This work represents an attempt at a comparative analysis of the theory of
discrete and visually presentable continuous symmetry groups in the plane *E*^{2} or in
*E*^{2}\{*O*}: Symmetry Groups of Rosettes, Friezes and
Ornaments (Chapter 2), Similarity Symmetry Groups in *E*^{2}
(Chapter 3), Conformal Symmetry Groups in *E*^{2}\{*O*}
(Chapter 4) and ornamental motifs found in ornamental art
that satisfy the afore mentioned forms of symmetry.

In each chapter symmetric forms are treated from the theory of groups point of view: generators, abstract definitions, structures, Cayley diagrams, data on enantiomorphism, form of the fundamental region... The analysis of the origin of corresponding symmetry structures in ornamental art: chronology of ornaments, construction problems, visual characteristics, and their relation to geometric-algebraic properties of the discussed symmetry is given. The discussions are followed by illustrations, such as Cayley diagrams and ornaments. Many of ornaments date from prehistoric or ancient cultures. In choosing their samples, chronology was respected as much as possible. Therefore, most of the examples date from the earliest periods - Paleolithic, Neolithic and the period of ancient civilizations. The problems caused by various datings of certain archaeological excavation sites have been solved by compromise, by quoting the different dates. The problem of symbols used in literature for denoting the symmetry groups has been solved in the same way.

The extension of the theory of symmetry to antisymmetry and colored symmetry was made only to facilitate a more detailed analysis of the symmetry groups by the desymmetrization method.

The surprisingly early appearance of certain symmetry structures in ornamental art of the Paleolithic and Neolithic led to attempts to interpret the causes of this phenomenon. Among the explanations we can note the existence of models in nature, and constructional possibilities. As the universal criterion, the principle of visual entropy was applied - maximal visual and constructional simplicity and maximal symmetry.

Somewhat different in concept is the chapter on Conformal Symmetry in
*E*^{2}\{*O*}. As opposed to the other chapters, where the
chronological priority of ornaments äs the oldest aspect of higher
mathematics given implicitly" (H. Weyl, 1952)
was stressed, in this chapter the emphasis is on
the path leading from the theory of symmetry (i.e., the derivation,
classification and analysis of conformal symmetry groups) toward ornaments
understood as the visual interpretations of abstract geometric-algebraic
structures. Such an approach is becoming increasingly more important, since it
makes possible the use of visually presented symmetry groups in all fields of
science where there is a need for the visualization of symmetry structures
(Crystallography, Solid State Physics, Chemistry, Quantum Physics, Particle
Physics,...). Also, by applying a comparative, multidisciplinary analysis
and by establishing the existence of parallelism between the theory of symmetry
and ornamental art, the research field of ornamental design can be enlarged. By
connecting the theory of symmetry and the theory of visual perception, more
precise aesthetic criteria in fine arts may be created. The possibilities to
apply these criteria when analyzing works of art (painting, sculptures,...)
could form the subject of a new study.

The closing chapter, The Theory of Symmetry and Ornamental Art, is an attempt at a survey synthesizing the relationship between the theory of symmetry and ornamental art, and a summary of the conclusions derived from individual chapters. Written as a compendium, this chapter could be considered as an independent entity.

The bibliography has been divided into two parts: one represents work in the field of the theory of symmetry and disciplines related to it, and the other work related to ornamental art.

I am especially thankful to Dr Dragomir Lopandic, Professor at the Faculty of Natural Sciences and Mathematics in Belgrade, under whose inspirational guidance this study came to life and to all the others who have helped give this study its final form.

A first version of this work was completed in 1981 and published by APXAIA (Belgrade, 1984). In the present version essential changes have been made in the discussion of the color- symmetry desymmetrizations, according to recent results in the field of colored symmetry. Certain definitions that were not sufficiently precise have been corrected and replaced by new ones. Important other contributions to the theory of symmetry, which have been made since 1981, have been also included, either in full, or through concise references to the original works. The symbols for the symmetry groups of friezes and antisymmetry have been simplified. The author is grateful to Professors H.S.M. Coxeter, B. Grünbaum, A.F. Palistrant, H. Stachel and W. Jank for their remarks, advice and suggestions, that were of immense value for the final version of the text.

Belgrade, 1989