Scaling Girih Tiles (Also see Girih Extended.)
Girih Extended and the scaling girih tiles project 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. I've extended the system for design purposes to allow for scale and density variations. A subset of tilings within the project is loosely based fractals from decagons and pentagons. The following describes my method for generating the tiling fractals.
The tilings in this project are not tessellations. They do not fill the plane, and there may be gaps. However, the boundaries of tiles which form the basic motif for this subset of the scaling girih tiles project are fractal. A single line traced around the border of equal decagons or pentagons creates a fractal boundary. The boundaries are much like a Koch snowflake or variant of the Koch curve. The basic tilings are frameworks from which I build girih tile patterns.
The first process is simply nesting or subdividing a decagon with five smaller decagons. Starting with a regular decagon, place five smaller regular decagons, edge-to-edge, within the original decagon. One vertex each of the smaller decagons should coincide with a vertex of the larger. Two sides each of the smaller decagons should be edge-to-edge with another small decagon. The lengths of the edges of the smaller decagon are 1/(2x(1+cosine(72))) of the larger. This is the same as calculating the sides of the five smaller decagons by dividing the larger decagon sides by the golden ratio, twice. That is, divide the length of a side of the original decagon by approximately 1.618, then divide the result again by approximately 1.618. The result is a measure for the side of a smaller decagon to nest and repeat inside a larger. The process of subdividing or nesting decagons with five smaller decagons may be repeated infinitely. Following are diagrams of nesting decagons, and a fractal border.
