Chapter 5


The extensiveness and universality of the theory of symmetry can be noted, though only partially, by considering those scientific fields in which it plays a significant role: Mathematics, Physics (especially Solid State Physics, Particle Physics, Quantum Physics), Crystallography, Chemistry, Biology, Aesthetics, Philosophy, etc. Owing to its universality and synthesizing role in the whole scientific system, certain modern-day authors give to the theory of symmetry the status of a philosophic category expressing the fundamental laws of organization in nature. According to that, there is the attitude of A.V. Shubnikov, who defined symmetry as "the law of construction of structural objects" (A.V. Shubnikov, V.A. Koptsik, 1974). The symmetry of natural laws, material and intellectual human creations represents a form of symmetry in nature.

An important, although apparently restricted area of the theory of symmetry is the field of the theory of symmetry and ornamental art. Throughout history, there have existed permanent links between geometry and painting, so that visual representations were often the basis for geometric investigations. This especially refers to ornamental art, termed by H. Weyl (1952) "the oldest aspect of higher mathematics expressed in an implicit form". More recently, especially for the needs of non-Euclidean geometry, visualizations of mathematical structures became their most usual model. The visual modeling of structures belonging to the field of natural sciences (Physics, Crystallography, Biology, Chemistry, etc.) by means of different visual representations - diagrams, graphics, graphic symbols of symmetry elements, Cayley diagrams, graphs, etc. - brought into being a complete visual language for the expression and representation of symmetry structures. The long lasting interaction between geometry and painting, especially present in the periods of scientific and art synthesis (e.g., in Egyptian, Greece, Renaissance science and art) is reflected in the simultaneity of the most significant epochs, tendencies and ideas in geometry and painting. In the modern period, this connection is expressed as a prolific exchange of experience between mathematicians, scientists and artists, especially in the period of the formation and domination of the geometric abstraction (e.g., in work of P. Mondrian, K. Malevich, V.  Vasarely). There are also examples of direct cooperation between artists and scientists (e.g., the long standing contacts between H.S.M. Coxeter and M.C. Escher, and the retrospective exhibition of M.C. Escher during the International Congress of the World Crystallographic Union in Cambridge, 1960) and their joint projects (M.C. Escher, 1971a, b, 1986; C.H. Macgillavry, 1976). Progress in the field of visual communications (press, photography, film, TV) and its further development open new possibilities for the visual representation of different symmetry structures - the subjects of scientific studies - thus making new scientific knowledge more accessible and expressible in a more comprehensible form.

The very roots of the theory of symmetry (in Greece) are inseparably linked to the establishment of the aesthetic principles - the canons and theory of proportions. The links between the theory of symmetry and aesthetics developed and were strengthened throughout history, where works of ornamental art represented the common ground between the theory of symmetry and painting. A new motive for the analysis and revision of aesthetic criteria was the appearance of abstract painting, especially involving geometric abstraction. Since figurative painting works are usually based on reality - on models found in nature - in the field of the aesthetics of painting there existed criteria not connected to visuality: e.g., contents of myths, degree of realism, etc. Non-figurative painting, abstract and ornamental, pointed out the inconsistency and incompleteness of aesthetics founded on the classical basis and the necessity to construct new visual-aesthetic criteria, formed according to the theory of symmetry (asymmetry, dissymmetry, antisymmetry, colored symmetry, curved symmetry, etc.).

The fundamental role of symmetry in the art is not exhausted by its connection with ornament or geometric abstraction. Art historians often used symmetry to characterize the formal qualities of a work of art, distinguishing symmetry as a basic principle of all artistic rules - the canons, laws of composition, criteria of well-balanced form... As the most significant property of harmony and regularity, symmetry is one of the main organizational principles in every art: painting, sculpture, architecture, music, dance, poetry... Even in the most extreme modern art - conceptualism or minimalism, it lays in their intellectual background.

This work is restricted to the area of ornamental art. This is so because of the fact that examples of visual interpretations of symmetry groups occur most frequently in their explicit form in this very area. They occur, also, in other fields of painting, but mostly in an implicit form. The possibilities of the theory of symmetry and ornamental art combined are, due to the extensiveness of their application (covering science, aesthetics, visual communications, etc.) far greater than those of ornamental art itself.

In this work, discrete symmetry groups formed by isometries and similarity symmetries in the plane E2 and also by conformal symmetries in the plane E2\{O}, are discussed. The condition of discreteness directly causes the existence of a bounded or unbounded fundamental region of a discrete symmetry group.

In the plane E2 and E2\{O}, the following categories of the symmetry groups of isometries were discussed: the symmetry groups of rosettes G20, friezes G21, ornaments G2 , similarity symmetry groups S20 and conformal symmetry groups C21 and C2. The maintenance of the metric properties of figures, holding for symmetry groups of isometries, for similarity symmetry groups is replaced by the similarity condition, so that the properties of equiformity and equiangularity remain preserved. In conformal symmetry groups, only the property of equiangularity holds. Similarity symmetry groups S20 and conformal symmetry groups C21 and C2 are characterized, respectively, by their isomorphism to symmetry groups of polar rods G31, tablets G320 and non-polar rods G31, making possible an adequate visual interpretation of these symmetry groups of the space E3 in the plane E2 (E2\{O}). Further extensions of the theory of symmetry leading toward elliptic and hyperbolic groups of symmetry and their interpretations in the inversive, conformal plane can be achieved without eliminating the condition of equiangularity, preserved by all the conformal symmetry transformations. A final consequences of such extensions are homology and curved symmetry groups belonging to a family of discrete affine and topological groups. That approach is in accord with the concept expressed in the Erlangen program by F. Klein (1872), who proposed a derivation of sequences of symmetry group extensions and their generalizations.

From the standpoint of the theory of symmetry and ornamental art, it is also considerable extension of the classical theory of symmetry to antisymmetry and colored symmetry. They are not the main topic of this work, so we have discussed only those cases where antisymmetry or colored symmetry is used for deriving classical-symmetry groups by desymmetrizations. Antisymmetry and colored symmetry have been discussed in detail by A.V. Shubnikov, N.V. Belov et al. (1964), A.L. Loeb (1971), A.V. Shubnikov, V.A. Koptsik (1974), A.M. Zamorzaev (1976), B. Grünbaum, G.C. Shephard (1977b, 1983, 1987), A.M. Zamorzaev, E.I. Galyarski, A.F. Palistrant (1978), M. Senechal (1979), J.D. Jarratt, R.L.E. Schwarzenberger (1980), T.W. Wieting (1982), R.L.E. Schwarzenberger (1984), S.V. Jablan (1984a, b, c, 1985, 1986a, b), H.S.M. Coxeter (1985, 1987), A.M. Zamorzaev, Yu.S. Karpova, A.P. Lungu, A.F. Palistrant (1986), etc.

When analyzing the visual characteristics of ornaments, the most fundamental appears to be the principle of visual entropy - maximal constructional and visual simplicity and maximal symmetry. From the examples and the accompanying arguments given in this work we can conclude that the appearance and frequency of occurrence of different symmetry groups in ornamental art are conditioned by the degree of agreement of corresponding ornaments with this principle. The principle of visual entropy is an affirmation of the universal natural principle of economy - its affirmation in the field of visuality. The individual components of this principle - the criterion of maximal visual and constructional simplicity and maximal symmetry and their role have been analyzed regarding all the categories of plane symmetry structures discussed. The maximal degree of symmetry is inseparably connected to constructional and visual simplicity. This is proved in ornamental art by the chronological priority and domination of examples of visually presentable continuous symmetry groups and maximal symmetry groups generated by reflections or their equivalents, containing all the other symmetry groups of the same category as subgroups. The principle of maximal symmetry represents the basis of the desymmetrization method, thus making possible the derivation of symmetry groups of a lower degree of symmetry by a desymmetrization of groups of a higher degree of symmetry, examples of which appear in ornamental art much earlier and occur more often. This can be achieved by classical-symmetry, antisymmetry and color-symmetry desymmetrizations. A consequent application of the desymmetrization method makes possible the realization of all symmetry groups as subgroups of the maximal symmetry groups of certain categories, where these subgroups can be derived directly from them or by sequences of successive desymmetrizations. In such a complementary approach to the theory of symmetry, asymmetry could be understood as a positive-defined property, in the sense of desymmetrization transformations, and not only as a negative-defined property - the absence of symmetry (I.D. Akopyan, 1980).

The principle of visual entropy is in accord with the standpoint of Gestalt psychology in considering the theory of visual perception (R. Arnheim, 1965, 1969). This point of view underlines the primary importance of the perception of a whole (Gestalt) and its essential structural organization laws, among which symmetry occupies an important position. That means the priority of the synthetic part of the visual perception. The position of Gestalt psychology has been greatly strengthened by recent research work on the physiological and psychological basis of visual perception. One of the arguments proving this is, for example, the study of chronology and frequency of occurrence of certain aspects of symmetry existing in ornamental art, outlined in this book.

By the term "visual simplicity", the stationariness or dynamism of symmetry structures is considered, where the dynamism of a visual object in many cases coincides with the complexity of its form and structure, resulting in difficulties when registering its symmetry. The enantiomorphism is the possibility of existence of the "left" or "right" form of discussed symmetry structure - their enantiomorphic modifications. The absence of enantiomorphism is the result of the existence of at least one indirect symmetry transformation within the symmetry group discussed. Somewhat more restricted, the term "polarity" introduces questions of polarity, non-polarity and bipolarity of symmetry elements and their corresponding invariant lines. The non-polarity of a certain symmetry element is conditioned by the existence of adequate reflections (inversions) commuting with it, while bipolarity is caused by the presence of adequate central reflections or their equivalents.

The polarity of generators of a symmetry group, especially of generators of the infinite order, introduces into ornamental art the time component - a suggestion of movement - that can be seen in polar rosettes, friezes and ornaments, or similarity symmetry rosettes producing the visual impression of centrifugal expansion. The polarity considerably affects the degree of visual dynamism. A dynamic visual effect produced by polar generators can be stressed or lessened by the right choice of the relevant visual parameters (e.g., by using acute-angular forms oriented toward the orientation of a polar element of symmetry, by changing the form of a fundamental region or the form of an elementary asymmetric figure belonging to it, or by choosing the coefficient of dilatation at the similarity symmetry groups S20, etc.).

The chronology and frequency of occurrence of certain visual examples of symmetry groups are caused also by the principle of maximal constructional simplicity. Because visually presentable continuous symmetry groups and maximal discrete symmetry groups generated by reflections or their equivalents are, also, the simplest ones in a constructional and in a visual sense, constructional simplicity is directly connected to the degree of symmetry.

In the oldest phases of ornamental art, after the intuitive and empirical perception of construction problems, the solutions to these problems and the forming of adequate construction methods came. According to the criterion of maximal constructional simplicity, direct, non-metric construction methods prevail in ornamental art in all the cases where their application is possible.

The problem of the exact construction of rosettes with symmetry groups of the category G20 is reduced to the question of the construction of regular polygons, which is possible for the polygons with 2mp1p2pn sides, where p1, p2,..., pn are the prime Fermat numbers and n N, m N{0}, while in other cases only approximate constructions are possible. Although approximate constructions are often used in ornamental art, in the history of rosettal ornaments there is an apparent domination of rosettes with rotations of the order n = 1,2,3,4,6,8,12, while, for instance, rosettes with rotations of the order n = 7,9, are extremely rare. A more detailed analysis of the causes of this, (e.g., the existence of natural models) is given in Chapter 2.

Symmetry groups of friezes G21 possess a high degree of constructional simplicity. Among other reasons, this caused the appearance of all seven symmetry groups of friezes in Paleolithic ornamental art. Usually, friezes are constructed by the method of rosette multiplication, where discrete friezes are derived multiplying by means of a discrete translation or glide reflection a certain rosette with the symmetry group C1 (1), C2 (2), D1 ( m), D2 (2m) - that means, by an extension from the "local symmetry" of the symmetry groups of finite friezes G210 to the "global symmetry" of the symmetry groups of friezes G21.

Apart from their independent use, friezes are used for constructing ornaments with symmetry groups of the category G2. In Paleolithic ornamental art, there occurred a multiplication of friezes by some other isometry - by a translation, reflection, half-turn or, maybe, by a glide reflection. The results obtained were superpositions of friezes - plane ornaments with the absence of a rotation of an order greater than 2. From the Paleolithic date the five plane Bravais lattices corresponding to the maximal symmetry groups of relevant syngonies, which are, at the same time, the basis for applying the method of rosette multiplication for constructing plane ornaments - the multiplication of a rosette with the symmetry group Cn (n), Dn (nm) (n = 1,2,3,4,6) by the discrete translational symmetry group of ornaments p1. This method for constructing ornaments, which demands a knowledge of plane Bravais lattices, was much more used in Neolithic ornamental art. Interesting empirical results obtained by its use can be traced in Moorish ornamental art, where experiments with rotations of the order 5 were made, representing an empirical analysis of the principle of crystallographic restriction (W. Barlow, 1894).

Constructions of similarity symmetry rosettes are based on the multiplication of a rosette with the symmetry group Cn (n), Dn (nm) by a similarity symmetry transformation K, L or M, with a common invariant point. Constructional difficulties occurring with the non-metric method of construction, artists try to solve by using also the metric method, but this often results in deviations from the similarity symmetry, caused by the inconsequent application of the metric construction method.

Since the non-metric construction method fully satisfies the criterion of maximal constructional simplicity, it prevails when constructing conformal symmetry rosettes with the symmetry groups of the category C21 or C2. By applying this method, visual examples of finite C21 and infinite C2 groups of conformal symmetry in E2\{O} can be derived by multiplying some rosette with the symmetry group Cn (n), Dn (nm), by means of conformal symmetry transformations RI, ZI, SI and similarity symmetry transformations K, L, M. When doing this, aiming to simplify the construction, we can use the correspondence between circles and lines as homologous figures of conformal symmetry transformations, the invariance of all the points of reflection lines and inversion circles, and the invariance of circles perpendicular to inversion circles.

The desymmetrization method is a universal, dependent construction method, consisting of classical-symmetry, antisymmetry and color-symmetry desymmetrizations, making it possible to obtain visual examples of all the subgroups of a given symmetry group. It is applied mostly on continuous visually presentable symmetry groups or on discrete groups of a high degree of symmetry - the symmetry groups of rosettes Dn (nm) , friezes mm, ornaments pmm, p4m, p6m, similarity symmetry groups DnK (nmK) and finite and infinite conformal symmetry groups DnRI and KDnRI. By antisymmetry desymmetrizations we can derive all the subgroups of the index 2 of a given group. Therefore, they can be used to find all its subgroups of the index 2. As for the period of origin, antisymmetry and color-symmetry desymmetrizations are somewhat younger than classic-symmetry desymmetrizations, and they first appear in the Neolithic, with the beginning of dichromatic and polychromatic ceramics. The consequent use of the desymmetrization method requires knowledge of the tables of group-subgroup relations.

Owing to their high degree of symmetry, and maximal visual and constructional simplicity, continuous symmetry groups belong to the oldest and most frequent symmetry groups used in art. Without using textures, only continuous symmetry groups with non-polar continuous elements of symmetry are visually presentable. Because of that, their application in ornamental art is strongly restricted. The first examples of visually presentable continuous rosettes, friezes, semicontinua and continua date from the Paleolithic and belong to the family of elementary geometric figures - circles, lines, parallels, spirals - representing the oldest expression of human geometric perception and knowledge. The possibilities for the visual representation of continuous symmetry groups can be extended with textures. Continuous visually presentable symmetry groups fully satisfy the criterion of visual entropy, so besides having a special role as independent symmetry groups, they serve as the most important source for deriving symmetry groups of a lower degree of symmetry by a desymmetrization. The link between the approaches existing in ornamental art and in the theory of symmetry is apparent regarding their similar origins and the use of similar construction methods.

In studying symmetry structures, besides objective elements conditioning a visual impression, subjective factors - e.g., physical, physiological, or psychological - also play their part. Regarding a symmetry structure itself, they can represent symmetrization or desymmetrization factors. The most important of them are - the influence of human plane symmetry and binocularity, the position of the symmetry structure regarding the fundamental natural directions, the effect of orientation ("left" or "right"), the influence of the symmetry of surrounding structures, etc. The fact that realistic ornaments represent finite parts of "ideal" ornaments - their factor groups - can also have a great influence on the visual impression produced by a certain ornament. Therefore, aiming for a more thorough analysis from the visual point of view, realistic plane symmetry structures should be considered as the result of the interaction between all the objective and subjective factors mentioned.

Similar problems appear in attempting to perceive all the symmetry substructures of a given symmetry structure. In this case, elements of symmetry of the larger structure represent, with respect to substructures, the secondary visual symmetrization or desymmetrization factors. Visual perception of substructures and of their symmetry can be immensely simplified by using tables of group-subgroup relations.

The theory of symmetry is one of the most efficient means for studying the principles of balance and harmony in art. Since symmetry is one of the basic structure-organization laws in nature, the existence of natural models was one of the origins of ornaments and an inexhaustible source of ideas during all the history of ornamental art. In the field of rosettal ornaments, frequently used basic models were objects with the mirror symmetry D1 (m), shapes with the symmetry group D2 (2m) expressing the relation between a vertical and horizontal line, and rosettes with the symmetry groups Cn ( n), Dn (nm), n = 1,2,3,4,5,6 corresponding to the symmetry of certain plants, flowers and some other forms of life (e.g., a starfish, jellyfish). The basis of friezes are the models found in nature, the distribution of leaves on plants, the shape of waves, the periodic character of many natural phenomena (the turn of day and night, the phases of the Moon, the seasons of the year, etc.). Resulting from there we have the primary calendar role of friezes, witnessed by the names of many friezes preserved in the art of primitive peoples (R. Smeets, 1975).

As complex plane symmetry structures, besides imitating natural models - honeycombs or net structures - many ornaments are the result of a human longing to express regularity and to construct perfect visual forms - discrete regular plane tilings. The similarity symmetry rosettes and infinite conformal symmetry rosettes produce, in the visual sense, the impression of centrifugal expansion, giving an adequate visual interpretation of the basic natural tendency of living matter - growth, directly connected with spiral structures. The spiral itself is one of oldest archetypical dynamic symbols used in art, occurring in nature in different types of snails, flowers and plants.

This study of the origin and development of ornamental art has been based mainly on examples of the oldest ornaments from the Paleolithic, Neolithic, the art of the ancient civilizations, and the native art of primitive peoples. After an intuitive-empirical perception of symmetry by the use of natural models (in the Paleolithic) and by solving elementary construction problems, the first symbolic meanings of ornaments were formed. In time, the visual-symbolic language of ornaments became a specific form of visual communication. After having solved constructional problems, in ornamental art there, then came the phase of an empirical analysis of the visual form of ornaments: by bringing into accord the visual form and the symbolic meaning, by solving the problems of visual stationariness or dynamism and orientation, and by studying tessellations and perfect ornamental forms. These were the first attempts to achieve a desired visual effect through the choice of relevant visual parameters.

The methods for obtaining different ornaments with the same symmetry group, by a change of the shape of the fundamental region or of the shape of an elementary asymmetric figure belonging to it, can be traced in the ornamental art of Neolithic, the ancient civilizations and primitive peoples. In this way, a special contribution to ornamental art is represented by Islamic ornaments (D. Hill, O. Grabar, 1964; K. Critchlow, 1976) and Moorish ornaments, and recently by the work of M.C. Escher (1971a, b, 1986).

Regarding the form of the fundamental region, we can distinguish visually static groups in the strictest sense - groups generated by reflections and their equivalents (circle inversions) - not allowing any kind of change of the boundary of a fundamental region, from the other symmetry groups allowing a change of boundaries not belonging to the reflection lines or inversion circles. Since the form of the fundamental region is fixed in the symmetry groups of rosettes Dn (nm), friezes m1, mm, ornaments pmm, p3m1, p4m, p6m and conformal symmetry groups DnRI, KDnRI generated by reflections (and inversions), the abundance and variety of such ornamental motifs can be exclusively achieved by changing the metric parameters - the dimensions and shape of an elementary asymmetric figure belonging to the fundamental region. At least one isohedral plane tiling corresponds to each of these groups, except to the symmetry group of ornaments p3m1 (D), whose corresponding regular tessellation {3,6} has the symmetry group p6m. This one and similar problems are an important part of the theory of plane tilings (B. Grünbaum, G.C. Shephard, 1987).

By solving the construction problems and by investigating the possibilities for obtaining different ornamental motifs, generations of artist and artisans have opened vast possibilities for decorativeness in ornamental art. In time, the symbolic meanings of ornaments were lost, and the role of ornaments was gradually reduced to pure decorativeness, almost without any symbolic meaning.

In ornamental art, by using an empirical approach, probably all the different possibilities for plane symmetry structures are exhausted. Symmetry groups present in ornamental art, their existence and uniqueness (completeness), have been recently scientifically verified by the theory of symmetry. So, we can divide all the discrete symmetry groups of the plane E2 and E2\{O} into the following categories: two types of symmetry groups of rosettes G20; seven symmetry groups of friezes G21; 17 symmetry groups of ornaments G2 ; five types of similarity symmetry groups S20; five types of finite C21 and ten types of infinite C2 conformal symmetry groups. Throughout history, the role of ornaments was mostly symbolic or decorative, but in our time, thanks to their link with science, especially with the natural sciences, ornaments have gained new meaning. Since they can be understood as models of structures that are the subject of scientific studies, ornaments today have outgrown the restricted area of ornamental art.

The history of ornamental art began in the period of the middle and late Paleolithic, around the tenth millennium B.C., when we have the first examples of discrete symmetry groups of rosettes G20 among which prevail visually static rosettes with the symmetry group Dn ( nm) (n = 1,2,3,4,6), examples of all seven discrete symmetry groups of friezes, all five plane Bravais lattices and examples of ornaments derived by elementary superpositions of friezes. From Paleolithic art we can date examples of almost all visually presentable continuous symmetry groups of the categories mentioned and the first intuitive premonitions of similarity symmetry - spirals, radial structures, series of concentric circles or squares and also the oldest examples of finite conformal symmetry groups of the type DnRI.

From the ornamental art of the Neolithic and the first ancient civilizations originate examples of all the 17 symmetry groups of ornaments G2 . After solving construction problems and creating ornaments that possessed a higher degree of constructional and visual complexity, they began the artistic experiments that opened the way to decorativeness, e.g., by changing the form of a fundamental region or the shape of an elementary asymmetric figure belonging to it. From the Neolithic date the oldest examples of antisymmetry and colored symmetry groups, being most completely realized in the ornamental art of Egypt. The ornamental art of Greece, Rome and Byzantium gave new results in similarity symmetry, antisymmetry, colored symmetry and conformal symmetry, while Gothic art almost completely exhausted the possibilities of conformal symmetry of rosettes, because dominating in this period were architectural constructions with rosettes constructed by circles and lines. Islamic and Moorish were the peaks of ornamental art. Renaissance and post-Renaissance ornamental art in Europe was almost completely reduced to decorativeness, which automatically reduced its status to that of a "second rate art". Lately, with geometric styles in painting, ornamental art gained a new affirmation with the work of V. Vasarely, M.C. Escher, by op-art (C. Barrett, 1970) and computer art (M.L. Prueitt, 1984). With the development of visual communications and accompanying visual design, ornamental art found a new place within the applied arts.

Interesting and not sufficiently investigated fields, referring to the chronology of ornamental art, appear to be the complete dating of the first appearance of all the symmetry groups in ornamental art, the evidence of the most important archaeological locations and civilizations that have achieved the most as for the completeness and variety of ornamental motifs, etc. Very important are comparative analyses aiming to find connections between civilizations, where ornaments can be the relevant indicators of these relations; either regarding repetition of details, elementary figures and the same forms of fundamental regions in ornamental art of different civilizations; or, regarding the use of the same symmetry groups. According to Gestalt theory, when visually perceiving an ornament the observer records and recognizes it as a whole, often abstracting details, but trying to understand and remember its law of organization - symmetry. Therefore, the use of the same symmetry groups by different civilizations can be a relevant indicator of their connections. Problems of the sense and symbolic meanings of ornaments, questions about relations between cosmological theories of different civilizations and their ornaments, etc., are only parts of this large field of investigation.

Even the elementary symmetry structures: rosettes Cn or Dn and the corresponding geometrical figures: square, circle, cross, have a symbolic meaning:

"Point: the primeval element, beginning, and kernel; symbol of the number. It is the symbol of the beginning (grain of seed) and of the end (grain of dust); it represents the smallest substance (atom, nucleus). The point is in fact imaginary: it occupies no space...

Vertical line: the sign of life, health, activity, certainty, effective stability, manliness. It is the symbol of spirit directed upward, of grandeur and loftiness, and of man running erect; it is the sign of right and might...

Horizontal line: the polar opposite of the vertical, and symbol of the earth, the passive, woman, death and rest; the material and the earthbound...

Cross: one of the oldest and most universal signs, uniting the polar contrasts of vertical and horizontal, of God and the world, of the spiritual and the material, of life and death, of man and woman. It indicates the four points of the compass and the point of intersection. After the Crucifixion it became a holy symbol in Christianity and was used in many variations...

Circle: together with the square and triangle, the primeval signs. Alike on all sides and the only geometric figure formed with one line with no beginning or end, it is a sign of infinity, eternity, perfection, and God. As a round form it is likewise a symbol of the Sun, the Cosmos, the Earth and the planets. As a pure form it is a sign of purity; as an embracing sign, a symbol of community...

Yang-yin: the symbol of perfect antithesis, ideal balance of opposites. Yin signifies womanly, dark, bound to the earth, cool, reticent, oppressed; yang manly, light, heavenly, aggressive, warm, governing. The white dot in the dark yin and the dark dot in the light yang signifies that each is always a part of the other...

Spiral: indicates that all life develops from one point and, still spinning from that one point, grows to adulthood. Also a symbol of the rising sun and the year. Much loved as an ornamental sign in many variations and combinations in all times and by all peoples...

Double spiral: the magnificent sign for perfection - in fact, a completed S-line. Symbol of the day between the rising and the setting sun and leap year (since a single spiral represents one year)...

Figure eight: the loop without end and therefore symbolic of endless, eternal time, which has no beginning or end...

Square: one of the three basic signs. Symbol of massiveness, sturdy peace, and stability: it stands fast and firm on the ground. It is the same on all the sides and the token for the number four, therefore, it symbolizes the four seasons, the four points of the compass, the four elements, the four rivers of Paradise, the four Evangelists...

Triangle: the third of the basic, primary signs. It is an aggressive sign with its points directed outwards. It is the symbol of the Trinity and, with a point in the middle, a sign of the all-seeing eye of God. A triangle standing with its base firmly planted on the ground and its point striving upwards has a womanly character, as opposed to a triangle balanced on its point with broad "shoulders" above, which has a more manly character...

Hexagram: two triangles passing through each other create a new sign, a beautiful symmetrical star. It is a magic sign of preservation and protection against destruction; also a very old Jewish sign, the Star of David, that crowns the synagogue and decorates the Torah rolls, as well as an emblem of the cosmos, the divine Creator, and His work...

Pentacle: another very old sign, known as the druid's foot for its magical meaning. Pointing upwards, it is a symbol of white magic; downwards, of black magic. The sign shows the five senses and indicates the powers and forms in nature. The lines intersect one another in golden-section proportions..." (R. Smeets, 1975, pp. 54-56).

Symbols move the deep, secret recesses of the human soul. They carry the mind over the borders of the finite into the realm of infinite: they are signs of unspeakable. Even only one of them - the spiral, may be the subject of the monograph (J. Purce, 1975). Some of them, e.g. Islamic patterns, are the artistic vision of cosmology (K. Critchlow, 1976), but many of them, e.g. the signs discovered by prehistoric archaeology, are still not decoded. Their symmetry and its accord with the message may be the keys for deciphering their symbolic meaning.

The word "symmetry" has its roots in Greek philosophy and aesthetics, where it was used to express balance, proportion, and to point out a whole spectrum of the philosophic-aesthetic synonym terms: harmony, accord, completeness, which were used in history. This term entered science in the 1830-ies, with the beginning of the study of crystal classes and their analysis based on the theory of groups, introduced by E. Galois (1831) in work published in 1848. The essence of the theory of symmetry, based on the theory of groups, is expressed in the Erlangen program of F. Klein (1872), distinguishing the theory of symmetry as a universal approach to different geometries by registering manifolds, their groups of transformations and invariants of these groups. Further development of the theory of symmetry cannot be separated from crystallography and the theory of groups. Of central interest to the problems covered by this work are the results of the theory of symmetry in the planes E2 and E2\{O}.

The answer to the question about the existence and completeness of the classification of symmetry groups of rosettes G20, H. Weyl (1952, pp.119) attributed to Leonardo. The derivation of the 32 crystal classes (J.F.Ch. Hessel, 1830) and 14 space Bravais lattices (O. Bravais, 1848) laid the basis for the complete derivation of the 230 G3 crystallographic space symmetry groups (E.S. Fedorov, 1891a; A. Schönflies, 1891), while W. Barlow (1894) proved the crystallographic restriction, showing that rotations of the symmetry group of a lattice can only have periods n = 1,2,3,4,6.

The derivation of 17 discrete symmetry groups of ornaments G2 , given incompletely by C. Jordan (1868/69), where the symmetry group pgg is omitted, and by L. Sohncke (1874), is completely realized as a partial result of the derivation of the 230 space groups G3 (E.S. Fedorov, 1891b). The 7 discrete symmetry groups of friezes G21 are derived by G. Pòlya (1924), P. Niggli (1924) and A. Speiser (1927). The first two of them derived independently also 17 symmetry groups of ornaments (G. Pòlya, 1924; P. Niggli, 1924).

Antisymmetry introduced by H. Heesch (1929), linked to the question of the visual interpretation of subperiodic symmetry groups of the space E3 - symmetry groups of bands G321 and layers G32 in the plane E2 by Weber black-white diagrams, was further developed by H.J. Woods (1935) and A.V. Shubnikov (1951). Recent development of the theory of antisymmetry, multiple antisymmetry, colored symmetry and its extensions has been seen in the contributions of many authors (e.g. A.V. Shubnikov, V.A. Koptsik, N.V. Belov, A.M. Zamorzaev, A.F. Palistrant, E.I. Galyarski, M. Senechal, A. Loeb, R.L.E. Schwarzenberger, T.W. Wieting, H.S.M. Coxeter, etc.).

The idea of similarity symmetry, put forward by H. Weyl (1952) was developed by A.V. Shubnikov (1960), E.I. Galyarski and A.M. Zamorzaev (1963). The discovery of the isomorphism between similarity symmetry groups S20, finite C21 and infinite C2 conformal symmetry groups and symmetry groups of polar rods G31, tablets G320 and non-polar rods G31, respectively, incited the development of the theory of similarity symmetry in E2 and conformal symmetry in E2\{O}.

Besides ornaments, as the most obvious visual models of symmetry groups in the plane E2 and E2\{O}, Cayley diagrams (A. Cayley, 1878; M. Dehn, 1910) and tables of graphic symbols of symmetry elements elaborated in crystallography, are used.

The generalized Niggli's categorization of symmetry groups, resulting in Bohm symbols of symmetry group categories (J. Bohm, K. Dornberger-Schiff, 1966) is consequently used in this work. Different systems for denoting symmetry groups, introduced by many authors (e.g., A. Schönflies, A.V. Shubnikov, M. Senechal, etc.) are unified for classical-symmetry groups by using a simplified version of the International symbols of symmetry groups of ornaments G2 (H.S.M. Coxeter, W.O.J. Moser, 1980) and by the symbols of symmetry groups of friezes G21 introduced by M. Senechal (1975). For denoting symmetry groups of rosettes G20, similarity symmetry groups S20 and conformal symmetry groups C21, C2, symbols derived according to those introduced by A. Schönflies and A.V. Shubnikov, are used.

Aiming for a more complete knowledge of the theory of symmetry, very inspiring might be the books by H. Weyl (1952), L. Fejes Tòth (1964), A.V. Shubnikov, N.V. Belov et al. (1964), A.V. Shubnikov, V.A. Koptsik (1974), A.M. Zamorzaev (1976), A.M. Zamorzaev, E.I. Galyarski, A.F. Palistrant (1978), E.H. Lockwood, R.H. Macmillan (1978), T.W. Wieting (1982), B. Grünbaum, G.C. Shephard (1987), A.M. Zamorzaev, Yu.S. Karpova, A.P. Lungu, A.F. Palistrant (1986) and the monograph "Generators and Relations for Discrete Groups" by H.S.M. Coxeter and W.O.J. Moser (1980).

The survey of ornamental art and the theory of symmetry given jointly in this book makes possible their comparison in the sense of common constructional approaches, methods and their final results - symmetry groups and their visual interpretations. In both fields, more sophisticated results are obtained by a similar approach - by extending symmetry groups of isometries in the plane E2 to similarity symmetry groups S20, conformal symmetry groups C21 and C2, antisymmetry and colored symmetry groups. Regarding the chronology, the history of ornamental art fully satisfies the inductive series of extensions leading from the symmetry groups of rosettes G20, over symmetry groups of friezes G21, ornaments G2 and similarity symmetry groups S20 in the plane E2, to conformal symmetry groups C21 and C2 in the plane E2\{O}. In the development of the theory of symmetry there are only a few exceptions from this sequence, mostly conditioned by the practical interests of crystallographers (e.g., the derivation of the symmetry groups of ornaments G2 before the symmetry groups of friezes G21, or the very early derivation of the 230 space symmetry groups G3). In this way, the connection between ornamental art and the theory of symmetry represents a component of the universal, eternal link between art and science.