About the Scaling Girih Tile Diagrams (Also see Girih Extended.)
Girih Extended and this project reveal the structure behind a series of scaling girih tile drawings that I subsequently created. I based both projects, including these diagrams, on the discovery by Peter J. Lu [1] of the girih tiles [2], a set of decorated tiles used by Islamic architects for centuries. I've taken the basic girih tile set and extended it to include scaling tiles. I decorate these extra tiles with lines (girih strapwork) like the five girih tiles, but unlike the girih tiles, the scaling tiles are not equilateral. These tiles differ also in that they usually lack girih lines extending from two sides, but otherwise I generally preserve the angles of girih tiles.
I've developed seven different scaling tiles, some more useful than others. They include a trapezoid, a triangle, kites, and concave hexagons. They scale the basic girih tile set by a variety of factors, including approximately 0.5, 0.618, 0.727,or 0.851.
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.
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 fill the plane or stay true to the historical methods.
