Design for Muqarnas

Here's a scaling girih tile design in the shape of muqarnas. First is the design with tile edges, then just the girih tile strapwork.

Girih tiling based on a decagonal fractal

My last entry described a girih tiling based on a pentagonal fractal. It's also possible to base a fractal tiling on decagons. This 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.


Here's a completed girih tiling.

Girih tiling based on a pentagonal fractal

These girih tile drawing are loosely based on a fractal, a variant of the Koch snowflake, but pentagonal. The inside edges of the outer border of kite-shaped polygons follow a fractal. The original pentagon edges are broken three times to generate the final image. Pentagons may be used in place of the kites.

Each successive transformation of an edge into four edges was done by creating two new edges that are 72 degrees to the prior edge, and the lengths of the four new edges are 1/(2x(1+cosine(72))) of the prior edge.

Starting with a pentagon and using an angle of 72 degrees allows us to fill the interior with girih tiles. In the examples below I used a couple of scaling girih tiles.

Here is the girih line drawing I generated from the tiling. Yes, it has gaps.


Here are two more images following the same process.

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).

Pentagons and Isosceles Trapezoids

This dot pattern diagram from the series, Self-similar Boundaries, is based on pentagons and isosceles trapezoids. The trapezoids are the scaling polygons. They share non-parallel sides with a pentagon and parallel sides with scaled pentagons. The trapezoids generate one pentagon that scales down and one that scales up. The initial pentagon generates one trapezoid, with two scaled pentagons. Each subsequent iteration could double the number of pentagons, except that many are duplicates. In this example I follow three scaling iterations which might have produced 1 + 2 + 4 + 8 = 15 different pentagons, but since some are equal the pentagons produced are 10.