CHAPTER 6

TRANSFORMATIONS, SHAPES AND PATTERNS ANALYSIS IN THE NEGEV BEDOUINS’ EMBROIDERIES

The handwork of Negev Bedouin women, which was based on historical techniques of embroidery and knitting, produced a wide range of motifs that demonstrate a special way of transformation and movement across plane and space. Frieze or wallpaper groups, identified with the world of mathemtaics, are featured in these various motifs. Consequence of these representative samples of the wallpaper groups are visible in embroidered parts of dresses and belongings, and the frieze patterns can be found in dress strips and home decorations. Also, mathematics finds its expression in the patterns in a range of colors selected by the Bedouin women to beautify and enrich their products. Each color extends the range of mathematical features on the one hand, and triggers various associations, feelings, and emotions in the observer on the other. Moreover, the colors add further emphasis to the symmetry types of the motifs, without which it may have been impossible to identify the symmetry or transformation in a specific motif. Chapter 6, which begins with a brief account of the symmetries groups, continues with identification of motifs without accounting for the various colors. In the course of the discussion, we show how the symmetry groups in the embroideries and weave motifs, constructed from identical patterns in different colors, have an impact on their classification in the International Union of Crystallography (IUC) notation. The chapter concludes with the description of patterns from Bedouin motifs for the purpose of symmetrical analysis.

6.1. A BRIEF ACCOUNT OF THE SYMMETRIES OF FRIEZE AND WALLPAPER GROUPS

Frieze Patterns and Groups An infinite strip with a repeating pattern is called a frieze pattern. In architecture, frieze is an architectural ornament consisting of a horizontal sculptured band between the architrave and the cornice. In cloth, fabric, textile, and material, frieze is an artifact made by weaving, felting, knitting, or crocheting natural or synthetic

69 Chapter 6 fibers. A frieze pattern is a of the plane or a linear pattern that repeats in one direction. All frieze patterns have translational symmetry. To clarify, the motions that leave the object appearing unchanged are called of the Euclidean plane or rigid motions. Isometries are kinds of geometric transformations that preserve distances, and there are only four types of them: , reflection, rotation, and . Translation is a popular in artifacts, that is, a motion where the object is translated to a certain distance in a particular direction, while length is the distance between the repeats of the pattern. In artifacts, there is usually a translational symmetry. Plane symmetry involves moving all points around the plane so that their relative positions to each other remain the same, although their absolute positions may change. Symmetries preserve distances, angles, sizes, and shapes. Geometrical symmetry means symmetry under a sub-group of isometries in two or three-dimensional Euclidean space. Each type of isometry defines a corresponding symmetry sub-group. A frieze pattern is a figure with one direction of translation symmetry, and a frieze group is a contains several frieze patterns. The possibilities of symmetries, other than the one direction of translation symmetry, give rise to different types of symmetries. Frieze patterns can be classified according to the types of symmetries that are accepted as mathematically valid. These are comprised of seven symmetry groups (see Table 6.1), where the essential elements are symmetry and repetition. A frieze group is a mathematical concept used to classify designs, which are characterized by repetition in one direction, based on the symmetries in the pattern. We shall use the names of frieze groups and the notation for the symmetry group that were standardized and adopted by the IUC (International Union of Crystallography) in 1952. The symmetry groups of frieze patterns are named in the four-symbol format of the IUC notation: p, the first constant symbol, and three additional symbols, letters m or a, and numbers 1 or 2. The different quartet combinations indicate the kinds of motions. Washburn and Crowe (1988) were the first to present the flow chart for the seven one-color symmetry groups of Frieze patterns. Following is the Washburn and Crowe flow chart as it was shown in paragraph 4.2, titled “Flow Chart for one-dimensional Patterns,” that appears in their book in a table 4.1, page 83 (see Figure 6.1). Use of this flow chart helps to classify the artifacts according to seven one- dimensional frieze groups of patterns, when every frieze pattern must be identical to one of them. In this book, we use this flow chart as follows.

Wallpaper Patterns and Groups The world of wallpaper, also called wall coverings, more generally, transforms walls of rooms, baths, and any other frame into a stylish art. Wallpapers use the math technique of the pattern repeat for creation of any composition based on floral, geometric, and other designs. In Mathematics, a wallpaper pattern is any subset of the plane whose translational symmetry group is repetitive in two independent

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