Scaling Girih Tiles (Also see Girih Extended.)
Girih Extended and the scaling girih tiles projects are inspired by a remarkable Islamic art patterning system that originated by the year 1200, and was rediscovered by Peter J. Lu [1] in 2005. Islamic artisans used girih [2], from the Persian word for "knot", to develop intricate patterns from just five tiles decorated with lines. The lines rather than the tile edges become the pattern. I've extended the system for design purposes to allow for scale and density variations. By expanding the system I hope to create patterns with the look and spirit of one facet of Islamic art while providing new design possibilities. The key is the simple addition of scaling tiles and a self-similar tile set.
By taking the basic girih tile set and extending it to include scaling tiles I've added a level of complexity that's easily managed with digital media. With up to 32 tiles in my tile set, this system would have been impractical before computers. I decorate the extra tiles with lines (girih strap work) like the five girih tiles, but unlike the girih tiles the scaling tiles are not equilateral. These tiles also differ in that they may lack girih lines on some sides, but otherwise I gnerally preserve the angles of girih tiles. I generate multiple sets of the girih tiles, scaled to match the two different sides of each scaling tile. With the complete set I can create fractal-like drawings in which tile patches at different scales are similar. I've posted example diagrams with both the girih lines and the tile outlines here.
Since this is art, not math nor historical architecture, I'm not constrained by custom. Besides the scaling tiles, sometimes I add a second rhombus that is not in the basic set of five girih tiles. My designs are seldom tessellations – gaps and boundaries are OK. I choose to work with tilings that are always edge-to-edge, meaning adjacent tiles always share full sides. I often create symmetrical designs, but infinitely changing asymmetrical patterns are also possible. I'm searching for interesting designs whether or not they include gaps, fill the plane, or stay true to the historical methods.
It's interesting though not surprising that in some of the tile sets in this project the long and short sides of the scaling tile are in the ratio of 1.61803…, the golden ratio. A subset of tilings within the project is loosely based on decagon and pentagon fractals.
