Five Tiles That Map the Infinite

In the twelfth century, craftsmen in Isfahan achieved a mathematical feat the West would not formalize for eight hundred years.

I spent three weeks last autumn in the Darb-i Imam shrine in Isfahan, crouching beneath the turquoise dome with a measuring tape and a notebook. The tilework above me, laid in 1453, covers every surface with interlocking decagons, hexagons, and elongated pentagons — a composition that repeats without ever settling into a simple periodic grid.

Five Prototiles, Infinite Variation

The girih system relies on just five tile shapes — the decagon, hexagon, bowtie, rhombus, and pentagon. Each edge carries strapwork marks that align when two tiles meet, forming continuous interlaced bands. In 2007, Lu and Steinhardt proved these tiles produce aperiodic tilings — predating Penrose tilings by five centuries.

"The geometry is not decoration applied to structure. The geometry is the structure — load-bearing, self-similar, infinite."