“No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by propositions five and six of Book two of Elements.”
– Omar Khayyam
Geometric patterns are a prevalent feature in Islamic art and architecture, the complexity of which ranges from simple polygon tiling to multi-layered, fractal structures. Here I look at some famous patterns which occur in Islamic geometric design, and try to break down some of the neat mathematical features behind each one. In addition, I’ve added in a few interactive modules to play around with, which hopefully are a) fun and b) provide a better visual intuition.
ESCHER AND THE ALHAMBRA PALACE
The first piece examined is a tessellation pattern from the Alhambra in Granada, Spain, which served as a stimulus for MC Escher to develop his own versions of the art form. In fact, tessellations are a subject of study themselves in mathematics, as they demonstrate the various ways in which one can cover a space completely, without gaps or overlaps. What stands out particularly about these tilings is that the irregular, curved features of the tiles contrast quite heavily with the usual polygonal forms seen in Islamic art, and yet the shapes still merge with one another perfectly. The construction of such a pattern relies on first finding regular shapes that already tile the plane perfectly; then by adding or subtracting from each side of the regular shape in a balanced fashion, irregular tiles are formed which maintain the ability to tessellate.
To begin, we need to base our pattern off of a regularly shaped tile; and the only polygons which can be used as the base for the template are triangles, quadrilaterals and hexagons… Why?
In order to cover the plane exactly, it means that any point on the plane is covered 360° about that point. The points we need to check would be the points of contact between the vertices of polygons; and we realize this means that the internal angle of the polygon must divide 360° evenly – otherwise there would be gaps or overlaps. We can use the fact that the internal angles of a regular polygon are given by: Interior Angle = ((n-2) × 180°) / n, where n is the number of sides in the polygon. From here we can see that the only acceptable values of n are 3,4 and 6, hence the need to begin with triangles, quadrilaterals or hexagons as a base for the regular tessellation.
Here’s an interactive graphic to play with:
- You start with a hexagonal tessellation.
- Move your mouse to modify a side in the hexagon; notice there is a symmetrical action on the opposite side of the tile in order to maintain the tessellating property.
- Click through to modify other sides (3 total)
- On the 4th click, a “parquet deformation” is formed, where you can see a seamless gradient from the regular hexagonal tessellation to the modified shape.
4-FOLD ROSETTES & THE IBN TULUN MOSQUE
The second style of pattern we examine are rosettes. These are style of tiling which often consist of some kind of star with petals radiating from its sides. The patterns follows strict proportionality rules and are generated via straight-edge and compass constructions; the motivations behind this become clear once we see how new patterns can be formed and the results of this can be astounding.
First, let’s outline the rules of straight-edge and compass constructions (which are the first three postulates of Euclidean geometry):
- A line may be constructed to connect any two points
- Any line segment may be extended continuously in a straight line
- A circle may be drawn given a point for its center and a radius
Given these axioms for construction in general, we can now outline the rules which govern the construction of a template tile in a rosette:
- All closed shapes must have at least one line of symmetry
- Any similar shapes must be congruent
- Any polygons which appear must be regular
While these rules may seem quite restricting in terms of pattern design, we’ll see how by following them we can construct and decompose patterns quite easily, with little modification to the base template. For example, looking at the animation on the right, we can see how the generating tile for the Ibn Tulun pattern takes two smaller tiles and seamlessly merges them (in fact this process is called خياطة in Arabic – literally meaning “sewing”).
Lets call the original generator tile A, the larger sub-generator tile B and the smaller sub-generator tile C. Note that tile A isn’t randomly divided into tile B and C, rather the diagonal of tile B is exactly the same side length as A. This proportion between the side length of tile A and its diagonal sets the scale for the rest of the pattern, and by looking at the basic construction lines of tile A, we can see that sub-tile B contains half of a regular octagon and sub-tile C has the four-fold construction lines.
Ultimately this means that tiles B and C are able to themselves turn into valid pattern generators independently, as well as combine with each other to generate another pattern, i.e. tile A.
Could we then go the other way and combine tile A with an appropriately sized tile to create a larger generator for a new pattern?
The answer, as you might have guessed, is yes, and I’ve added another little interactive visual below to get a sense of how this proportionality works:
- You start with an overall generator tile A, outlined in the black box, which is a combination of tile B (red box – initially scaled to match tile A) and tile C, which is the complement of B (so initially scaled to 0)
- By moving your mouse along horizontally, you can scale down tile B and scale up tile C; the two sub-tiles are combined like the pattern in the Ibn Tulun mosque, and the resulting generator tile is outlined by the black box
- By moving your mouse vertically, you can see how the combined pattern would tile the plane
- A mouse click toggles between a simple 4-fold pattern and the basic construction lines for any 4-fold pattern
Some things to consider… by scaling the construction lines, what scales produce tiles with interlocking lines (e.g., try 0.5, or √2/2 ~ 0.707)? What does the corresponding 4-fold pattern look like and is it valid based on the rules outlined above? If not, can you modify the pattern based on the construction lines to create a valid tiling?
PENROSE TILINGS & THE DARB-E-IMAM
The final patterning style we look at is rather exceptional, and is in fact a fractal pattern in the form of a Penrose tiling. As we noted earlier when trying to tessellate the plane, we would need to begin with a triangle, quadrilateral or hexagon and modify the shape symmetrically in order to create a tiling. As a result, we end up with a “periodic” tiling, meaning we can translate copies of our tessellation onto itself perfectly; the reason for this is simply because the triangles, quadrilaterals or hexagons we started with also have this translational symmetry.
Penrose tilings, named after Roger Penrose who introduced them in a 1974 paper, are tilings which are “aperiodic”; in this case meaning they don’t have that property of translational symmetry. These tilings are also fractal, or self-similar, so patterns repeat themselves at different scales. Those two sentences might seem contradictory, but while the aperiodicity implies a lack of translational symmetry, rotational and reflectional symmetry is still often apparent. This means zooming in/out of the tiling pattern as opposed to panning across it will reveal an infinitely repeating pattern.
That self-similarity is also useful because it gives us a way to actually generate tilings, once we have an initial set of “seed” tiles and a “subdivision” rule telling us how to cut each of our tiles so we can iterate our pattern. For example, one type of Penrose tiling is built from two different rhombi… which are quadrilaterals… which should tile the plane periodically based on what we said earlier… except they don’t, and we make sure of that by giving our rhombi particular dimensions so that they never are able to form a single parallelogram. And those dimensions are (of course) related to the Golden Ratio.
Essentially, our rhombi are made up of the acute and obtuse Robinson triangles: triangles with either (duplicate side/base) or (base/duplicate side) equal to the Golden Ratio. The acute Robinson triangle can be further split into one acute and one obtuse Robinson triangle, and these subdivision rules, along with our original rhombi in a configuration, become all we need to produce a Penrose tiling.
Below I show the divisions of the rhombi and Robinson triangles, as well as the far more involved subdivision rules needed for the Darb-e-Imam tiling, as determined by Lu and Steinhardt in their 2007 paper, Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture; this tiling makes use of three shapes known as girih tiles: a decagon, a bowtie and a slim hexagon made by splitting the decagon in two with the bowtie. The pattern is only iterated at two levels (presumably for practical reasons), with the first level only consisting of the decagon and bowtie, and the second level is generated by sub-dividing the decagon and bowtie as shown.
And finally, one last interactive visual demonstrates the generation of a Penrose tiling using the rhombus tiles described above:
- Clicking in the top-right corner of the module iterates the pattern by dividing each tile according to the rule (as shown in the figure below); you can iterate twice before needing to zoom out
- Clicking the bottom-left corner will “zoom out” of the pattern; you can zoom out twice before needing to divide the pattern again
- Hovering in the bottom-right shows construction lines, making the divisions easier to see
- Hovering in the top-left fills in the pattern.