ISLAMIC GEOMETRIC PATTERNS

Before going any further, I would like to first explain some mathematical characteristics of Islamic geometric patterns in order to give a better and technical picture of these patterns. 
The base of Islamic geometric patterns is tessellations. 
According to the website Wolfram Mathworld, 
“A regular tiling of polygons (in two dimensions), polyhedra (three dimensions), or polytopes (n dimensions) is called a tessellation” (Wolfram MathWorld, 2008), where by regular tiling we mean that a regular polygon (an n-gon with equal sides, e.g. a triangle, a square or a hexagon) fills up the whole 2- dimensional space (which is basically just a plane) without any gaps. 

However, there are only three polygons which can fill a plane without any gaps: triangles, squares and hexagons (Wolfram MathWorld, 2008).

Islamic geometric art employs these tessellations as an underlying grid and develops its own patterns. Main content of the Islamic geometric patterns is Islamic stars. 
The typical construction of an Islamic geometric pattern consists of a major star in the middle and accompanying stars or other geometric elements around this central star. 

According to El-Said and Parman (El-Said & Parman, 1976), there are 3 main classes of Islamic geometric patterns based on the proportions in the geometric figures :

These patterns are based on the square tiling of a plane.

These patterns are based on the hexagonal tiling of a plane. 

The most common examples for this class are the patterns with 6 or 12 pointed stars.

The patterns of √5 relations class are based on ten pointed stars and ten pointed stars are based on pentagons. This class is named after the ratio of a pentagon’s diagonal to its side (1+ √5)/2 (~1.168). The ratio of a pentagon’s diagonal to its side is also known as golden ratio since the antiquity (Lawlor, 1982).