## Geometry in Art (Week 6)

In week 6, we read about Islamic art, including the geometric designs on the mosques.  The geometric tiling on the walls fascinated me. I had seen examples when I visited Iran, but I have never studied them in great depth. This project gave me the chance to do that. I knew that it was possible to have a periodic tiling using certain shapes.  For example, we can use shapes such as rhombuses and triangles, but we cannot use pentagons or decagons. However, if you examine Islamic architecture, very often you see pentagons and decagons.  Even so, there are no gaps in the tiling, and the pattern appeared to repeat seamlessly forever.  It was not repeating, but rather aperiodic, meaning that it almost repeated, in the sense that some of the shapes used did not repeat.  That is why these patterns are called “quasi-crystalline” instead of “crystalline”.

In my project, I tried to construct such a pattern in the form of a collage. Initially, I used only decagons and bow-ties, and I attained this pattern:

So far so good. It seemed that at this point, I could actually make the pattern periodic. Pretty shortly afterwards, however, I realized that I had to insert hexagons. Then, we see that we end up with this picture, which is very clearly not periodic, but it does seem to be able to continue:

I learned that this process was quite difficult, and what they did centuries ago was not easy.  Here, I knew in advance what shapes would work.  The shapes I used are girih tiles, where in Farsi, girih means “knot”.  The 5 girih tiles are:

Unlike 4-fold or 6-fold symmetries, Islamic architects realized that periodic patterns with exact 5-fold or 10-fold symmetries cannot be realized. They managed, nevertheless, to design intricate patterns using the above tiles that almost has 5-fold or 10-fold rotational symmetries.

Unaware of these old discoveries of Islamic architects, scientists have rediscovered these quasi-crystalline patterns in the 1980s.  The fact that these patterns were known to Islamic architects was discovered in 2007 (by Harvard student Peter Lu and his mentor Paul Steinhardt). In fact, quasi-crystals have also been discovered in nature and the scientist predicting the existence of quasi-crystals has been recognized in the 2011 Nobel Prize in Chemistry.

In the readings, there was debate about what Islamic art is.  I think it is safe to say that geometric tilings on buildings built after the advent of Islam in areas that had Islamic influence can be classified as Islamic art.  Why would the artists and architects spend so much time and effort on geometric patterns? It has to be for a greater meaning. For instance, perhaps they were trying to represent the infiniteness of God through the repeated patterns, and the majesty and elegance of God through the fact that the patterns are aperiodic.  Perhaps they were honoring God, who is too majestic and wise to be understood.  There has also been speculation that since the numbers 5 and 10 are special in Islam (we have 5 pillars in usul-e-deen and 10 in furu-e-deen), it is natural that the Islamic artists tried to incorporate these numbers into their arts, which would naturally lead to quasi-crystalline 5-fold and 10-fold rotational symmetries, which is not possible to obtain using regular crystals.  Regarding these patterns, in Islamic Art and Spirituality Nasr argues that the fact that these patterns that they used are found in nature “illustrate an important aspect of the Islamic revelation, which is to bring out the reality of the cosmos itself as God’s primordial revelation” (p. 49).  He does not think that their mathematical prowess is unnatural but actually quite natural in this context.