Girih tiles

The Darb-e Imam shrine in Iran contains an early and exciting example of non-periodic tiling that was only mathematically appreciated five hundred years later.

Darbi-e Imam
Unknown artisan, 15th century [Public domain]
I could easily write a dozen blog posts about the geometric patterns seen in Islamic architecture: they have a level of complexity and beauty that awes me. Today, though, I want to write about aperiodic tiling and girih.

Girih is that pattern of geometric knots that you see all the time in Islamic art and architecture. It can be planned out in a number of ways, for example by using a compass and a ruler and a strongly mathematically-inclined artisan, but one of the most ingenious methods use the girih tiles.

Girih tiles come in five different shapes, and are painted with a specific set of patterns:

Girih tiles
Cronholm144 [CC BY-SA]

Connect the tiles together so that they fit snuggly against each other. The joints between them disappear, leaving the painted patterns to form the intricate knotwork design:

Girih tiles
Cronholm144 [CC BY-SA]
The lines and patterns don’t need to be drawn individually or planned out: you just fit the tiles together and they do all the work.

There’s something really exciting about these patterns: they do not repeat themselves. The technical description is that they are aperiodic: you can fit the tiles together without any gaps, but the resulting pattern has no translational symmetry.

I should clarify that second part of the description with an example:

  1. Take a a picture of the pattern.
  2. Put it on a transparent piece of plastic.
  3. Overlay that transparent pattern on the original.

If the tiles are aperiodic, you could not slide that plastic in any direction so that it matches up with the underlying pattern – unless you rotate it. The pattern could keep on extending in all directions without falling into an endless repeat. The most famous aperiodic patterns are those formed by Penrose tiles, discovered in the 1970s:

Penrose tiling
Inductiveload [Public domain]
Girih tiles date back to the 12th century CE, and they bear more than a passing resemblance to Penrose tiling. The shrine of Darb-e Imam in Isfahan, Iran, is a significant early example of girih tiles in action. Constructed in 1453, its patterns carry the same aperiodic properties that were so significant 500 years later.

(End note: these patterns also have great significance for crystallography, but that’s a post for another day.)

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