Girih Tiles - Mathematics of Girih Tilings

Mathematics of Girih Tilings

In 2007, Peter J. Lu of Harvard University and Professor Paul J. Steinhardt of Princeton University published a paper in the journal Science suggesting that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilings (presentation 1974, predecessor works starting in about 1964) predating them by five centuries.

This finding was supported both by analysis of patterns on surviving structures, and by examination of 15th century Persian scrolls. However, we have no indication of how much more the architects may have known about the mathematics involved. It is generally believed that such designs were constructed by drafting zigzag outlines with only a straightedge and a compass. Templates found on scrolls such as the 97 foot (29.5 metres) long Topkapi Scroll may have been consulted. Found in the Topkapi Palace in Istanbul, the administrative center of the Ottoman Empire and believed to date from the late 15th century, the scroll shows a succession of two- and three- dimensional geometric patterns. There is no text, but there is a grid pattern and color-coding used to highlight symmetries and distinguish three-dimensional projections. Drawings such as shown on this scroll would have served as pattern-books for the artisans who fabricated the tiles, and the shapes of the girih tiles then would dictate how they could be combined into large patterns. In this way, craftsmen could make highly complex designs without resorting to mathematics and without necessarily understanding their underlying principles.

This use of repeating patterns created from a limited number of geometric shapes available to craftsmen of the day is similar to the practice of contemporary European Gothic artisans. Designers of both styles were concerned with using their inventories of geometrical shapes to create the maximum diversity of forms. This is using a skill and practice very different from mathematics.