Video: A mysterious design that appears across millennia

What can we make of a design that shows up over and over in disparate cultures throughout history? Theorist Terry Moore explores “Penrose tiling” — two shapes that fit together in infinite combinations without ever repeating — and ponders what it might mean.

Running time: six minutes

Terry Moore is the director of the Radius Foundation which seeks new ways of exploring and understanding dissimilar conceptual systems or paradigms — scientific, religious, philosophical, and aesthetic — with the aim to find a world view of more complete insight and innovation.

~

In 2007 Peter Lu and Paul Steinhardt demonstrated that the tiling on a 15th century shrine in Isfahan was an example of a quasicrystalline pattern, a geometrical structure that was not understood in the West until the 1970’s when Sir Roger Penrose of Oxford University defined several new sets of aperiodic tiling based on five-fold symmetry. These tile sets can form patterns that are perfectly ordered and yet never repeat, and that contain different expressions of the golden ratio and the property of self-similarity. Much to the amazement of physicists and engineers, these patterns were then found in the molecular structure of certain natural materials, where they became known as “quasi-crystals.” Prior to that time, it was thought that five-fold symmetry could not occur in the arrangement of atoms in solids.

Working with Princeton cosmologist Paul Steinhardt, Lu first uncovered the hidden technique by which medieval Islamic craftsmen constructed patterns in five-fold symmetry: they used a set of tessellating tiles as hidden templates. This set of tiles, which Lu and Steinhardt dubbed “girih tiles,” has the same mathematical properties as the aperiodic tiling sets that Penrose defined hundreds of years later. The underlying girih-tile structure of the Islamic patterns is veiled, however, because the lines of the finished patterns are not taken from the edges of the girh tiles, but from lines inscribed on them.

Steinhardt and Lu published their discoveries in the February 23, 2007 issue of Science magazine in an article entitled “Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture.” The article received international attention—not only because they had rediscovered the technique used so effectively by the medieval craftsmen, but because they showed that girih tiles form a previously unknown Penrose set. Steinhardt and Lu further showed that medieval Islamic craftsmen had a sophisticated understanding of the mathematical properties of this tiling set, even constructing true quasi-crystalline tiling patterns and highlighting their expressions of self-similarity. The craftsmen achieved this feat of geometry hundreds of years before quasi-crystalline patterns were understood in the West.

Yet, beyond the expression of complex mathematics, there are deeper meanings to Islamic tiling patterns: they express the worldview and spiritual insights of the culture and the tradition that created them. More than simply decorations, they are an expression of some of the principles that the culture holds most sacred.