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.