More Scaling Girih Tiles

My scaling girih tiles project is 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. This innovation would have been inappropriate before computers. The remarkable complexity of design in Islamic girih tiles doesn’t suggest the need for extension by multiple levels of self-similarity. Nevertheless, 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 would have been impractical and unnecessary for Islamic artisans. I decorate these 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 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.

Since this is art, not math nor historical architecture, I’m free to add tiles to the origianl girih tile set. Sometimes I add a second rhombus that is not in the basic set of five girih tiles. My designs are not all always 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. I’m not attempting to 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.






References

1. Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". http://peterlu.org/content/decagonal-and-quasicrystalline-tilings-medieval-islamic-architecture.

2. Wikipedia contributors, "Girih tiles," Wikipedia, The Free Encyclopedia,http://en.wikipedia.org/wiki/Girih_tiles(accessed November 11, 2010).

Asymmetrical Scaling Girih Tiling

Below are three renderings of an asymmetrical scaling girih tiling. This tiling demonstrates that the scaling girih tiles might be extended to fill the plane, and that the scaling could continue as well. In this case I have intentionally ended the boundary with pentagons and decagons at the same scale to indicate the degree of control possible. Depending on requirements, a scaling girih tiling could be made to create controlled patterns, fill a variety of forms, or control a range of line-shape densities.

Scaling Girih Tilings

This series of drawings is based on the discovery by Peter J. Lu [1] of 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. The scaling tiles lack girih lines on two sides, but otherwise preserve the angles of girih tiles.


The scaling tile sides match the sides of two sets of girih tiles. I use multiple scaled sets of the tiles to generate fractal-like drawings in which tile patches at different scales are similar.

Since this is art, not math nor archeology, I’m free to add tiles, ignore symmetry, and disregard normal restrictions on tessellations. I’m searching for interesting designs whether or not they fill the plane. I’m not attempting to stay true to the historical methods.





References
1. Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". http://peterlu.org/content/decagonal-and-quasicrystalline-tilings-medieval-islamic-architecture.
2. Wikipedia contributors, "Girih tiles," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Girih_tiles (accessed November 11, 2010).