Almost all geometric compositions are set in a rectangle: door panels, book pages, dados etc. You can see that these compositions were made-to-measure because you can see quarter star patterns on the corners and half stars on the borders.
So, how did craftsmen manage to do this? How did they make a composition fit perfectly in a rectangle? How did they know the scale of the pattern they needed before they started?
The answer lies in square root rectangles. A composition like the above koran box of Ottoman Sultan Selim II has a central tenpointed star. This tells us it is a tenfold pattern. Therefore it is based on a circle divided into 5 (or 10) equal sections, Therefore, the proportion of the box is a square root of 5 rectangle.
Some Basic Design Principles
All geometrical compositions are embellished in one way or another. They are never just a composition of single lines. How these compositions have been embellished over the centuries depends to a large extent on regional design traditions, and on available materials. So, for example, if you see ceramic tiles with a turquoise glaze, such as on this dome in Iran, it is because the craftsmen were able to obtain the mineral turquoise to make that pigment.
Below left is a much more complex grid. Nevertheless there are only three different shapes: the blue wedge shapes, the green shape with the arrow inside and the red shape with the five pointed star inside. Using shapes like these, it becomes possible to be creative and experiment. Shapes can be arranged in many different ways. This composition can be found on a minbar in the Great Mosque of Isfahan, Iran. Below image shows a recent interpretation of that same composition.
Islamic Geometric Family Tree
About 90% of all patterns and compositions you will ever see, can be categorised as either fourfold, fivefold or sixfold. The remaining 10% is essentially “everything else”.
In Islamic geometric design, there are essentially only three variables:
1) The grid that forms the invisible structure
2) The size of the grid
3) The content/pattern that is placed inside the shapes that form the grid
These variables can be combined in thousands of different ways to make thousands of different compositions.
Repeating a pattern to make a bigger composition is called tessellation. For centuries, it has enabled craftsmen to plan their composition and to work efficiently. This brief animation shows tessellation in action (made by Professor D.R. Aguilera of the University of the Balearics, using Geogebra software)