Tiling designs utilized in the Wearable works are inspired by Islamic architecture. Photographs of plant details, mostly floral, are extracted from their backgrounds, montaged into still life compositions and embedded into tiles called girih, as shown in Figure 1.
Figure 1: Girih tiles with strapping lines.
The word girih literally translates to “knot” in Arabic, and was first used by Peter J. Lu to describe a set of five tiles: a regular decagon, a regular pentagon, a concave hexagon (bowtie), an elongated hexagon, and a rhombus decorated with zigzagging lines called strapwork. Still life pattern groupings are set into the tiles following the lines of the strapwork as guides for repetitions. The tiles are then arranged in a composition that is bilaterally symmetrical, but nonperiodic.
At close viewing distances, the floral forms are visible and distinct, but these dissolve into the broader context of geometric pattern at more typical viewing distances, consistent with the aesthetic of dematerialization observed in Islamic architecture.
At Generative Art 2012, we introduced Tilings made from still life compositions bounded by girih shapes. Each girih tile is divided further by strapping lines, which are reserved as negative space. As the tiles are joined, the strapping lines reveal larger fivefold symmetries. For this work we introduce self-similarity, a two-level design in which the girih forms appear at two scales with a variation on the subdivision rule used on the Darb-i Imam shrine built in 1453 Isfahan.
Smaller scaled girih tiles inflate to create larger tiles through subdivision and substitution. What makes this process more complex is that the tiles themselves are not filled with flat color. In most girih tilings, individual tiles are solid colored, making orientation irrelevant under reflections or rotations. As shown in Figure 2, each of our tiles is filled with an asymmetrical floral composition, with a clear orientation or handedness. As the larger second level tiles rotate to create a new pattern, the boundary tiles between them where they abut are bisected and flipped along the axis of rotation. This is where new interior configurations of the tiles are created (Figure 3), and hence the process becomes generative.
Figure 2: Girih tiles with handedness
Figure 3: Center tiles are asymmetrical, tiles on outer edges are symmetrical variations.
History
According to Lu and Steinhardt, the first known use of the girih tiles was in the thirteenth century. One such example is the spandrel from the Abbasid Al-Mutansiriya Madrassa in Baghdad, Iraq (1227–34). 1 As shown in Figure 4, perhaps most well–known, the drawing of girih construction known as panel 28 of the Topkapi Scroll, was likely used as a guideline by artisans and is a “documented example of girih-tile subdivision.” 2 The spandrel of the Darb-i Imam Shrine also demonstrates the use of subdivision (Figure 5). Figure 6 shows the subdivision rule used to create each large tile from small tiles 3 and Figure 7 shows the configuration of large tiles that was used to construct the spandrel. 4
Figure 4: (top) Panel 28 of Topkapi Scroll showing subdivision. (middle) Girih tiles filled with color. (bottom) Application of subdivision rule (blue lines) with larger scale girih (dotted red lines). From Lu and Steinhardt supporting online material. 2
Figure 5: Darb-i Imam shrine spandrel, Isfahan, Iran, 1453 C.E. 1
Figure 6: Subdivision rule used to create small-scale girih patterns from the Darb-i Imam shrine pattern. 3
Figure 7: Construction of large-scale girih pattern of spandrel. Yellow dotted ine shows boundary of spandrel. 4
Previous Work
Common themes of aperiodic tilings were explored in our previous work: Generative Texture Maps for Animation (Generative Art 1999), TOLAS: Terrain of Light and Sound (Generative Art 2004) and Tilings (Generative Art 2012). The earlier work used kite-and-dart Penrose Tilings (ca. 1974) using Roger Penrose’s rules for tiling a plane. Current work and Tilings of GA2012 is based on the recent mathematical exposition by Lu and Steinhardt on Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture.
In the work for GA2012 several iterations of a basic Venus of Willendorf figure/architectural plan were explored. What is at the head is the ambulatory or crossing; this motif exhibits radial symmetry. The torso corresponds to two aisles, evocative of a nave. The Venus’ arms and legs are repeats or partial repeats of the radial configuration. This overall configuration evolved in response to the natural limits of pattern growth imposed the girih tilings themselves, and by the width of fabric boundaries. Pattern lengths on the other hand, could be more flexible and were typically determined by aesthetic sensibilities for proportion or by the moment in the pattern when tiling rules were broken. This was a temporal consideration as the pattern was generated from top down. At the moment when a gap or overlap was going to be introduced (and thus breaking the tiling rules) a decision is forced to backtrack and rework (a choice rarely made), to add a new prototile or leave negative space in the event of a gap, or to introduce an overlap. Typically the composition is finalized shortly after the introduction of a rule-breaking interjection. In the samples shown in figures 8 and 9, there are two close-ups, one of the head/crossing/rose window to show the multiple variations of tile configurations used, and to demonstrate the rule-breaking overlaps or new prototiles utilized. Then, for purposes of showing examples of each Venus figure configuration, all of which are different variations on arranging the girih, the full images are shown in Figures 10-11.
Figure 8: Variations on rose window or "Venus head" girih configuration. (L) hexagonal star surrounded by rombus-stars; (C) decagon surrounded by pentagons; (R) decagon surrounded by alternating hexagons and bowties. Each is then surrounded by decagons.
Figure 9: (L) New prototile created from tiling rule-breaking in Fig. 8L above; (R) Tile overlaps imposed in response to rule-breaking in Fig. 8R above.
Figure 10: First variation on Venus form; uses all five tiles, has gaps; there is one gap that remains empty at lower center.
Figure 11: (L) Second iteration of Venus form; uses all five tiles, no overlaps, no gaps. (R) Third iteration of Venus form; uses all tiles except pentagon; has overlaps.
Generative Process
Our process entails the tiling of a plane using a set of five prototiles: a regular decagon, a regular pentagon, a rhombus, and two hexagons. One hexagon is a concave “bowtie,” while the other is a convex elongated configuration (hereafter, “hexagon”). In all of the tiles, edges have the same length, and all angles are multiples of 36 degrees. Strapping lines (girih) subdivide the basic shapes at an angle of 54 degrees. By assembling tilings of these prototiles, nonperiodic five-fold and ten-fold axial symmetries emerge. Using transformation rules of inflation and deflation via substitution of larger and smaller tile sets, self-similarity is exploited. “An object is said to be self similar if it looks ‘roughly’ the same on any scale.” A tiling with self similarity, using decagons that are divided into smaller decagons, hexagons and bowties, and using subdivision rules results in a quasicrystalline (in 2D) tiling.
In addition to the generative partition of the plane into regions defined by prototiles and subdivision rules, we impose additional aesthetic variations on the tiles. Instead of using flat colorings of the tiles, each tile is textured with a still life composition, typically with a clear handedness and therefore symmetry breaking in its use (see Figures 2 and 12). To reintroduce and/or preserve axial symmetries that are thus broken, asymmetric tiles are reflected across an axis, or new lines of reflection are introduced within a tile to reintroduce lost symmetries (See figures 3, 13-15).
Figure 12: Girih tiles for background to jacket design in Wearable.
Figure 13: Star constructed from five rhombi.
Figure 14: Rhomb tile subdivision into ten decagons, ten rhombii, eight hexagons and six bowties.
Figure 15: Five rhombs (see Figure 13) subdivided; new decagons constructed to preserve rotational symmetry.
The strapwork originally used in the girih tilings as a means to guide the placement of and cutting of ceramic tiles, is more often preserved in our tilings as negative space. Thus, the girih reinforce lines of symmetry as a kind of interlace design. In figure 16, strapping lines are emphasized in red.
Figure 16: Strapping lines over a configuration of a decagon, hexagons, and bowties.
Subdivision tilings (see Figure 17) were generated on textured backgrounds for use in jacket design. The subdivided decagon and bowtie are sourced from the Darb-i Imam shrine, and the hexagon subdivision was independently constructed.
Figure 17: Subdivision tilings used in jacket design.
Figurative overlays at a still larger scale incorporate the textural elements used within the tiles but float above them independent of the girih structure altogether. This final ornamentation of a jacket design is shown in Figure 18.
Figure 18: Jacket design from rhomb subdivision (see Fig. 14) and with free-floating floral patterns (also employed within tiling) floating on top of geometric composition.
Acknowledgements
Anna Chupa thanks Lehigh University College of Arts and Sciences and the Provost’s Office for travel support to exhibit and present at Generative Art 2014. She also acknowledges the Lehigh University Faculty Research Grant program for their support of textile acquisition and printing.
References
1 Peter J. Lu and Paul J. Steinhardt, “Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture.” Science 315 (2007): pp. 1106–1110.
2 Peter J. Lu and Paul J. Steinhardt, “Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture.” Science 315 (2007): supporting online materials.
3 Darbeimam subdivision rule. CC-BY-SA InfoCan. 13 February 2012 Wikimedia Commons. Accessed January 21, 2015. http://commons.wikimedia.org/wiki/File:Darbeimam_subdivision_rule.svg
4 Spandrel-large scale pattern. CC-BY-SA InfoCan. 13 February 2012. Wikimedia Commons. Accessed January 21, 2015. http://commons.wikimedia.org/wiki/File:Spandrel-large_scale_pattern.svg
5 “Self-Similarity.” Wolfram Mathworld. Accessed September 24, 2013. http://wathworld.wolfram.com/search/?query=Similarity.
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