Antisymmetry and Modularity in Ornamental Art

Abstract

BRIDGES Mathematical Connections in Art, Music, and Science A recapitulation of the development of anti symmetry theory is given in the introduction. Black-white ornamental motifs occurring in ornamental art are classified by using symmetry criteria, according to the corresponding anti symmetry groups. Antisymmetry groups of rosettes, friezes and ornaments are illustrated by examples from Neolithic and ancient ornamental art. The principle of modularity is used in order to explain possible methods of construction of certain ornaments, and to reconstruct some of them only from their parts preserved in archaeological material. 1. Antisymmetry Historical Remarks All kinds of art use geometry directly or indirectly. Even the most complicated painting composition may have a geometric structure in its basis. Sometimes, this structure is static, stable, based mostly on bilateral symmetry, but in many situations a rhythm of forms, lines and colors is an element or even the background of a whole structure. According to M. Ghyka [1], rhythm is observed and recorded periodicity. One of the simplest tools to suggest rhythm and contrast is the use of alternating black and white patterns. Periodical repetition is often used in decorative arts. A separated motif is static, but by repetition it imposes itself and creates a dynamic structure. Every periodical repetition suggests some kind of motion. We distinguish several types of periodic patterns: 1. Basic repeation (rotation, translation), suggesting oriented motion; 2. Alternation (glide reflection), producing stronger dynamic visual effect of double motion; 3. Inversion (convex-concave alternation); 4. Overlapping and interlacing (mostly alternating), which introduce a space component. Alternatively, a very strong dynamic component may be introduced by coloring. Using contrast, complementary colors, black-white, light-dark, over-under, above-below, positive-negative, convex-concave, the same object can be turned into its opposite, increasing the rhythm and dynamics. 56 Ljlljana Radovic and Slavik Jablan The idea of studying ornaments of different cultures from the point of view of the theory of symmetry originated with A. Speiser (1927) [2]. Among the early works in that field we can distinguish the analysis of Alhambra patterns given in the Ph.D. Thesis by E. Muller (1944) [3] and the papers of A Shepard (1948) [4]. The very influential monograph by H. Weyl [5] inspired the appearance of a whole series of works dedicated mostly to the ornamental art of ancient civilizations, to the cultures which contributed the most to the development of ornamental art (Egyptian, Arab, Moorish, etc.), and to ethnic ornamental art. Only in some recent works (e.g., by D.K. Washburn [6,7]), and S. Jablan [8]) has research turned to the very roots, the origins of ornamental art to the ornamental art of the Paleolithic and Neolithic, or to ethnic ornamental art. Thanks to the intensive development of the theory of symmetry and mathematical crystallography, the analysis of ornamental art completely followed this development. The more recent generalizations of the theory of symmetry, antisymmetry and colored symmetry are present in the graphic work of M.C. Escher [9, 10], as well as in several symmetry-related books and papers (e.g., in the books Symmetry in Science and by AV. Shubnikov & V.A Koptsik [11], Symmetries of Culture by D.K. Washburn & D.W. Crowe [12], etc.). Beginning with intuitively recognized regularities, and probably from very simple construction methods based on several basic (antisymmetric) prototiles (e.g., from Truchet tiles [13, 14] or similar elements), the historical development of antisymmetry ornaments proceeds by the use of modularity [15]. This is reason to believe that the theory of symmetry, literally taken from mathematical crystallography is probably not the only way, and maybe not the best explanation for the constructioll of ancient antisymmetric patterns. We believe that their basic concept and construction method was mostly derived from such usual working technologies, as matting, weaving, printing, and production of textiles or fabrics, rather then from the regular multiplication of a fundamental region by some (anti)symmetry group. Antisymmetry introduced in ornamental art the possibility of expressing, in a symbolical sense, a dynamic conflict, duality, and illustrated alternating natural phenomena (day-night, tides, phases of the Moon, a change of seasons). Treating the color change as a space property, a suggestion of two-sidedness (over-under, above-below) antisymmetry introduces also a 3D space component in ornamental art. This way, in both cases, as a kind of time component or space component, it introduces a new dimension, making possible a dimensional transition from a 2D plane image to 3-dimensionality. If we identify the color reversing transformation with reflection in the plane of the pattern, 2D anti symmetry groups of rosettes, friezes and plane ornaments are models of 3D symmetry groups of tablets, bands and layers, respectively. Exactly that idea was the origin of the mathematical theory of antisymmetry. Such visualization in a 2D plane, using black-white diagrams, was proposed in 1927 by A Speiser, and presented by L.Weber in 1929 [16]. The black-white diagrams of bands from his paper (Fig. 1), where the alternation of colors is used to denote figures above and below the invariant plane of the pattern, suggested the possibility for a more general dimensional transition from the symmetry groups of n-dimensional space, using the antisymmetry groups, to the symmetry groups of (n+ 1 )-dimensional space. That natural idea of a more sophisticated dimensional transition from 3D to 4D space resulted in one of the first and the most remarkable early results of H. Heesch [17] the approximate number of fourdimensional groups preserving invariant 3D-space (less then 2000). The 1651 3D-space antisymmetry groups, modeling the mentioned four-dimensional groups, were derived for the first time more then 30 years later by A.M. Zamorzaev in 1953 [18]. Unfortunately, the work of H. Heesh published in a crystallographic journal, as well as the paper of H.J. Woods [19] giving the derivation of the 46 blackwhite symmetry groups of plane patterns, published in the Manchester Journal o/the Textile Institute in 1935, never attracted the attention of readers they deserved. ~~~ ~~~~ -4<1-4<1 Antisymmetry and Modularity in Ornamental Art 57

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