Mathematics Behind the Construction of Islamic Geometric Patterns. Islamic art was born in a unique culture that predisposed it towards expression

Size: px

Start display at page:

Download "Mathematics Behind the Construction of Islamic Geometric Patterns. Islamic art was born in a unique culture that predisposed it towards expression"

Chad Martin
5 years ago
Views:

1 Theo Smith 391A-Math Gems Jenia Tevelev 4/28/15 Mathematics Behind the Construction of Islamic Geometric Patterns Islamic art was born in a unique culture that predisposed it towards expression through geometric patterns. Islamic teaching precludes the use of images of living things, yet this is not the case with all forms of Islamic art (Abas, Salman 9). Patterns of flowers and plants are common, and images of animals can also be found. Therefore, it is likely that the predominance of geometric patterns extends deeper than this simple explanation. Unlike Western or Far Eastern traditions, in Islam God never appears in physical form. Thus, Allah is a more abstract concept, which leads to the use of geometry as a mode of expression. Beyond simple abstraction, the geometric patterns used in Islamic art possess traits which make them particularly applicable as a medium for expressing God. The patterns used possess both an underlying unity, as well as a sense of the infinite, both of which create a meditative aspect (Abas, Salman 9). The result of this emphasis on geometry is a wealth of incredibly complex patterns. It is likely that the artists did not have a formal understanding of the mathematical concepts underpinning their art, yet the repeated use of patterns without errors suggests an impressive conceptual comprehension. The designs contain mathematical principles which would not be explicitly formulated by western mathematicians for hundreds of years. The incredible breadth of patterns and mathematical concepts far exceeds the scope of this review, so this paper will focus on several of the common underlying principles of geometric patterns as well as the more complex formulations of several specific patterns. In addition, there are many different methods of approaching a geometrical analysis, so this paper will focus primarily on the construction and symmetries of these geometric patterns.

2 Most Islamic geometric patterns are composed of repeating unit cells that contain multiple lines of symmetry (Abas, Salman 47). Figure 1 is an example of a pattern found in the 12th century Great Mosque in Seville, Spain. Despite the apparent simplicity of the pattern, when it is broken down a wealth of other patterns appear. Outlined in red is the unit cell, which contains all of the points of symmetry and can be used to reconstruct the entire pattern. One way to analyze this pattern would be to continue to overlay a grid of equilaterals triangles which are outlined in black. At the center of each triangle is a point with 120 rotational symmetry, shown by the blue dot and lines. A scaled version of the sixpointed star appears within the pattern when connecting the lines delineating points of threefold rotational symmetry. At the center of each six-pointed star is a point of sixfold symmetry, which is represented by the black lines and dots. Surrounding each point of sixfold symmetry is a hexagon which can be rotated 60 without any apparent change to the pattern within. At the midpoint of each black line is a point of twofold symmetry which can be rotated 180. Connecting every point of threefold symmetry yields the hexagon outlined in green which is a scale of the larger black hexagon with the same symmetrical properties. Either of these could be propagated into a grid like much like the equilateral triangles that provides another way to view the pattern. Another hexagon, outlined in orange, can be created by connecting alternating points of six and threefold symmetry. The orange hexagon is centered on a point of threefold symmetry, and the interior can consequently be rotated 120 with no apparent change. Alternating the hexagon outlined in orange and the blue sixpointed star creates an additional sub-pattern. The properties of the unit cell warrant further consideration. The unit cell can actually be reduced further to the area outlined in pink. It is depicted outside the unit cell in order to avoid color, but since it lies on a twofold symmetry line it is a simple matter to visualize rotating it inside the unit cell. This area is the smallest area that can be used to recreate the entire unit cell, using only the symmetrical properties contained within the cell itself. One way to do this would be to rotate it three times around the threefold symmetry point, and to then rotate the result 180 about the central twofold symmetry point. Thus the entire pattern can be understood if the pattern contained within the pink area and the points of symmetry are known. The unit cell contains a condensed group of all of the symmetries that could be found in the infinite pattern. This pattern, in addition to the rotational symmetries, also has translational symmetry, which was not depicted but can be imagined by picking a point, and sliding it to an equivalent point when the pattern repeats. By the application of translational symmetry a complete, infinite group of symmetries can be acquired. These

3 groups would not be formally defined for almost one thousand years until group theory was introduced by Evariste Galois (Abas, Salman 69). Despite the lack of a formal understanding of group theory, the complete set of rotational and translational symmetries found in this work of Islamic Art satisfies the requirements of closure, associativity, identity, and inverse when combined together (Abas, Salman 69). An additional property of the unit cell found in this pattern is the presence of only two, three, and six-fold symmetries. According to the crystallographic restriction, the only other rotational symmetry allowed in infinite repeat patterns is fourfold symmetry (Abas, Salman 63). This property is particularly important as we approach the next set of patterns. Patterns containing star shaped, pentagonal, and decagonal figures present interesting properties. These introduce the potential to break the previously mentioned crystallographic restriction. Not all patterns containing decagons or ten pointed stars possess five and tenfold symmetry, but when they do the result is the creation of quasi-crystalline geometric forms, which use repeating figures and shapes but in a non-periodical manner (Hwang). This phenomenon was first studied carefully by a physicist, Peter Lu, in 2007 (Lu, Steinhardt 1106). Prior to his publication scholars assumed that most work on these patterns, named girih patterns, were composed with a ruler and compass. Lu proposed that the level of precision found in examples of artwork could not be achieved through such methods (Lu, Steinhardt 1106). He discovered a set of 5 tiles which could have been used to produce a wide variety of patterns without repetition, termed girih tiles. His theory was supported by the Topkapi scroll which outlines patterns for such tiles seen in figure 2A (Lu, Steinhardt

$Figure SEQ Figure \* ARABIC 2 A-Girih Tiles B- 15 th century Topkapi Scroll with images of girih tiles overlaid (Lu, Steinhardt 1107) These shapes consist of a decagon, a pentagon, a rhombus, a$ hexagon, and a bowtie, which can be seen traced on the scroll in figure 2B.

hexagon, and a bowtie, which can be seen traced on the scroll in figure 2B.

4 1106). Figure SEQ Figure \* ARABIC 2 A-Girih Tiles B- 15 th century Topkapi Scroll with images of girih tiles overlaid (Lu, Steinhardt 1107) These shapes consist of a decagon, a pentagon, a rhombus, a hexagon, and a bowtie, which can be seen traced on the scroll in figure 2B. The side of each tile is the same length and is bisected by a decorative line at a line that forms a 72 and 108 angle (Lu, Steinhardt The use of these tiles can be traced to the 13 th century as more complex geometric patterns were developed, although they may have existed earlier (Lu, Steinhardt 1106). The use of a set of polygons to create a non-periodic pattern has profound mathematical importance as it presents a solution to a problem which would not be formally solved in the west for another 700 years. In 1961 Hao Wang proposed that a set of tiles which demand a non-periodic arrangement is impossible. He was proven wrong by several different sets of tiles which demonstrated this non-periodic property, although the smallest set was discovered by Roger Penrose in 1974 (Hwang). The set consists of a kite and a dart as shown in Figure 3, which must be connected in such a way that the points marked heads and tails must coincide with other points with the same marking (Schultz). The five girih tiles discovered by Lu in the patterns he observed can be decomposed into kites and darts and display the same aperiodical behavior as the Penrose tiles (Lu, Steinhardt 1106). Penrose tilings have applications in physics and chemistry where certain materials contain a quasi-crystalline structure due to non-periodic tiling (Hwang). This was first seen in 1984 when Dany Schechtman found crystals composed of aluminum and manganese which displayed five-fold symmetry (Hwang). This discovery defied the rules that had previously been followed which dictated that all crystals must follow

5 a periodic arrangement. The complexity of these geometric theories which would not be fully understood until centuries later underscores the accomplishments of the Islamic artists who were able to create these patterns.

Medieval Islamic Architecture, Quasicrystals, and Penrose and Girih Tiles: Questions from the Classroom

Medieval Islamic Architecture, Quasicrystals, and Penrose and Girih Tiles: Questions from the Classroom Raymond Tennant Professor of Mathematics Zayed University P.O. Box 4783 Abu Dhabi, United Arab Emirates