The symmetry existing in nature and its reflection in human artifacts has been present, from the earliest times, in all that has been done by man. Visual structures, that are the common elements of geometry and painting, were often arranged according to the laws of symmetry. However, the relatively independent development of geometry and painting resulted in the formation of two different languages. Even when talking about the same object (such as the laws of symmetry in the visual organization of a painting, that are most explicitly expressed within ornamental art) these languages use quite different terms. In fine art, the expression "symmetry" preserved for centuries the meaning it had in Greek aesthetics: in its wider sense it indicated harmony, accord, regularity, while in the more narrow sense it was identified with mirror symmetry in a vertical reflection line. The descriptive language used in most discussions on ornaments, drifted apart from the exact language of geometry.
With the development of natural sciences (Crystallography, Chemistry, Physics,¼) symmetry structures have become an important area of geometric studies; here the key words are transformation groups, invariance, isometry,¼ When painting and sculpture were differentiated from the decorative arts, began the period of a relative stagnation of ornamental art, which acquired a rather subordinate role and remained on the margins of the dominant aesthetics. On the other hand, the dynamic progress of the mathematical theory of symmetry caused the fact that the first more significant incitement for the study of ornamental art came from mathematicians (A. Speiser, 1927).
The approach to the classification and analysis of ornaments based on symmetries was enriched by the contributions of different authors (E. Müller, 1944; A.O. Shepard, 1948; J. Garrido, 1952; N.V. Belov, 1956a; L. Fejes Tòth, 1964; D.W. Crowe, 1971, 1975; D. Washburn, 1977;¼). In these works the descriptive language was replaced by more precise geometric-crystallographic terminology, and the theory of symmetry was established as a powerful tool for the study of ornamental art. In time, symmetry analysis of ornamental art became a reliable method, used mainly to study ancient ornamental art or that of primitive peoples.
The new approach to ornamental art leads to many new questions: which aspects of symmetry, when and where, appear in the history of ornamental art; which are the dominant forms; how to classify colored ornaments; etc. Following attempts to provide answers to all these and similar questions of the "how", "where" and "when" type, the question "why" arises naturally: why is man creating ornaments at all, why do some ornaments appear earlier or more often,¼ The questions in the first group do not penetrate the field of aesthetics and the psychology of visual perception, and so the language of geometry is almost sufficient for their discussion. In contrast, the second group of questions points to the necessity for a more profound understanding of the links between visuality and symmetry and also to the necessity to compare the language of the theory of symmetry with that of the theory of visual perception. Therefore, besides the question about the classification of ornamental motifs, the chronology of ornamental art, problems of colored ornaments etc., one of the aims of this work is to study the possibilities of translating geometric properties into the language of visuality, and vice versa. When discussing the numerous problems that ornamental art raises, special attention is paid to its roots. They are to be found in the ornamental art of the prehistoric period, which represents the most complete record of the beginnings of human understanding of regularity. In turn, regularity is the underlying basis of all scientific knowledge, so that in contemporary science, visualization of symmetry structures often represents the simplest way of their modeling and interpreting. This is an additional stimulus to strengthen the ties between science and art.
Knowledge of the terminology of the theory of symmetry is necessary for its application to the study of ornamental art. This is the reason for giving the basic technical terms and methods in the Introduction. The remaining mathematical terms with which the amateur reader is less familiar, are given later, in the relevant chapters.