The repeating patterns in Islamic design are based on the exploration of geometric principles - most importantly, the relation between circles and hexagons, and certain happy ratios.
Eva Wilson explains (cobbled together from her book Islamic Designs):-
The circumference of a circle can be divide into six equal parts by its radius to form a hexagon. When alternative points of the figure are joined by straight lines two triangles are produced which together make up a six-pointed star.
The ratio between the diameters of hexagons based on concentric six-pointed stars drawn from the hexagon's corners is 1:2. When the concentric stars are drawn from the hexagon's mid points, the ratio between the heights of the hexagons is also 1:2.
The hexagon, together with its associated six-pointed star, is the most important geometric construction in Islamic art. It is a common basis for repeat patterns and many designs of apparently different character are drawn on hexagon-based grids.
A hexagon-based pattern is also used in circular designs. A circle surrounded by six other circles of the same size forms a hexagon. Allowing the peripheral circles to encroach on the inner circle produces a variety of geometric and decorative characteristics.
Tassellatura from the Alhambra, Granada