Classifying Hexagonal Tilings in Islamic Architecture with a Single Numerical Parameter (original) (raw)

Islamic designers drafted a large variety of hexagonal patterns in various media, including stone, tile, wood and cloth, throughout the medieval Islamic world. To highlight the geometric similarities among these patterns, we present a simple scheme to classify different types of hexagonal tilings in Islamic architecture, using a single numerical parameter. Islamic artisans and designers exploited geometry to create a wide variety of intricate patterns in a range of media, including brick, stone, ceramic tile, wood and cloth. In many cases, the underlying structure of these patterns was derived from compass and straightedge techniques [1]; on other cases, particularly decagonal and pentagonal geometry, a set of prototiles known as " girih tiles " was used to lay out geometric patterns [2]. One of the simplest periodic structures is based on tiling the plane with hexagons, forming a honeycomb lattice. In this paper, we illustrate how this simple hexagonal honeycomb structure was elaborated to form a range of line patterns that appear throughout Islamic art and architecture; these patterns are expressed in a number of different media: brick, stone, ceramic and cloth. Although these patterns may appear at first glance to be quite different, we show that in fact these patterns all share not only the same symmetry, but also the same underlying structure: hexagons placed at the vertices of a tessellation of hexagons of a different size. We present a number of patterns that can be classified in part by a single dimensionless numerical parameter: the ratio of the length of the hexagons that appear in the final pattern relative to that of the underlying honeycomb lattice, with only some minor variations. This simple scheme allows a large number of patterns to be compared with only a single number, facilitating comparison and contrast among this collection of historical Islamic art and architecture. The drafting of a hexagonal honeycomb grid using a compass and straightedge was well understood both by the ancient Greeks and by medieval Islamic architects and designers. The specific sequence of steps using the compass and straightedge to create the honeycomb tessellation of hexagons has been well illustrated in a number of publications [1, 3]. Line patterns with underlying hexagonal geometry have appeared throughout Islamic art and architecture. Perhaps the simplest way to decorate this grid is to place a motif at the vertex of each hexagon in the grid; one common motif is another hexagon, which appears in a large number of patterns throughout Islamic history. Examples are expressed in a wide variety of materials, including brick, stone, ceramic and cloth, and cover a range of dates and locations throughout the Islamic world, as shown in Fig. 1, where the underlying hexagonal honeycomb grid is delineated with black dotted lines, and the decorating hexagon motifs are marked in red.