Chapter 2.6

 Ornaments G₂ and
 Ornamental Art

A plane is called homogeneous iff all the plane points are translationaly equivalent, and isotropic iff all directions containing some fixed point of the plane are equivalent. The symmetry group of the homogeneous and isotropic plane E² is the maximal continuous symmetry group of ornaments p₀₀¥m (s ^¥¥). The continua with the symmetry group p₀₀mm (s^¥¥) can be understood as the result of the multiplication of a point or circle - a rosette with the continuous symmetry group D_¥ (¥m) - by means of the continuous symmetry group of translations p₀₀. A plane continuum with the symmetry group p₀₀¥m ( s^¥¥) represents the area where all the other plane symmetry groups exist. Apart from this continuous symmetry group of ornaments, also visually presentable are the symmetry groups of semicontinua p₁₀1m (s1m) and p₁₀mm (smm). They are derived, respectively, as the extensions of the visually presentable continuous symmetry groups of friezes m₀1 and m₀m (Figure 2.29) by a translation perpendicular to the frieze axis. In ornamental art, the symmetry groups of semicontinua usually are presented by adequate systems of parallel lines, constructed by such a procedure. The other continuous symmetry groups of ornaments may be visually interpreted by using textures (A.V. Shubnikov, N.V. Belov et al., 1964).

From the common point of view of geometry and ornamental art, the most interesting are the crystallographic discrete symmetry groups of ornaments. The idea to analyze the ornaments of different cultures by applying the theory of symmetry and by registering the symmetry groups of ornaments, developed by G. Pòlya (1924), has been abundantly used by A. Speiser (1927), E. Müller (1944), A.O. Shepard (1948), H. Weyl (1952), J. Garrido (1952), N.V. Belov (1956a), L. Fejes Tòth (1964), D.W. Crowe (1971, 1975), A.V. Shubnikov and V.A. Koptsik (1974), D.K. Washburn (1977), E. Makovicky and M. Makovicky (1977), B. Grünbaum (1984b), I. Hargittai and G. Lengyel (1985), D.W. Crowe and D.K. Washburn (1988) and by many other authors. The quoted works mainly analyze the appearance of 17 discrete symmetry groups of ornaments in the art of the ancient cultures (Egypt, China, etc.), in Moorish ornamental art or in ethnical art. How difficult it is to exhaust all the symmetry possibilities for plane ornaments and to discover all the symmetry groups of ornaments, is illustrated by the fact that many nations, even those with a rich ornamental tradition, in their early art do not have such examples (B. Grünbaum, 1984b; B. Grünbaum, Z. Grünbaum, G.C. Shephard, 1986).

In the mathematical theory of symmetry, the first complete list of the discrete symmetry groups of ornaments was given by E.S. Fedorov (1891b), although this problem was, even before that, the subject of study of many important mathematicians.

Therefore, the fact that examples of most of the discrete symmetry groups of ornaments, given in bone and stone engravings or drawings, date from the Paleolithic ornamental art is very surprising. In the Neolithic there came the further development of ornamental art, mainly related to the decoration of ceramics. Neolithic ornamental art is characterized by paraphrasing, variation, enrichment of already existent ornaments and by the discovery of those symmetry groups of ornaments which remained undiscovered.

Ornaments with the symmetry group p1 (Figure 2.58, 2.59) for the first time occur in Paleolithic art (Figure 2.58a, b). The origin of these ornaments, obtained by multiplying an asymmetric figure by means of a discrete group of translations, may be interpreted also as a translational repetition of a frieze with the symmetry group 11, already existent in Paleolithic ornamental art. Both of the axes of generating translations are polar. Since the symmetry group p1 does not contain indirect isometries, the enantiomorphism occurs. A fundamental region usually has an arbitrary parallelogramic form. Due to their low degree of symmetry, ornaments with the symmetry group p1 are relatively rare. Mostly, they occur with stylized asymmetric motifs inspired by asymmetric models in nature, rather than by using asymmetric geometric figures.

Ornaments with the symmetry group p2 (Figure 2.60-2.63) also date from the Paleolithic, occurring in their most elementary form - as a lattice of parallelograms (Figure 2.60b). Already in the late Paleolithic (the Magdalenian), there are different ornaments with the symmetry group p2. Some of them are very rich, as, for example, the meander motif from Ukraine, dating to the XI millennium B.C. Despite some deviations from the exact symmetry of ornaments, this record of Paleolithic ornamental art is of an unexpected scope (Figure 2.60a). In Neolithic ornamental art, besides the meander motifs, very popular were motifs based on the double spiral - rosette with the symmetry group C₂ (2) - occurring for the first time in the Paleolithic (Figure 2.60c). It is probable that, the double spiral can be found in ornamental art of all the Neolithic cultures, often with wave motifs. Layered patterns (B. Grünbaum, G.C. Shephard, 1987) with the symmetry group p2 (Figure 2.62a) originating from the Paleolithic, are frequent (Figure 2.60b). Since a central reflection is the element of the symmetry group p2, in the corresponding ornaments both the generating translation axes will be bipolar. Therefore, in a visual sense, such ornaments produce an impression of two-way motion. Enantiomorphic modifications exist. Since in nature the symmetry group C₂ (2) occurs relatively seldom, ornaments with the symmetry group p2 mostly are geometric ones. The simplest construction of ornaments with the symmetry group p2, probably used in Paleolithic ornamental art, is a multiplication of a frieze with the symmetry group 12 by a non-parallel translation. A fundamental region is often triangular and offers the use of curvilinear boundaries.

In the same way, a construction of ornaments with the symmetry groups pm, pg, pmg and pmm can be interpreted as a multiplication of the corresponding friezes by means of a translation perpendicular to the frieze axis. All the afore mentioned symmetry groups of friezes originated from Paleolithic ornamental art.

From the Paleolithic onward, ornaments with the symmetry group pm (Figure 2.64-2.67) are frequently used in ornamental art, as "geometric", "plant" and "animal" ornaments. By having one polar and one non-polar generating translation axis, they produce an impression of directed motion in the direction parallel to the reflection lines, which usually coincides with a vertical or horizontal line. In the same way as in friezes with the symmetry group 1m, the visual dynamism of these ornaments can be increased by choosing an adequate fundamental region or elementary asymmetric figure belonging to it. There are no enantiomorphic modifications. The form of the fundamental region may be arbitrary, but requires at least one rectilinear boundary.

In ornamental art, friezes with the symmetry group 1g occur relatively seldom, so the same holds for ornaments with the symmetry group pg (Figure 2.68, 2.69). The appearance of friezes with the symmetry group 1g in Paleolithic ornamental art (e.g., in stylized plant motifs) offers some evidence to believe that also ornaments with the symmetry group pg originate from the Paleolithic. In Neolithic ornamental art this symmetry group mostly occurs in geometric ornaments, while in the pre-dynastic period of Egypt and Mesopotamia it was frequently used with zoomorphic motifs. In ornaments with the symmetry group pg both generating translation axes are polar. Since the symmetry group pg contains indirect isometries - glide reflections - enantiomorphic modifications do not occur. In the visual sense, ornaments with the symmetry group pg produce a visual impression of one-way alternating motion.

Ornaments with the symmetry group pmg (Figure 2.70-2.73), and corresponding friezes with the symmetry group mg, originate from the Paleolithic. Most probably, the symmetry group pmg is one of the oldest symmetry groups of ornaments used in ornamental art. Their first and most frequent visual interpretations are as stylized motifs of waves. The symmetry group pmg is the most frequent symmetry group of ornaments in Paleolithic and Neolithic ornamental art throughout the world.

Regarding the frequency and variety of corresponding ornaments, only the symmetry group of ornaments p2 can be compared with it. It is probably impossible to find a culture and its prehistoric ornamental art without ornaments with the symmetry group pmg. Also, by having many various ornaments (Figure 2.71), it offers the possibility to analyze the connections between different Paleolithic and Neolithic cultures, distant both in space and time, by the similarity of the motifs they used. The symmetry group pmg contains reflections with the reflection lines perpendicular to one generating translation axis and parallel to the other, and central reflections, so that the first generating translation axis will be non-polar and the second bipolar. Enantiomorphic modifications do not exist. Owing to the bipolarity of the other generating translation axis and to glide reflections that produce the visual impression of two-way alternating motion, ornaments with the symmetry group pmg will be dynamic ones. On the other hand, as their static component, the reflections produce the impression of balance. A fundamental region is usually rectangular, and requires one rectilinear boundary that belongs to a reflection line. In Paleolithic and Neolithic ornamental art, ornaments with the symmetry group pmg mostly occur as geometric ornaments with stylized "water" and meander motifs. Later, the symmetry group pmg occurs in plant or even zoomorphic ornaments, where an initial figure - a rosette with the symmetry group D₁ (m) - is multiplied by means of a glide reflection, perpendicular to the reflection line of the rosette. Afterward, the constructed frieze with the symmetry group mg is repeated by a translation perpendicular to the frieze axis. Most frequently, ornaments with the symmetry group pmg are constructed by a direct translational repetition of the corresponding frieze with the symmetry group mg by means of a translation perpendicular to the frieze axis. A spectrum of the symbolic meanings of ornaments with the symmetry group pmg is defined by their geometric and visual properties. Therefore, such ornaments are suitable for presenting periodic alternating non-polar phenomena.

Static ornaments with the symmetry group pmm (Figure 2.74-2.76) date from Paleolithic art, occuring as cave wall paintings and bone or stone engravings. Originally, they were used in their simplest form - as a rectangular lattice - and later, as its various paraphrases realized by an elementary asymmetric figure belonging to the fundamental region. Reflections with the reflection lines perpendicular to the generating translation axes cause the non-polarity of both generating translation axes, and a high degree of stationariness and balance. The impression of stability can be stressed if fundamental natural directions - a vertical and horizontal line - coincide with the generating translation axes. In ornamental art, ornaments with the symmetry group pmm usually are in that position. Also, the existence of horizontal reflections restricts the area of suitable ornamental motifs to rosettes with a symmetry group having as a subgroup the symmetry group D₂ (2m). Among them, rosettes with geometric and plant (e.g., flower) motifs prevail. The use of zoomorphic motifs is restricted to a minimum. This is because the symmetry group D₂ (2m) occurs as a subgroup of the complete symmetry group only in sessile living organisms, while all non-sessile forms of life are characterized by their polarity - their orientation in space. The polarity of living things in the vertical direction and their upward orientation makes impossible the use of zoomorphic ornaments with the symmetry group pmm. The stated polarity contradicts the existence of horizontal mirror symmetry, so that in such an ornament half of the figures will be in an unnatural position. Since the symmetry group \bold p\bold m\bold m is generated by reflections (\bold p\bold m\bold m = {R₁,R₃} ×{R₂,R₄} = D_¥ ×D_¥), its fundamental region must be a rectangle.

The reasons for a relatively rare use of the symmetry group pgg in ornamental art (Figure 2.77-2.79) are connected with the difficulty in recognizing the regularities these ornaments are based on and their constructional complexity. As opposed to ornaments with symmetry groups possessing "evident" symmetry elements consisting of one elementary symmetry transformation, or to symmetry groups, derived as extensions of corresponding symmetry groups of friezes by means of a non-parallel translation, the symmetry group pgg is generated by two complex, composite symmetry transformations - perpendicular glide reflections. Just how difficult it is to perceive these "hidden symmetries", recognize them, construct the corresponding ornaments and discover the symmetry group pgg, is proved by the fact that it is the only symmetry group of ornaments omitted by C. Jordan (1868/69). Ornaments with the symmetry group pgg are usually realized by the multiplication of a frieze with the symmetry group 1g by a glide reflection perpendicular to the frieze axis. Since in Paleolithic and Neolithic ornamental art friezes with the symmetry group 1g occur seldom, this especially refers to ornaments with the symmetry group pgg. The oldest examples of these ornaments can be found in the Neolithic (Figure 2.77) and in the ornamental art of ancient civilizations (Figure 2.78). Besides layered patterns with the resulting symmetry group pgg, antisymmetry ornaments with the antisymmetry group pmg/ pgg (Figure 2.102l) are treated by the classical theory of symmetry as ornaments with the symmetry group pgg. Namely, the symmetry group pgg can be realized by a desymmetrization of the symmetry group pmg, where the reflections are replaced by glide reflections. In the classical theory of symmetry this can be achieved by a complex, and in a technical sense inconvenient way, by eliminating the mirror symmetry and by changing the shape of a fundamental region. By the antisymmetry desymmetrization - black-white coloring resulting in the antisymmetry group pmg/pgg - this may be realized more easily. A fundamental region of the symmetry group pgg usually is a rectangle, offering a change of the shape. In ornaments with the symmetry group pgg both generating translation axes are bipolar, producing the impression of maximal visual dynamism. On the other hand, the visual dynamism of these ornaments represents a drawback in the sense of perceiving symmetry elements, constructing corresponding ornaments and defining a fundamental region. Thanks to a high degree of dynamism and bipolarity of generating translation axes, ornaments with the symmetry group pgg are mainly used with geometric and plant ornamental motifs, as symbolic interpretations of double-alternating motions. As singular directions, the natural perpendiculars - vertical and horizontal lines - may be taken. So that, the recognition of symmetry elements may be made easier - this being difficult in a slanting position of glide reflection axis with respect to the observer.

Ornaments with the symmetry group cm or cmm (Figure 2.80-2.88) are based on a rhombic lattice, the lattice with equal sides, appearing in Paleolithic ornamental art on bone engravings. The origin of a rhombic lattice and its use may be interpreted as a manifestation of the principle of visual entropy - maximal symmetry and maximal simplicity in the visual and constructional sense. The replacement of the unequal sides of a lattice of parallelograms with equal sides results in the symmetrization of the lattice, changing its cell symmetry from the symmetry group C₂ (2) to the symmetry group D₂ (2m). Ornaments with the symmetry group cmm originated earlier, in their most elementary form as a rhombic lattice with a rectilinear fundamental region (Figure 2.85c). Ornaments with the symmetry group cm or cmm maybe originate, respectively, from ornaments with the symmetry group pm or pmm constructed by a translational repetition of a rosette with the symmetry group D₁ (m) or D₂ (2m) by two perpendicular translations, where "gaps" between the rosettes are filled with the same rosettes in the same position - by centering.

Ornaments with the symmetry group cm (Figure 2.80-2.84) possess one polar diagonal generating translation. They are convenient for suggesting directed motion. Owing to the indirect isometries - reflections - enantiomorphic modifications do not occur. A fundamental region is often triangular, with one rectilinear side. Apart from by the construction proposed above - by means of rosettes with the symmetry group D₁ (m) - such ornaments may be also obtained by a desymmetrization of the symmetry group cmm, where one reflection must be eliminated either by changing the shape of the fundamental region or by a coloring. In the classical theory of symmetry, antisymmetry ornaments with the antisymmetry group cmm/ cm are included in the class of ornaments with the symmetry group cm (Figure 2.102g). Ornaments with the symmetry group cm may also be constructed by multiplying a frieze with the symmetry group 1g by a reflection with the reflection line parallel to the frieze axis.

Ornaments with the symmetry group cmm (Figure 2.85-2.88) possess one non-polar diagonal and one bipolar generating translation axis. There are no enantiomorphic modifications. A fundamental region is often triangular, with two rectilinear sides belonging to reflection lines. Since in ornamental art, especially in the Paleolithic and Neolithic, friezes with the symmetry group mg are the most frequent, the origin of ornaments with the symmetry group cmm can be understood as a multiplication of a frieze with the symmetry group mg by a reflection with the reflection line parallel to the frieze axis. This is the reason that ornaments with the symmetry group cmm are used so much and are in such variety. In the classical theory of symmetry, antisymmetry ornaments with the antisymmetry group pmm/cmm (Figure 2.102b) are included in the class of ornaments with the symmetry group cmm, thus enriching it considerably.

The class of ornaments appearing much later and occurring in ornamental art seldom consists of ornaments with the symmetry groups p3, p3m and p31m, belonging to the hexagonal crystal system (Figure 2.89, 2.90). All previously discussed symmetry groups of ornaments can be realized in a simple way, as primary plane lattices (Bravais lattices) or by composing the symmetry groups of friezes with the simplest symmetry groups of rosettes D₁ (m) or C₂ (2), generated by a reflection or by a central reflection.

Figure 2.89

Ornamental motifs that suggest the symmetry of plane ornaments with three-fold rotations: (a) Hallaf, around 6000 B.C. (7600-6900 B.C.?); (b) Mohenjo Daro, around 4500-4000 B.C.; (c) Tripolian culture, around 4000-3500 B.C.; (d) the interlaced motif, Egypt, early dynastic period.

Figure 2.90

Ornaments with three-fold rotations: (a) p3; (b) p31m; (c) p3m1.

Ornaments with the symmetry group p3 may be constructed by multiplying a rosette with the symmetry group C₃ ( 3) by two non-parallel translations with translation vectors of the same intensity, constructing a 60^o-angle or by multiplying a frieze with the symmetry group 11 by a trigonal rotation. Since natural forms with the symmetry group C₃ (3) occur seldom, the absence of these natural models accounted for the absence of ornaments with the symmetry group p3 in the earliest periods of ornamental art. A few older examples of ornaments containing rosettes with the symmetry group C₃ (3) deviate from the exact rules of symmetry (Figure 2.89) and only suggest the p3-symmetry, which would be used consequently in the ornamental art of the ancient civilizations (of Egypt, China, etc.) (Figure 2.90). The usual shape of a fundamental region of the symmetry group p3 is a rhombus with a 60^o-angle, offering great possibilities for variations and for curvilinear boundaries. Ornaments with the symmetry group p3 possess enantiomorphic modifications. Since they contain two dynamic polar components - polar translation axes and trigonal polar oriented rotations, such ornaments are dynamic ones. Their visual dynamism can be stressed by using a suitable shape of the fundamental region or of an elementary asymmetric figure belonging to the fundamental region. Representing the result of a superposition of the symmetry groups 11 and C₃ (3), the symmetry group p3 possesses three singular polar directions, so that the corresponding ornaments produce an impression of a three-way directed motion. As a subgroup of the index 4 of the symmetry group p6, the symmetry group p3 may be derived by a desymmetrization of the symmetry group p6m, or even by a desymmetrization of the symmetry group p6. The symmetry group p6 is the subgroup of the index 2 of the symmetry group p6m, and the symmetry group p3 is the subgroup of the index 2 of the symmetry group p6. This offers many possibilities for classical-symmetry or antisymmetry desymmetrizations, in particular, for the antisymmetry desymmetrization resulting in the antisymmetry group p6/p3, treated by the classical theory of symmetry as the symmetry group p3 (Figure 2.90a, 2.102d).

Ornaments with the symmetry groups p31m and p3m1 differ among themselves by the position of reflection lines. In ornaments with the symmetry group p31m reflection lines are parallel to the generating translation axis, while in ornaments with the symmetry group p3m1 they are perpendicular to it. Regarding their origin, both groups can be understood as a result of superposition of the symmetry group of rosettes D₃ (3m) and the symmetry group of friezes 11, where in ornaments with the symmetry group p31m the frieze axis is parallel to the reflection line of the rosette, while in ornaments with the symmetry group p3m1 it is perpendicular to the reflection line. Also, we may consider the symmetry group p31m as the superposition of the symmetry group of rosettes C₃ (3) and the symmetry group of friezes 1m, and the symmetry group p3m1 as the superposition of the same symmetry group C₃ (3) and the symmetry group of friezes m1. From there result the following common properties of these two groups: the symmetry groups p31m and p3m1 are the subgroups of the index 2 of the symmetry group p6m, so they can be derived by desymmetrizations of the symmetry group p6m, in particular, by antisymmetry desymmetrizations. Owing to reflections with reflection lines incident with trigonal rotation centers, the symmetry group p3m1 is characterized by the non-polarity of all trigonal rotations. In ornaments with the symmetry group p31m half of the trigonal rotation centers are incident to reflection lines, and the others are not, so the symmetry group p31m contains the polar and non-polar trigonal rotations. In both symmetry groups, the indirect transformations - reflections - cause the absence of the enantiomorphism. In ornaments with the symmetry groups p31m and p3m1 glide reflections parallel with reflection lines appear as secondary elements of symmetry. The different position of reflection lines with respect to the generating translation axes causes certain relevant constructional and visual-symbolic differences between the corresponding ornaments.

Same as their generating friezes with the symmetry group 1m, ornaments with the symmetry group p31m, that contain a reflection parallel to the generating translation axis, have a polar, oriented generating translation axis. From this the visual dynamism of ornaments with the symmetry group p31m results. They produce the impression of three-way directed motion. A fundamental region of the symmetry group p31m is usually defined by a longer diagonal and by two sides of the rhombic fundamental region already mentioned in the case of the symmetry group p3, where these two sides can be replaced with adequate curvilinear contours. This makes possible different variations and the emphasis or alleviation of the dynamic visual impression produced by the corresponding ornaments. Despite the existing conditions for a variety of ornaments with the symmetry group p31m, in old ornamental art they occur relatively seldom, mainly because of their constructional complexity and the absence of natural models. Their full affirmation will come in the ornamental art of the developed ancient civilizations (Figure 2.90). Examples of the symmetry group p31m may be obtained by a desymmetrization of the symmetry group p6m in which the symmetry group p31m is the subgroup of the index 2. Owing to the constructional complexity of the corresponding antisymmetry mosaics, the antisymmetry group p6m/p31m, treated by the classical theory of symmetry as the symmetry group p31m , is poorly represented in ornamental art (A.V. Shubnikov, N.V. Belov et al., 1964, pp. 220).

Similar to the corresponding symmetry group of friezes m1, the symmetry group of ornaments p3m1 contains reflections with reflection lines perpendicular to the generating translation axis, that cause the non-polarity of the axis. Therefore, in a geometric and visual-symbolic sense, the symmetry group p3m1 can be considered as a static equivalent of the symmetry group p31m. The fundamental region of the symmetry group p3m1 is an equilateral triangle. The symmetry group p3m1 (D) is generated by reflections, that require a rectilinear triangular fundamental region and restrict the area of the corresponding classical-symmetry ornaments to the use of some elementary asymmetric figure belonging to a fundamental region, resulting in a marked isohedral tiling (B. Grünbaum, G.C. Shephard, 1987). No desymmetrization of a fundamental region of the symmetry group D₃ (3m) can be achieved by changing its shape. Therefore, an internal desymmetrization of the fundamental region becomes indispensable, making impossible a visual interpretation of the symmetry group p3m1 accompanied by an isohedral tiling, without a previous internal desymmetrization of the fundamental region. A fundamental region must be rectilinear in all the groups generated by reflections: pmm, p4m, p6m and p3m1. Except the symmetry group p3m1, this fact has no influence on the possibility to construct an isohedral tiling with a tile serving as the fundamental region of the symmetry group discussed. Only in the symmetry group p3m1, where this tiling is a regular tessellation {3,6} , there comes the symmetrization by composing six equilateral triangles with a common vertex and each with the symmetry group D₃ (3m). The result is the isohedral tiling with the symmetry group p6m. Therefore, in ornamental art, the symmetry group p3m1 occurs seldom, especially in the earlier period (Figure 2.90c). Distinct from classical-symmetry ornaments with the symmetry group p3m1, antisymmetry ornaments with the antisymmetry group p6m/p3m1, in the classical theory of symmetry treated as ornaments with the symmetry group p3m1, are some most frequent antisymmetry ornaments. The antisymmetry group p6m/p3m1 can be derived by the antisymmetry desymmetrization of the symmetry group p6m, where the adjacent equilateral triangles of a regular tessellation {3,6} are colored oppositely (Figure 2.102n). Their generating symmetry group p6m is one of the most frequent symmetry groups in the whole of ornamental art, while their constructional and visual simplicity caused the frequent occurrence of ornaments with the antisymmetry group p6m/p3m1.

The symmetry group p6 belongs to the hexagonal crystal system. The corresponding ornaments (Figure 2.91) may be constructed by multiplying a rosette with the symmetry group C₆ (6) by means of two non-parallel translations with translation vectors of the same intensity, constructing a 60^o-angle. The symmetry group p6 can be obtained by superposing the symmetry group of friezes 11 and the symmetry group of rosettes C₆ (6), or by composing the symmetry group of friezes 12 and the symmetry group of rosettes C₃ (3). The oldest examples of ornaments with the symmetry group p6 date from the ancient civilizations (Figure 2.91c,d). Owing to their constructional complexity and the lack of models in nature, in ornamental art they are rather rare. A fundamental region of the symmetry group p6 may by an equilateral triangle with sides that can be replaced by curved lines. This results in the variety of ornaments with the symmetry group p6. The enantiomorphism occurs. Since central reflections are elements of the symmetry group p6, the generating translation axes will be bipolar. Although their dynamic visual effect is alleviated by central reflections and by the bipolarity of the generating translation axes, ornaments with the symmetry group p6 belong to a family of dynamic ornaments. In them, it is possible to recognize three singular bipolar directions. The symmetry group p6 is the subgroup of the index 2 of the symmetry group p6m, so that examples of the symmetry group p6 can be constructed by a desymmetrization of the symmetry group p6m, this means, of regular tessellations {3,6} or {6,3} . Aiming to eliminate reflections and to maintain the symmetry p6, besides classical-symmetry desymmetrizations, antisymmetry desymmetrizations are used. As a result, antisymmetry ornaments with the antisymmetry group p6m/p6, considered by the classical theory of symmetry as ornaments with the symmetry group p6, may be obtained (Figure 2.102o).

The symmetry group p6m (Figure 2.92, 2.93) is the maximal symmetry group of the hexagonal crystal system. The oldest and most frequent ornaments with the symmetry group p6m are regular tessellations {3,6} and {6,3} . The tessellation {6,3} dates from the Paleolithic (Figure 2.92a), occurring on a bone engraving. This tessellation and its corresponding symmetry group p6m probably originated from natural models - honeycombs - the regularity and symmetry of which has always attracted the attention of artists and mathematicians. Its dual tessellation { 3,6} also dates from Paleolithic ornamental art (Figure 2.92b). Both ornamental motifs mentioned, either in their elementary form or with various paraphrases, occured in ornamental art of different civilizations. Ornaments with the symmetry group p6m may be constructed by multiplying a rosette with the symmetry group D₆ (6m) by means of two non-parallel translations with translation vectors of the same intensity, constructing a 60^o-angle, or by composing the symmetry group of friezes mm and the symmetry group of rosettes C₆ (6). Owing to its constructional complexity, the other construction is not as frequent as the first. A fundamental region of the symmetry group p6m is a right-angled triangle defined by the sides and altitude of an equilateral triangle that corresponds to a regular tessellation {3,6}. Since the symmetry group p6m is generated by reflections, its fundamental region must be rectilinear. The variety of ornaments with the symmetry group p6m can be realized only by using some asymmetric figure that belongs to a fundamental region. There are no enantiomorphic modifications. The generating translations and 6-fold rotations are non-polar. A static visual component caused by reflections is dominant. Besides having an important role in ornamental art, the symmetry group p6m can serve as a basis for the derivation of all the symmetry groups of the hexagonal crystal system by classical-symmetry, antisymmetry and colored-symmetry desymmetrizations. All the groups of the hexagonal crystal system are subgroups of the symmetry group p6m. The symmetry groups p31m and p3m1 are its subgroups of the index 2, and p3 is its subgroup of the index 4 (H.S.M. Coxeter, W.O.J. Moser, 1980). Each subgroup of the index 2 may be derived from its supergroup by an antisymmetry desymmetrization. Regarding the frequency of occurrence, the symmetry group p6m is one of the most frequent symmetry groups in ornamental art, proving that the principle of visual entropy - the aim toward maximal constructional and visual simplicity and maximal symmetry - is fully respected.

The square crystal system consists of the symmetry groups of ornaments p4 (Figure 2.94), p4g (Figure 2.95-2.97) and p4m (Figure 2.98-2.101). Frequent examples of ornaments with the symmetry groups of the square crystal system are mainly a result of the constructional and visual simplicity of a square lattice - regular tessellation {4,4} - on which these ornaments are based. The oldest examples of a square lattice date from the Paleolithic stone or bone engravings. By uniting two fundamental properties - division of a plane into squares, and perpendicularity - a regular tessellation {4,4} possesses the maximal symmetry group p4m of the square crystal system and serves as a basis for the construction of all ornaments with the symmetry groups belonging to this crystal system. Because all discrete symmetry groups of ornaments, except the groups of the hexagonal crystal system based on regular tessellations {3,6} or {6,3} , are its subgroups, the symmetry group p4m and its corresponding square regular tessellation {4,4} can serve as a starting point for the derivation and visual interpretation of all the subgroups mentioned by the desymmetrization method. The symmetry group p6m corresponding to regular tessellations {3,6} and {6,3} possesses a similar property and contains as subgroups, except the groups of the square crystal system, all the remaining discrete symmetry groups of ornaments. Owing to its visual and constructional simplicity, a regular tessellation {4,4} and the corresponding symmetry group p4m has become the richest source for deriving symmetry groups of ornaments by the desymmetrization method. Regular tessellations {3,6} , {6,3} and the corresponding symmetry group p6m will serve only for constructing ornaments with symmetry groups of a lower degree of symmetry, which cannot be realized by the desymmetrization method from a regular tessellation {4,4} - this means, for deriving by the desymmetrization method, the symmetry groups p3, p31m, p3m, p6 and p6m of the hexagonal crystal system.

Figure 2.94

Ornaments with symmetry group p4: (a) Starchevo, Yugoslavia, around 5500-5000 B.C.; (b) Egypt and Aegean cultures; (c) Akhbar school, India; (d) the ethnical art, Africa.

Figure 2.95

Examples of ornaments with the symmetry group p4g in Neolithic art: (a) Tripolian culture, USSR, around 4000 B.C.; (b) Hallaf, around 6000 B.C.; (c) Catal Hüjük, around 6400-5800 B.C.

Figure 2.96

Examples of ornaments with the symmetry group p4g in the art of the dynastic period of Egypt.

Figure 2.97

Ornaments with the symmetry group p4g: (a) Crete; (b) Aegina, around 5000 B.C.; (c) Thebes, Egypt, around 1500 B.C.; (d) Greco-Roman mosaic.

The symmetry group p4 is the minimal symmetry group of the square crystal system. It can be derived as an extension of the symmetry group of rosettes C₄ (4) by means of a discrete group of translations generated by two perpendicular translations with translation vectors of the same intensity. Also, it can be derived as a superposition of the symmetry group of friezes 12 and the symmetry group of rosettes C₄ (4) or by a desymmetrization of the symmetry group p4m in which the symmetry group p4 is the subgroup of the index 2 . Since the symmetry group p4 does not contain indirect isometries, there is a possibility for enantiomorphism. Owing to the visual dynamism caused by the absence of reflections, the possibility of enantiomorphism, and the bipolarity of generating translation axes and polarity of 4-rotations, the visual and constructional simplicity of a regular tessellation {4,4} is not so expressed in ornaments with the symmetry group p4. In constructing ornaments with the symmetry group p4, all construction methods will have almost equal importance. Owing to their low degree of symmetry and complex construction, ornaments with the symmetry group p4 are not as frequent as ornaments with the symmetry groups p4g and p4m belonging to the same crystal system. The oldest examples of ornaments with the symmetry group p4 originate from Neolithic ceramics (Figure 2.94). These ornaments are constructed by multiplying a frieze with the symmetry group 12, with the "wave" motif based on a double spiral, by the transformations of the symmetry group C₄ (4). Usually, a fundamental region of the symmetry group p4 is an isosceles right-angled triangle defined by the immediate vertices and the center of a square of the regular tessellation {4,4} . Aiming for a variety of the corresponding ornaments, a curvilinear fundamental region may be used. Polar, oriented four-fold rotations are a dynamic visual component of ornaments with the symmetry group p4. Therefore, although they do produce a visual suggestion of motion, and this is somewhat alleviated by the visual effect produced by the central reflections and by the bipolarity of the generating translation axes, such ornaments belong to a family of visually dynamic ornaments. They offer the possibility for the visual distinction of four singular bipolar directions that correspond to the generating frieze with the symmetry group 12. Since the symmetry group p4 is the subgroup of the index 2 of the maximal symmetry group p4m of the square crystal system, it is possible to derive the symmetry group p4 by a desymmetrization of the symmetry group p4m. Since it is also the subgroup of the index 2 of the group p4g, desymmetrizations of the symmetry group p4g may be used to obtain ornaments with the symmetry group p4. That especially refers to the antisymmetry desymmetrization resulting in the antisymmetry group p4g/p4, by the classical theory of symmetry discussed as the symmetry group p4.

Ornaments with the symmetry group p4g (Figure 2.95-2.97) may be constructed by the multiplication of a rosette with the symmetry group C₄ (4) by means of two perpendicular reflections that do not contain the center of this rosette. The same symmetry group can be derived as a superposition of the symmetry group of friezes 1g and the symmetry group of rosettes C₄ (4) or by a desymmetrization of the symmetry group of ornaments p4m, in which the group p4g is the subgroup of the index 2. The symmetry group p4g corresponds to a uniform tessellation s{4,4} , i.e. (4.8²) (H.S.M. Coxeter, W.O.J. Moser, 1980; B. Grünbaum, G.C. Shephard, 1987). According to the criterion of maximal constructional simplicity, a desymmetrization of the symmetry group p4m or the multiplication of a rosette with the symmetry group C₄ (4) by means of two perpendicular reflections, non incident with the rosette center, will be the prevailing methods for constructing ornaments with the symmetry group p4g. The oldest ornaments with the symmetry group p4g appear in Neolithic ceramics (Figure 2.95). In ornamental art of the ancient civilizations such ornaments, with somewhat changed Neolithic motifs, are very frequent. This especially refers to the ornaments with the symmetry group p4g using "swastika" motifs, which appeared as early as the Neolithic (Figure 2.95b).

Therefore, a detailed comparative analysis of the repetition of that and some other ornamental motifs can be useful in the study of the inter-cultural relations and influences that occurred in the Neolithic period and at the beginning of ancient civilizations. Usually, a fundamental region of the symmetry group p4g is an isosceles right-angled triangle defined by the centers of adjacent sides and by the corresponding vertex of a square of the regular tessellation {4,4} . Aiming to increase the variety of ornaments with the symmetry group p4g and to emphasize or decrease the intensity of the dynamic visual effect produced by them, its edges can be replaced by curved lines. The generating translation axes are bipolar. Despite the secondary reflections and the absence of the enantiomorphism, the four-fold rotations are polar, since these reflections do not contain the four-fold rotation centers. Owing to the four-fold polar, oriented rotations and glide reflections, ornaments with the symmetry group p4g belong to a family of extremely dynamic ornaments in the visual sense. They suggest a motion by their parts and by the whole, which is only partly offset by the static visual effect produced by the secondary reflections. Ornaments with the symmetry group p4g offer the possibility for the visual distinction of generating rosettes with the symmetry group C₄ (4) and generating friezes with the symmetry group mg. According to the principle of visual entropy, a visual recognition of their symmetry substructures will be hindered by the visual dynamism of these ornaments and by the visual dominance of glide reflections as the elements of symmetry. The symmetry group p4g is the subgroup of the index 2 of the symmetry group p4m, so that desymmetrizations of the symmetry group p4m and of a regular tessellation {4,4} can be efficiently used for constructing ornaments with the symmetry group p4g. Besides classical-symmetry desymmetrizations, antisymmetry desymmetrizations resulting in the antisymmetry group p4m/p4g, by the classical theory of symmetry considered as the symmetry group p4g, frequently occur (Figure 2.102h).

The most frequent symmetry group of ornaments is the maximal symmetry group p4m of the square crystal system (Figure 2.98-2.101), which corresponds to a regular tessellation {4,4} . Besides their large independent application, the symmetry group p4m plays an important role in the construction of all ornaments, except those with the symmetry groups of the hexagonal crystal system, by the desymmetrization method. Besides the property of regularity, a regular tessellation {4,4} possesses another fundamental property - the existence of two perpendicular generating translation axes incident to reflection lines. Since all the discrete symmetry groups of ornaments, except the groups of the hexagonal crystal system, are subgroups of the symmetry group p4m, it can serve as a basis for the derivation of all the discrete symmetry groups of ornaments by the desymmetrization method, in the first place for those symmetry groups with perpendicular generating translation axes: pm , pg, pmm, pmg, pgg, cm, cmm, p4, p4g. Besides classical-symmetry desymmetrizations, antisymmetry desymmetrizations resulting in all the subgroups of the index 2 of the symmetry group p4m - pmm, cmm, p4, p4g, p4m and color-symmetry desymmetrizations, may also be used. Ornaments with the symmetry group p4m can be constructed by multiplying a frieze with the maximal symmetry group of friezes mm by means of a four-fold rotation, by multiplying a rosette with the symmetry group D₄ (4m) by means of two perpendicular translations with translation vectors of the same intensity, or by means of a regular tessellation {4,4} . According to the principle of maximal constructional simplicity, the dominant methods for constructing ornaments with the symmetry group p4m will be by a square regular tessellation {4,4} or the translational multiplication of a rosette with the symmetry group D₄ (4m) . Early ornaments with the symmetry group p4m date from the Paleolithic, occurring as bone engravings representing an elementary square lattice - regular tessellation {4,4} (Figure 2.98). The further development of ornamental art tended toward the enrichment of ornamental motifs (Figure 2.99, 2.100). A fundamental region of the symmetry group p4m is an isosceles right-angled triangle defined by the center of a side, its belonging vertex, and the center of a square that corresponds to the regular tessellation {4,4} . A fundamental region of all the groups generated by reflections must be rectilinear, so that the variety of ornaments with the symmetry group p4m is reduced to the use of an elementary asymmetric figure belonging to a fundamental region. Four-fold rotations and generating translations are non-polar. Enantiomorphic modifications do not occur. Therefore, ornaments with the symmetry group p4m produce an impression of stationariness and balance, caused by perpendicular reflections incident to the generating translation axes.

Secondary diagonal reflections contribute to the impression of stationariness. Since taken for the directions of reflection lines usually are the fundamental natural directions - vertical and horizontal line - the dynamic visual component produced by secondary glide reflections with the axes parallel to the diagonals is almost irrelevant. It will come to its full expression in ornaments with symmetry group p4m placed in such a position that the diagonals coincide with the natural fundamental directions - the vertical and horizontal line. Besides the classical-symmetry desymmetrizations, antisymmetry desymmetrizations resulting in all the subgroups of the index 2 of the symmetry group p4m, and the colored-symmetry desymmetrizations, are used. Among antisymmetry desymmetrizations the most frequent is the antisymmetry group p4m/p4m ("chess board") (Figure 2.102j) discussed in the classical theory of symmetry as the symmetry group p4m.

Basic data on antisymmetry desymmetrizations is given in the table of the most frequent antisymmetry groups of ornaments G₂', i.e. the most frequent classical-symmetry groups that may be derived by antisymmetry desymmetrizations:

All of them occurred in Neolithic ornamental art (Figure 2.103).

Analyzing connections between ornamental art and the theory of symmetry, we can use the chronology of ornaments, considering as the main characteristics the time, the methods and the origin of ornaments. The oldest examples of ornaments date from the Paleolithic and Neolithic and belong to ornaments with the symmetry groups p1, p2, (pg), pm, pmg, pmm, cm, cmm, p4m and p6m. For all the symmetry groups of ornaments, except the symmetry group pg, we have concrete examples of the corresponding ornaments, occurring in the Paleolithic stone and bone engravings or cave drawings and Neolithic ceramic decorations. Since friezes with the symmetry group 1g originate from the Paleolithic, and by their translational repetition ornaments with the symmetry group pg can be constructed, probably the same dating holds for ornaments with the symmetry group pg. Another argument in favor of their early appearance is the existence of models in nature - the arrangements of leaves in some plants. On the other hand, the visual dynamism and low degree of symmetry of ornaments with the symmetry group pg could be account for their absence in Paleolithic ornamental art and for their somewhat later appearance in the Neolithic. It is also possible that the symmetry group pg, according to the principle of visual entropy, was replaced by the symmetry group pmg, very frequent in Paleolithic and Neolithic ornamental art. The first eight of the symmetry groups of ornaments mentioned can be derived as superpositions of symmetry groups of friezes and rosettes, without using rotations of the order greater than 2. Because the examples of all seven discrete symmetry groups of friezes 11, 1g, 12, m1, 1m, mg and mm occur in Paleolithic, this is the simplest method for constructing ornaments. Five of these symmetry groups of ornaments appear in the Paleolithic as the Bravais lattices (a lattice of parallelograms with the symmetry group p2, rectangular pmm, rhombic (cmm), square (p4m) and hexagonal (p6m)) - the simplest visual interpretations of the maximal symmetry groups of the crystal systems bearing the same names. The importance of reflections is proved by their presence in all the symmetry groups of ornaments mentioned, except p1, p2 (and pg). This illustrates the dominance of static ornaments in the visual sense, expressed by the absence of almost all the dynamic symmetry elements - polar rotations, glide reflections and by a relative dominance of non-polar and bipolar generating translations over the polar ones. As generating rosettes there are those with the most frequent symmetry groups of rosettes: D₁ (m), C₂ (2), D₂ (2m), D₄ (4m) and D₆ (6m). The symmetry group of rosettes D₂ (2m), besides the presence of reflections also possesses the other fundamental property - perpendicularity of reflection lines that coincide to the fundamental natural directions - the vertical and horizontal line. This symmetry group is a subgroup of four of the symmetry groups of ornaments mentioned: pmm, cmm, p4m, p6m. The two stated ornaments, p4m and p6m, correspond to regular tessellations {4,4} and {3,6} or {6,3} , i.e. to the perfect ornamental forms. With isogons, isohedrons, etc., regular tessellations belong together to the general tiling theory, discussed by E.S. Fedorov (1916), B.N. Delone (1959), H. Heesch, O. Kienzle (1963), L. Fejes Tòth (1964), H. Heesch (1968), A.V. Shubnikov, V.A. Koptsik (1974), B. Grünbaum, D. Lockenhoff, G.C. Shephard, A.H. Temesvari (1985), in many works by B. Grünbaum and G.C. Shephard (e.g., 1977c, 1978, 1983) and in their monograph Tilings and Patterns (1987).

Generating rosettes and friezes serve as the basis for constructing ornaments. They caused the time of appearance and the frequency of occurrence of particular symmetry groups of ornaments. Such constructions are extensions from the "local symmetry" of rosettes and friezes to the "global symmetry" of ornaments. As a common denominator for all the characteristics of Paleolithic ornaments we can use the principle of visual entropy - maximal constructional and visual simplicity and maximal symmetry.

Models existing in nature are prerequisites for the early appearance of some symmetry groups of ornaments (p2 - waves, pmg - water, p6m - a honeycomb). On the other hand, they impose certain restrictions in ornamental motifs. Since almost all animals and many plants are mirror-symmetrical, the same holds for ornaments inspired by those natural models. Even in geometric ornaments, where such restrictions have no influence, the importance of mirror symmetry in nature served as the implicit model and caused the dominance of ornaments containing reflections. The causes of this phenomenon can be found also in the constructional simplicity of ornaments with reflections, in human mirror symmetry and binocularity.

In the lather periods - in the Neolithic and in the period of the ancient civilizations - ornaments with the symmetry groups pg, pgg, p3, p31m, p3m1, p4, p4g, p6, appeared. Ornaments with the symmetry groups p3, p31m, p3m1, p4, p4g, p6 contain rotations of the higher order - 3, 4, 6 - occurring, until till then, only in regular tessellations {4,4} , {3,6} and {6,3} , i.e. in the symmetry groups p4m, p6m. The new symmetry groups caused new construction problems, in that the construction of ornaments required a very complicated multiplication of friezes. In these symmetry groups of ornaments, dynamic symmetry elements - polar rotations, polar generating translations and glide reflections - prevail. Therefore, with the constructional complexity of the corresponding ornaments, there is the problem of their visual complexity - i.e. difficulties in perceiving the regularities and symmetry principles they are based on. Also, these symmetry groups of ornaments presented the most problems to mathematicians.

The exception in this class is the symmetry group p3m1 (D), belonging to the family of the symmetry groups generated by reflections, and containing non-polar three-fold rotations, non-polar translations and a high degree of symmetry - favourable conditions for the early appearance in ornamental art. Its fundamental region is an equilateral triangle of a regular tessellation {3,6} , which must be rectilinear because the group p3m1 (D) is generated by reflections. The symmetry group of a regular tessellation {3,6} is the group p6m, so that ornaments with the symmetry group p3m1 can be constructed by a translational multiplication of rosettes with the symmetry group D₃ (3m) or by means of an internal classical-symmetry desymmetrization of the fundamental region. The use of such an elementary asymmetric figure belonging to the fundamental region results in a marked isohedral tiling with the symmetry group p3m1. However, both methods were not used in the Paleolithic.

Distinct from classical-symmetry ornaments with the symmetry group p3m1 occurring relatively seldom in ornamental art, antisymmetry ornaments with the antisymmetry group p6m/p3m1, considered in the classical theory of symmetry as the symmetry group p3m1, are some oldest and most frequent antisymmetry ornaments, because they can be obtained by the antisymmetry desymmetrization of the symmetry group p6m, in which the symmetry group p3m1 is the subgroup of the index 2, this means, from a regular tessellation {3,6} .

The stated chronological analysis offers an insight into all construction problems, the methods of forming ornaments and their origin. It points out the parallelism between the mathematical approach to ornaments from the theory of symmetry, and their origins and construction by methods developed in ornamental art. As basic common construction methods we can distinguish a frieze multiplication, a rosette multiplication and different aspects of the desymmetrization method - classical-symmetry, antisymmetry and color-symmetry desymmetrizations.

Owing to its simplicity, the construction of ornaments by using friezes (A.V. Shubnikov, V.A. Koptsik, 1974) is, most probably, the oldest method of constructing ornaments, since the conditions for so-doing - the existence of examples of all the symmetry groups of friezes - were fulfilled already in Paleolithic ornamental art. Very successful in the construction of Paleolithic ornaments, this method shows its deficiency in the constructional complexity for the ornaments with rotations of the higher order (n = 3, 4, 6). Somewhat more complex is the method of rosette multiplication in which we must consider, besides rosettes with a definite symmetry, directions and intensities of the vectors of generating discrete translations multiplying these rosettes. Therefore, such a construction method implicitly introduces problems of plane Bravais lattices, crystal systems with the same names, and tessellations.

Occurring in Paleolithic ornamental art are examples of all the five Bravais lattices corresponding to the symmetry groups p2, pmm, cmm, p4m and p6m, i.e. to the maximal symmetry groups of the crystal systems bearing the same names, including regular tessellations {4,4} (p4m), {3,6} and {6,3} (p6m). Till the Neolithic, the method of rosette multiplication remained on the level of elementary forms - Bravais lattices. Later, it was used for the construction of all kinds of ornaments. In the mathematical theory of symmetry this approach is discussed by different authors, e.g., by A.V. Shubnikov, N.V. Belov et al. (1964), and A.V. Shubnikov, V.A. Koptsik (1974).

The desymmetrization method is based on the elimination of adequate symmetry elements of a given symmetry group, aiming to obtain one of its subgroups. All the discrete symmetry groups of ornaments are subgroups of the groups p4m and p6m generated by reflections (H.S.M. Coxeter, W.O.J. Moser, 1980). Knowledge of group-subgroup relations between the symmetry groups of ornaments makes possible the consequent application of desymmetrizations. In ornamental art, this method appears in the Neolithic, with the wider application of colors and decorative elements, to become in time, along with the method of rosette and frieze multiplication, one of the dominant construction methods.

Antisymmetry desymmetrizations originate from the use of colors in ornamental art, beginning with Neolithic ceramics. Although colors, as an artistic means, were used even in the Paleolithic, they were mainly used for figurative contour drawings. Paleolithic ornaments mostly occur as stone and bone engravings or drawings, so that antisymmetry desymmetrizations came to their full expression with Neolithic colored, dichromatic ceramics. They can serve as the basis for the derivation of all subgroups of the index 2 of a given symmetry group. In cases when classical-symmetry desymmetrizations are complex, this is the most fruitful method for the derivation of certain less frequent symmetry groups of ornaments (cmm/pgg, pmg/ pg, p4m/p4g, pmg/pgg, p6m/p3m1, p6m/p6), because the resulting symmetry groups (pgg, pg, p4g, pgg, p3m1, p6) are derived by antisymmetry desymmetrizations of the frequently used generating symmetry groups ( cmm, pmg, p4m, pmg, p6m, p6m).

The list of antisymmetry desymmetrizations of the symmetry groups of ornaments, i.e. the list of antisymmetry groups of ornaments G₂', can serve as the basis for antisymmetry desymmetrizations. A survey of the minimal indexes of subgroups in the symmetry groups of ornaments (H.S.M. Coxeter, W.O.J. Moser, 1980, pp. 136), mosaics for dichromatic plane groups (A.V. Shubnikov, N.V. Belov et al., 1964, pp. 220) and numerous works on the theory of the antisymmetry of ornaments (H.J. Woods, 1935; A.M. Zamorzaev, A.F. Palistrant, 1960, 1961; A.V. Shubnikov, N.V. Belov et al., 1964; A.M. Zamorzaev, 1976; S.V. Jablan, 1986a) are some of the possible sources for future consideration of antisymmetry desymmetrizations.

In the table of antisymmetry desymmetrizations the antisymmetry groups of ornaments G₂' are given in the group/subgroup notation G/H giving information on the generating symmetry group G and its subgroup H of the index 2 obtained by the antisymmetry desymmetrization. Aiming to differentiate between two antisymmetry groups possessing the same group/subgroup symbol and to denote two different possible positions of reflections in the symmetry group pm, the symbols pm1 and p1m, are used.

The table of antisymmetry desymmetrizations of symmetry groups of ornaments G₂:

Antisymmetry and color-symmetry desymmetrizations are the very efficient tools for finding all subgroups of the index N of a given symmetry group. The complete list of group-subgroup relations between 17 discrete symmetry groups of ornaments G₂ and the minimal indexes of subgroups in groups are given in the monograph by H.S.M. Coxeter and W.O.J. Moser (1980, pp. 136). From the table of color-symmetry desymmetrizations we can conclude that [cm:cm] = 3 (pp. 182, the color-symmetry desymmetrization cm/cm, N = 3) and [ cmm:cmm] = 3 (pp. 182, the color-symmetry desymmetrization cmm/cmm/cm, N = 3). After these corrections and the corrections: [pmg:pmg] = 2, [p6:p6] = 3, already given by the authors in the reprint of their monograph, this table reads as follows.

Because the construction of antisymmetry ornaments and the application of antisymmetry desymmetrizations is followed by various construction problems, their use in ornamental art is usually restricted to the antisymmetry groups stated in the table listing the most frequent antisymmetry desymmetrizations (pp. 172).

The use of color-symmetry desymmetrizations in ornamental art began with Neolithic polychromatic ceramics, to reach its peak in the ornamental art of the ancient civilizations (Egypt), in Roman floor mosaics and in Moorish ornaments. When considering color-symmetry desymmetrizations one should distinguish, as in antisymmetry desymmetrizations, two different possibilities. In the first case the use of colors results in a desymmetrization of the symmetry group of the ornament, but neither an antisymmetry nor color-symmetry group is obtained. Although the color is a means of such desymmetrization, this is, in fact, a classical-symmetry desymmetrization, as opposed to real antisymmetry or color-symmetry desymmetrizations resulting in antisymmetry and colored symmetry groups of ornaments. Only those desymmetrizations resulting in antisymmetry or color-symmetry groups of ornaments will be accepted and analyzed under the term of antisymmetry or color-symmetry desymmetrizations, while all the other desymmetrizations using colors only as a technical means, will be discussed as classical-symmetry desymmetrizations.

In line with the principle of visual entropy - maximal constructional and visual simplicity and maximal symmetry - from the point of view of ornamental art, especially important will be color-symmetry desymmetrizations of the maximal symmetry groups of the crystal systems - desymmetrizations of Bravais lattices.

The discussion on color-symmetry desymmetrizations of the symmetry groups of ornaments relied on the works of A.M. Zamorzaev, E.I. Galyarski, A.F. Palistrant (1978), M. Senechal (1979), A.F. Palistrant (1980a), J.D. Jarratt, R.L.E.  Schwarzenberger (1980), R.L.E. Schwarzenberger (1984), A.M. Zamorzaev, Yu.S. Karpova, A.P. Lungu, A.F. Palistrant (1986) and the monograph by T.W. Wieting The Mathematical Theory of Chromatic Plane Ornaments (1982). In this monograph the numbers of the color-symmetry groups of ornaments for N £ 60 colors, the catalogue of the color-symmetry groups of ornaments for N £ 8 colors and the colored mosaics for N = 4 colors, are given.

Because of the large number of the color-symmetry groups of ornaments, in the table of the color-symmetry desymmetrizations of the symmetry groups of ornaments, the restriction N £ 8 is accepted. In that way, the usual practical needs for the construction and analysis of colored ornaments, where the number of colors N does not exceed 8, are also satisfied. According to the definition of color-symmetry groups given by R.L.E. Schwarzenberger (1984), all the color-symmetry groups discussed require the even use of colors. The uneven use of colors or the use of colors in a given ratio, e.g., 2:1:1, 4:2:1:1, 6:2:1, 6:3:1:1:1 (B. Grünbaum, Z. Grünbaum, G.C. Shephard, 1986), which occur in ornamental art and demand a special mathematical approach, represents an open research field.

In the table of the color-symmetry desymmetrizations of the symmetry groups of ornaments, by analogy to the corresponding tables of the color-symmetry desymmetrizations of the symmetry groups of rosettes and friezes, every color-symmetry group is denoted, for a fixed N, by a symbol G/H/H₁. For H = H₁, i.e. iff H is a normal subgroup of the group G, such a symbol is reduced to the symbol G/H. If the first or the second symbol does not uniquely determine the color-symmetry group, i.e. the corresponding color-symmetry desymmetrization, such symbols supplemented by the symbol of the translational group that corresponds to the subgroup H, are used (D. Harker, 1981). In the supplementary symbols -a is denoted by a, and the symbol of the 2×2 matrix with the colomns a, b and c, d is [ab,cd]. The incidence of the symmetry element S of the group G to the symmetry element S₁ of the subgroup H is denoted by the supplementary symbol (S º S₁). To differentiate between the symmetry groups pg with the glide reflections g_x, g_y, the symbols pg1, p1g are used respectively. The symbols cm1, c1m, and pmg, pgm, are used analogously for denoting the two different possible positions of the reflection m.

The table of color-symmetry desymmetrizations of symmetry groups of ornaments G₂:

A homogeneous and isotropic plane possesses the maximal continuous symmetry group of ornaments p₀₀¥m (s^¥¥), while all the other symmetry groups of ornaments are its subgroups. In ornamental art, it may be identified as a ground representing the environment where the remaining symmetry groups of ornaments exist. Among the symmetry groups of semicontinua, only the symmetry groups p₁₀1m (s1m) and p₁₀mm (smm) possess their adequate visual interpretations. The condition for the visual presentability of the continuous symmetry groups of ornaments is the non-polarity of their continuous translations and continuous rotations. For the symmetry groups of semicontinua this condition is equivalent to the existence of the corresponding visually presentable continuous friezes, by the translational multiplication of which the corresponding semicontinua can be derived.

Between the visually presentable continuous symmetry groups of ornaments, the following group-subgroup relations hold (Figure 2.104). These relations point out desymmetrizations suitable to derive continuous symmetry groups of a lower degree of symmetry and to recognize symmetry substructures of visually presentable plane continua and semicontinua. Since all discrete symmetry groups of ornaments G₂ are subgroups of the symmetry groups p4m and p6m, for further work on group-subgroup relations between the continuous and discrete symmetry groups of ornaments, it is possible to use this data.

The geometric-algebraic properties of the symmetry groups of ornaments G₂ - their presentations, data on their structure, properties of generators, polarity, non-polarity and bipolarity, enantiomorphism, form of the fundamental region, tables of group-subgroup relations, Cayley diagrams, etc. - offer the possibility to plan the visual properties of ornaments before their construction. Besides the usual approach to ornamental art, where visual structures serve as the objects for analyses from the point of view of the theory of symmetry, there is also an opposite approach - from abstract geometric-algebraic structures to the anticipation of their visual properties. Then ornaments may be understood as visual models of the corresponding symmetry groups.

Generators define a possible form of a fundamental region, so that the incidence of a reflection line with a segment of the boundary of the fundamental region means that such a part of the boundary must be rectilinear. A fundamental region must be rectilinear in the symmetry groups of ornaments generated by reflections - pmm, p3m1, p4m, p6m. All the other symmetry groups of ornaments offer the use of curvilinear boundaries or curvilinear parts of boundaries of a fundamental region, which do not belong to reflection lines. The question of the form of a fundamental region is directly linked to the perfect plane forms - monohedral tilings, plane tilings by congruent tiles. This problem constantly attracts mathematicians and artists (B. Grünbaum, G.C. Shephard, 1987). In ornamental art it came to its fullest expression in Egyptian and Moorish ornamental art and in the graphic works by M.C. Escher (M.C. Escher, 1971a, b, 1986; B. Ernst, 1976; C.H. Macgillavry, 1976) . For the symmetry groups of ornaments generated by reflections pmm, p4m and p6m, the problem of isohedral tilings is simply reduced to a rectangular lattice (pmm) and regular tessellations {4,4} (p4m), {3,6} or {6,3} (p6m). There is a certain anomaly in the symmetry group p3m1 (D). Owing to a symmetrization, its corresponding tessellation {3,6} possesses the symmetry group p6m. Therefore, this is the only symmetry group of ornaments not offering any possibility for the corresponding isohedral unmarked tilings and requiring the use of an asymmetric figure within a fundamental region, this means, a marked tiling. This contradiction can be solved by taking rotation centers always inside the tiles, so that the symmetry group p3m1 will correspond to a regular tessellation {3,6} , and the symmetry group p6m to a regular tessellation {6,3} . Under such conditions the symmetry group p3m1 belongs to the family of the symmetry groups of ornaments appearing in Paleolithic ornamental art, since the regular tessellation {3,6} dates from the Paleolithic (Figure 2.92b).

A generating translation axis of an ornament is non-polar iff there exists a reflection perpendicular to it. If a central reflection center belongs to a translation axis, this axis will be bipolar, while in the other cases that axis will be polar. There will be no enantiomorphic modifications of an ornament iff its symmetry group contains indirect isometries. A rotation of an ornament will be non-polar iff there exists a reflection incident to the center of this rotation n (n = 2, 3, 4, 6), i.e. iffD_n (nm) is a subgroup of the symmetry group of the ornament.

The visual stationariness of an ornament is conditioned by the non-polarity of its axes and rotations, by the presence of reflections, especially perpendicular ones, by the absence of glide reflections suggesting alternating motions and by the absence of the enantiomorphism. On the other hand, the factors that determine whether an ornament will be a dynamic one in the visual sense, will be the polarity of translation axes suggesting one-way directed motions, the polarity of rotations, the presence of glide reflections, the absence of reflections and the existence of the enantiomorphism.

Besides the stated objective geometric-visual elements defining visual stationariness or dynamism of certain ornament, the "subjective" elements are also important. By changing them, it is possible to influence the visual impression that the ornament gives to the observer. Under these changeable factors - the "visual parameters" of an ornament - we can include the form of a fundamental region or the shape of an asymmetric figure within a fundamental region. By changing this it is possible to decrease or emphasize the visual dynamism. The use of acuteangular forms in dynamic ornaments, the change of a position of symmetry elements with respect to the observer and to the fundamental natural directions, the visual role of the äscending" and "descending" diagonal, etc., can have the same function.

Real ornaments occurring in ornamental art are bounded parts of "ideal" unbounded geometric ornaments, i.e. their finite factor groups, derived by the identifications p_x^m = E, p_yⁿ = E (m, n Î N). Such a bounded part of an ornament is defined by the intensities of the generating translations p_x, p_y, their mutual position and by the parameters m, n. In ornaments with the p_x- and p_y-translation vectors of the same intensity, especially interesting is the case of m = n.

The visual effect given to the observer by a real ornament is the result of the interaction between the symmetry group of the ornament, the plane symmetry of the observer and binocularity, the symmetry of a bounded part of the "ideal" ornament, symmetry of the environment, etc. It is also formed under the visual-physiological influence of the fundamental natural directions (the vertical and horizontal line, gravitation, etc.). All these "subjective" factors participate in the formation of the primary visual impression, as desymmetrization or symmetrization factors. Some of them (e.g., the vertical and horizontal line), may occur as a specific implicit coordinate system having a relevant positive or negative influence on the recognition of the symmetry of ornament. In the further course of the process of visual perception, the observer tends to recognize the symmetry of the ornament itself. Even after the elimination of disturbing elements, the eye often perceives a visually dominant structure, reducing it to the simplest one possible. Sometimes, according to the principle of visual entropy, the visual introduction of new symmetry elements - symmetrization of the structure - and visual substitution of dynamic elements of symmetry by their static equivalents takes its place. This is often manifested in experiments to recognize or reproduce certain motifs.

One comes across similar problems during the visual perception of symmetry substructures - rosettes, friezes or ornaments - belonging to certain ornament. Their symmetry groups appear as the subgroups of the symmetry group of the ornament. The primary factors relevant for the visual perception of substructures will be the visual simplicity of substructures themselves (their visual stationariness or dynamism), the degree of visual simplicity of the ornament and the position of substructures with respect to the fundamental directions of the ornament, to the fundamental natural directions, etc.

Visually simpler, static non-polar forms with a high degree of symmetry, occurring as substructures, will be more easily perceived and visually recognized. The possibility for recognizing its symmetry substructures will depend also on the visual simplicity of the ornament itself. A high degree of symmetry of the ornament in such a case is the aggravating factor for registering subentities, so that the same symmetry subgroup will be easily recognized in ornaments with a low degree of symmetry. For recognizing the symmetry of ornaments and their substructures, it is very important to visually recognize and discern a fundamental region or an elementary asymmetric figure belonging to the fundamental region. Otherwise, a slow recognition of symmetry elements and the symmetry group of the ornament, is unavoidable. Problems with the recognition of symmetry substructures of one ornament can be efficiently solved by using the table of subgroups of a given symmetry group of ornaments, and especially, by using their visual interpretations - tables of the graphic symbols of symmetry elements (A.V. Shubnikov, V.A. Koptsik, 1974; B. Grünbaum, G.C. Shephard, 1987) and Cayley diagrams of ornaments (H.S.M. Coxeter, W.O.J. Moser, 1980).

Visualization was an important element in the development of the theory of symmetry of ornaments, so that the visual characteristics of ornaments caused the occurrence of similar ideas, construction methods and even a chronological parallelism between these fields. In ornamental art and in the theory of symmetry the oldest methods for the construction of ornaments, used the frieze and rosette multiplication, including the problem of Bravais lattices, crystal systems and tessellations. The criteria of maximal constructional and visual simplicity and maximal symmetry, united by the principle of visual entropy, played the same important role in both fields, where the symmetry structures with emphasized visual simplicity, in which prevail static elements, were dominant. Considering dynamic structures, mathematicians and artists were faced with the problem of perceiving the regularities on which they were based, when defining the elements of symmetry, generating and other symmetry substructures.

Throughout history, visuality was usually the cause and the basis for geometric discussions. Only lately, with non-Euclidean geometries, have the roles been partly replaced, so that visualization is becoming more and more the way of modeling already existing theories. From a methodological aspect, this is the evolution from the empirical-inductive to the deductive approach. Similarly, we can note the way ornaments progressed from their origin linked to concrete meanings - models in nature - or the symbolic meanings of ornamental motifs. After their meaning and form were harmonized, ornaments became a means of communication. In the final phase, after having complete insight into the geometric and formal properties of ornaments, the question of the meaning of ornaments is almost solved, but new possibilities for artistic investigations of the variety and decorativeness of ornaments, are opened.

This analysis of the visual properties of ornaments and their connection with the symmetry of ornaments, points out the inseparability of these two fields. Besides the possibility for the exact analysis of ornaments, "the theory of symmetry approach" to ornamental art provides the groundwork for planning, constructing and considering ornaments as visual creations with desired visual properties. These visual properties can be anticipated from the corresponding symmetry groups of ornaments, their presentations, properties of generators, structures, etc. Ornaments, treated as visual models of different symmetry structures, can be the subject of scientific studies, and can be used in all scientific fields requiring a visualization of such symmetry structures.