselves.

So when 'crystals' that appeared to have these symmetries were discovered in 1984, they seemed to violate the basic rules of geometry. But it became clear that these quasicrystals aren't perfectly periodic. In the same year, Steinhardt pointed out how patterns with the same geometric properties as quasicrystyals could be constructed from the tiling scheme devised by Penrose.

Steinhardt and Lu say that, while there is no sign that the Islamic artists knew of the Penrose tiling, their girih tiling method provides an alternative way to make the same quasicrystalline patterns. The researchers say that a design on the Darb-i-Imam shrine in Isfahan, Iran, made in 1453, is virtually equivalent to a Penrose tiling. One of the mesmerizing features of this pattern is that, like a true quasicrystal, it looks regular but never repeats exactly.

“I’d conjecture that this was quite deliberate”, says Lu. “They wanted to extend the pattern without it repeating. While they were not likely aware 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 quasicrystal.”

Reference
Lu, P. J. & Steinhardt, P. J. Science 315, 1106 - 1110 (2007).

Postscript
I have received some comments from Roger Penrose on this work, sadly too late for inclusion in the Nature piece but which provide some valuable perspective on the discovery. This is what he says:
"The patterns are fascinating, and very beautiful, and it is remarkable how much these ancient architects were able to anticipate concerning 5-fold quasi-symmetric organization. But, as Steinhardt (and, in effect, Lu) have confirmed directly with me, the Islamic patterns are not the same as my patterns (on several counts: different basic shapes, no matching rules, no evidence that they used anything like a "Penrose pattern" to guide them, the hierarchical structure indicated by their subdivision of large shapes into smaller ones is not strictly followed, and would not, in any case, enable the patterns to map precisely to a "Penrose tiling"). I do, however, regard this work of Steinhardt and Lu as a most intriguing and significant discovery, and one wonders what more the ancient Islamic designers may have known about such things. I should perhaps add that the great Astronomer Johannes Kepler, in his Harmonice Mundi (vol.2), published in 1619, had independently produced a regular pentagon tiling that is much closer to my own tilings than anything that I have seen so far in this admittedly wonderful Islamic work."

Peter Lu, incidentally, has indicated that he agrees with everything that Penrose says here. The relationship between the Darb-i-Imam pattern and a Penrose tiling is subtle - much more so, it seems, than media reports of this work have tended to imply.