The Islamic Decorative Canon, 2 out of 3

Girih Arabesque is another extension of the Islamic patterning system, girih tiles. These designs were developed much like Girih Extended, but this time every line is curvilinear. So now I've addressed 2 of the 3 elements of the Islamic decorative canon -- the geometric and the arabesque.

Girih Arabesque, Described

Girih patterns are composed of straight lines. The girih lines continue uninterrupted across polygon boundaries, and terminate at intersections within the polygons. This means that they are ideally suited for a conversion to arabesque using Bézier curves.

Where arabesque girih lines cross polygon boundaries the curves are tangent, creating a continuous flow of line. Girih boundary points define quadratic Bézier curve end points, and girih interior points become Bézier curve control points.

This technique of converting girih patterns to arabesque could be applied to the basic five girih polygons. I’ve applied it to my extended girih designs. As far as I know, this is the first application of arabesque to girih tilings.

Girih Arabesque

I've added Girih Arabesque to my web site. Girih Arabesque is another extension of the Islamic patterning system, girih tiles. These designs were developed much like Girih Extended, but this time every line is curvilinear. So now I've addressed 2 of the 3 elements of the Islamic decorative canon – the geometric and the arabesque.

Girih Arabesque #2

This is my second example of girih arabesque. The underlying structure is from a finite tile set based on Girih Extended. I've converted the girih strapwork to Bézier curves. As in example #1 I'm using an extension of the standard girih tiles, including some scaling tiles that give me the ability to vary density. There are two versions of the pentagon strapwork in this example, adding to the variety of pattern. This arrangement of girih polygons is without gaps and overlaps. It could be repeated indefinitely.

Girih Arabesque #1

Girih patterns start with simple straight-line decorations on the five girih tiles. I've extended the system with scaling polygons to get variation in density of line. In July and August of 2011 I showed examples of Girih Extended at Chambers @ 916. The image below is the first example of another variation on girih patterns. I've converted the geometric, straight-line patterns to arabesque. This is more a working example than a finished design.

It's possible to apply this same technique to just the five girih polygons. The underlying structure of the example includes four of the five. As in my girih extended series I've added an additional rhombus, plus the scaling polygons: a kite and trapezoid. I've been using up to six scaled copies of the extended polygon set within a single design. The scaling polygons allow me to create edge-to-edge tilings or tile patches with considerable variation in scale and density of line. Sometimes I design with inner gaps, and sometimes not. Sometimes the designs are symmetric, sometimes asymmetric. Since my interest is art, not math, I'm not constrained by a requirement to develop tessellations that might fill the plane. The design above could be filled and extended to fill the plane, without gaps or overlaps.

As far as I know this is the first example of girih arabesque. The decoration or strapwork on the polygons has been converted to Bézier curves. At the polygon boundaries the curves meet tangentially, or nearly so. This gives the appearance of continuously flowing lines. There are numerous possible ways to elaborate the curves. In the example I show the curves were easily programmable, and more or less true to the original girih patterns.

The Theoretical Versus Trial-and-Error

When math and art overlap it seems that the role of and relative complexity of the math involved becomes exaggerated. It's as though the art can't be good enough without a complex theory. Witness the never ending false attribution of powers of the golden ratio to art.

I'm aware of a couple of controversies surrounding Islamic patterns and math. There is the general problem of how big a role mathematicians played in the development of patterns. Also, there's the small controversy over whether or not an Islamic pattern predicted the Penrose quasi-crystal pattern. (See Peter Lu and Paul Steinhardt's answer to Emil Makovicky here. [1])

Previously, I referred to the fact that my Girih Extended drawings are developed largely through trial and error. In A.J. Lee's 1987 paper, "Islamic Star Patterns", he addresses the part played by mathematicians in the development of Islamic patterns, and shows how many of the patterns could be developed simply. So simply, in fact, that little more than trial and error was needed. Lee states:

"It may be an advantage for a modern author to develop a systematic analysis of Islamic patterns in purely mathematical terms, but a knowledge of pure mathematics or geometry is unnecessary for those who wish merely to draw Islamic patterns or invent new ones. A theoretical background will often allow the artist to see a number of combinatorial possibilities more quickly than the use of trial-and-error methods, but it forms no substitute for true creativity." [2]

Trial and error can take you a long way. My Girih Extended patterns are not truly Islamic. They only resemble some of the historical patterns. However, they are new art, not a mathematical analysis, cataloging, or repetition of Islamic patterns.

I should add that there’s no simple or recursive definition at work in the Girih Extended designs. After settling on a general goal, I reach a graphic solution through trial and error. Though digital tools were used and custom programs were developed, the designs are not a programmatic approach to drawing. It is still drawing with polygons.

Reza Sarhangi's paper presented to the Bridges 2004 conference provides the best insight into the possible role of the mathematician in Islamic design. He cites a a treatise written by Buzjani in the 10th century. Sarhangi writes:

"Buzjani wrote in On Those Parts of Geometry Needed by Craftsmen that he participated in meetings between artisans and mathematicians. 'At some sessions, mathematicians gave instructions on certain principles and practices of geometry. At others, they worked on geometric constructions of two- or three- dimensional ornamental patterns or gave advice on the application of geometry to architectural construction.'" [3]

So Islamic mathematicians instructed artisans on basic geometry. Did a 15th century Islamic mathematician understand principles behind quasicrystals? (See "The Tiles of Infinity" by Sebastian Prange. [4]) Or, is it possible that the singular example, the Darb-i Imam shrine in Isfahan, Iran, was created by chance, that is by trial and error?

References:
1. Lu, Peter J., et al. Response to Comment on "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture". Science 318, 1383 (2007). [The full comment is available at www.sciencemag.org. here.]

2. Lee, A.J. 1987. "Islamic Star Patterns. In Muqarnas IV: An Annual on Islamic Art and Architecture". Oleg Grabar (ed.) Leiden: E.J. Brill. [The full document is available from ArchNet.]

3. Sarhangi, Reza. "Modularity in Medieval Persian Mosaics: Textual, Empirical, Analytical, and Theoretical Considerations". Presented at the Bridges 2004 conference. [The full document is available here.]

4. Prange, Sebastian. (September/October 2009). "The Tiles of Infinity". Saudi Aramco World: 24–31. [The full document is available here.]

Art as Catalyst for Change

The following exchange is from "Richard Serra: The Coagula Interview", by Mark Simmons, from Coagula, Issue #36 (1998). (See: http://coagula.com/richard-serra/.)

MS: What would you hope that the people who assist in the production of your work would get from having experienced working on Richard Serra's vision, from idea through fruition? And I'm talking about the people who do the computer thing and the steel workers, and the riggers.

RS: I think this, I think basically I'm not interested in people following my work or making work like my work. But what does interest me is the notion that if you do a lot of work it means there's a potential for other people to understand that a lot of things are possible with a sustained effort and that the broadening of experiences is possible and I think that's all art can be. A little catalyst for change. It's not going to change the world. But it can be a catalyst for thought and thought can change and how people think about what's possible can change and I think that if the work has any value at all on its interpretive level I can't get into how people are going to experience it but if it has any value at all I think it stands for one person understanding that the potential for change is in all of us.

The image below represents my latest attempt to broaden experience, seconding Serra's proposal. I've been generating patterns with polygons and code. Last year, I realized I wanted to add density and scale changes within the patterns. Up to that point my diagrams were like many patterns, overall consistent and therefore a bit boring. Scale and density changes outside of patterning are usually simple because you can control graphite, ink, paint in any way you choose to get the full range of densities possible within a medium. But with polygons as your drawing unit and programming as your tool, it's different. Then too, I imposed a not so arbitrary restriction that the polygons had to be from a finite set, placed edge-to-edge, with no overlaps, with boundaries, but not necessarily gaps. So I encountered the density and scale problem, and found a solution that changed the way I make patterns.

Now, I wouldn't expect anyone to repeat this same process, or get into all the particulars of how to generate girih patterns with scale changes. This is my own solution, and not likely of interest to another artist. On the other hand, patterning is frequently subject to innovation by artists and designers. What might be of interest to other artists is Serra's idea about the potential for change. Rather than simply personalize existing techniques, we should promote change.


Here's the last exchange from the Coagula interview:

MS: One last question. Do you have any advice for sculptors and artists?

RS: Work out of your work. Don't work out of anybody else's work.