*About geometric decoration in Islamic architecture, with a focus on the Islamic star patterns and quasiperiodicity by using the girih tiles. In this post I touch on the great mystery of the Islamic tiling, the pattern works that can ‘touch’ the viewer on a spiritual way. *

**History and background**

One of the world’s great ornamental design traditions, the Islamic star pattern, has been a fascination for viewers and mathematicians since they started to occur in the Arabic architecture and art well over a century ago. Characteristic for the Islamic star patterns is their rigidly geometric design that features star-shaped polygonal regions shaping a tessellation of balance, symmetry, and ingenuity. Islamic star patterns tend to be very mathematical and technical, while also being spiritual art works. The patterns seem to be design to creates a visual wonder for the viewer, a wonder that can deliver a message – religiously connected (traditionally).

The tiling in its self is not functional through a technical point of view, but in a cultural and spiritual perspective it does make a great function and importance in an Arabic context. We find a lot of different types of tiling in the traditional Islamic pattern works, and they vary in complexity and variety. But there are two key elements to them all, *geometry *and* symmetry. *

Unfortunately, we don’t know very much about how they constructed these tiling. The masters of the tiling techniques were very restricted to keep their techniques for themselves and their apprentice. Overtime it got ultimately lost and today we know very little about it. The quest to design an irregular tiling, with its ability to create a visual confusion and illusion, has therefor being an intriguing mystery for mathematicians for many years. But there is some physicists and mathematicians showing that behind the complexity you’ll probably find less complex tiles in the end, the girih tiles is one example.

**Tiling **

A tiling could be described as a family of sets (the *tiles*) witch cover the plane whiteout *any* gaps or overlaps. If all the tiles are congruent to one of a minimal set of tiles, then these are called *the prototiles *of the tiling. The amount and complexity of the prototiles will decide if the tiling is a periodic (regular), non-periodic (regular or irregular) or a quasiperiodic/aperiodic (irregular) pattern. I will continue my post by focusing on quasiperiodic patterns.

It’s called quasiperiodic because it has the same properties as a quasicrystal witch is sometimes called “the impossible atomic arrangement”, discovered by Dan Shechtman in 1982.

*“… Basically, a quasicrystal is a crystalline structure that breaks the periodicity (meaning it has translational symmetry, or the ability to shift the crystal one unit cell without changing the pattern) of a normal crystal for an ordered, yet aperiodic arrangement. This means that quasicrystalline patterns will fill all available space, but in such a way that the pattern of its atomic arrangement never repeats…” *(D. Oberhaus, 2015)

Dan Shechtman got a Nobel Prize in 2011 for his discovery and today we find quasicrystals in most of our homes, including frying pans and the LED lights.

*Starting from the left top: Darb-i Imam shrine, Isfahan, Iran (1453 C.E.), sience.siencemag.or. Quasicrystals, nickyskye.blogspot.de. Quasicrystals, microscopy.ethz.ch. Both images in the bottom right is taken at Darb-i Imam shrine, Isfahan, Iran (1453 C.E.), from siencenews.org and al-qanaa.blogspot.se*

The Penrose tiling is maybe one of the most well known quasiperiodic tiling, investigated by mathematician and physicist Roger Penrose in the 1970s. The tiling has a fivefold-rotated symmetry and can come together in a long-range of orders. In 2007, the physicist Peter J. Lu and Paul J. Steinhardt clamed that they had discovered a new set of tiles that possessed properties consistent with self-similar fractal quasicrystale tilings, such as the Penrose tilings, they named it the Girih tiling.

**Girih tiling**

Before, researchers said that girih (geometric star-and-polygon, or strap work) patterns in medieval Islamic architecture were conceived as a network of zigzagging lines by their designers after the tiling was set. Peter J. Lu and Paul J. Steinhardt discovered that by 1200 C.E. a conceptual breakthrough occurred in witch girih patterns were preconceived as tessellations of a special set of regular polygons (“girih tiles”) decorated with lines. By the 15th century, the tessellation approach was combined with self-similar transformations to create nearly perfect quasi-crystalline Penrose patterns. This was five centuries before West discovered it.

Girih tiling tends to look complex for the viewers, but behind the lines there is maximum 5 types of tiles and they are all regular: (1.) a decagon, (2.) a regular pentagon, (3.) an elongated hexagon, (4.) a rhombus and (5.) a bow tie. All the edges of these polygons have the same length; and all their angles are multiples of 36°. The tiles also have a fixed set of decorative lines, the “girih”, they will together create the pattern.

By just observing Islamic architecture we can discover the the built up, while knowing the “girih rules”. Based on one of Peter Lu’s lecture, I will in the following diagrams give you some examples of the girih tiling and show you how they are designed and used.

Here you can follow how to draw the cell for this pattern in the example to the left. This example is a periodic pattern, that means that you can copy and paste the cell to the infinity periodically. This type of pattern is wide spread and used in many other locations and works.

To read the tiles behind the black lines we will in the following diagram mark every intersection of black lines that creates an X, and subdivide it with a red line (outside the original star). We will then see an underlying grid, witch is the girih tiles of this tiling.

The girih patterns also allows a multiple overlapping pattern, that’s built over by the same system, but in different scales. The small and the large tiling can be subdivided in the same way, they are in fact related.

They’ve probably constructed these tiling by placing out the large polygons first, and from there filling with the small tiles. In this example they’ve highlighted the large pattern black, witch gives the architecture a unique and mysterious expression.

So now we know how we can construct some of the Islamic star patterns, and luckily there is a pretty understandable logic behind it. When designing a building with a complex facade, we should make sure that the worker men can handle the complexity. By using the girih tiles, you can maximum use five different types of tiles witch makes it easy with labeling. So, lets not forget the regional culture when we start new projects because of “complexity”, lets engage it!

**References:**

*Harvard University Physics Department Colloquium Lecture, presented on 3 Dec 2007 by Peter J. Lu: “Quasicrystals in Medieval Islamic Architecture”. **https://www.youtube.com/watch?v=rldnu9rNpH8*

*Kaplan, Craig S. . Islamic Star Patterns from Polygons in Contact. In GI ’05: Proceedings of the 2005 conference on Graphics Interface, 2005.*

*Lu, Peter J., and Paul J. Steinhardt. “Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture.” Science 315 (2007): 1106-1110.*

*Oberhaus, Daniel. ** “Quasicrystals Are Nature’s Impossible Matter”. Motherboard.vice.com (2015) http://motherboard.vice.com/read/quasicrystals-are-natures-impossible-matter*