How can a ruler and a compass cover a whole mosque in patterns?

A ruler and a compass are enough because the patterns are built from a few simple shapes on hidden grids. Many designs are made from repeated squares and circles, overlapped and interlaced, and they include a wide variety of tessellations, patterns of shapes that fit together without gaps. Such patterns can extend indefinitely, which suggests infinity, and they are constructed on grids that need only a ruler and compass to draw.

Take the 8-pointed star, a favourite motif of Islamic tilework. It is made of two squares, one rotated 45 degrees with respect to the other. Many designs start with a circle, which symbolizes unity and diversity in nature. At the 15th-century mosque in Yazd, a circle with six touching circles around it makes a regular hexagon, and from that grid grows a six-pointed star pattern. Then comes a twist: the finished tile pattern is overlaid with a blue tracery that partly hides the plan underneath.

Why geometry? Islamic art mostly avoids figurative images, and this aniconism pushed artists to explore non-figural art and caused an aesthetic shift toward mathematically based decoration. The patterns grew more complex over time, from simple stars and lozenges in the ninth century, through 6- to 13-point patterns by the 13th century, to 14- and 16-point stars in the sixteenth.

The artist Roman Verostko argues that such constructions are in effect algorithms, making them forerunners of modern algorithmic art. A small set of rules, repeated with care, covers a whole wall.