Chapter 2.4

  Friezes and
  Ornamental Art

A straight line is, most probably, the first and simplest frieze occurring in the history of visual arts. From the point of view of the theory of symmetry, it belongs to the family of friezes with the continuous symmetry group m₀m.

Recent investigations of the process of visual perception have discovered that in a visual sense, the perceived symmetry of a line will depend on its position.

A horizontal line has the symmetry m₀m - maximal continuous symmetry group of friezes (Figure 2.29a).

Regarded from the visual point of view, a vertical line has a polar, oriented continuous translation axis and the continuous symmetry group p₀1m (0m). The phenomenon of a visual, subjective desymmetrization of a vertical line is probably implied by gravitation. It is possible that the tendency of a vertical line to go upward results from the objective narrowing of vertical objects in the upper part; from the visual convergence of parallel lines with high objects observed from a lower position; and from the habit conditioned by a constant application of the central perspective, where the center of the perspective is usually placed in the upper part of the picture. A vertical line has the "directed tension", so that along with the objective symmetry m₀m, occurs its visual, subjective desymmetrization and the reduction to the symmetry 0m. The tendency to accept a vertical form as longer in comparison with the horizontal form of the same length is conditioned by the polarity of the vertical axis (Figure 2.30).

A similar visual, subjective desymmetrization - a reduction to a lower degree of symmetry and the polarity of the axis - occurs also with an "ascending" and "descending" diagonal. This phenomenon can be also treated as the difference between the "right" and "left" (H. Weyl, 1952, pp. 16; R. Arnheim, 1965, pp. 22), in the theory of visual perception discussed by H. Wölfflin, M. Gaffron, A. Dean and S. Cobb. Along with different physiological interpretations, there is an interesting Weyl's question about whether this phenomenon is connected to writing from the left to right, and whether it is expressed by nations writing in the opposite direction. M.Gaffron tried to explain it by the dominance of the left brain cortex containing brain centers for several activities in right-handed people, i.e. in the dominance of righthandness. About this, S. Cobb says (R. Arnheim, 1965, pp. 23): "Many fanciful ideas have been put forward, from the theory that the left hemisphere has a better blood supply than the right, to the heliocentric theory that the right hand dominates because man originated north of the equator and, looking at the sun, was impressed with the fact that great things move toward the right! Thus right became the symbol of rectitude and dexterity, and things on the left were sinister. It is an interesting observation that about 70 per cent of human foetuses lie in the uterus in the "left occiput posterior" position, i.e. facing to the right. No one has ever found out whether or not these become the right majority of babies. Probably the dominance of right-handness is due to chance in heredity.".

All the continuous symmetry groups of friezes have physical interpretations (A.V. Shubnikov, V.A. Koptsik, 1974). In the visual arts, besides visual models of the continuous symmetry groups of friezes m₀1 and m₀m (Figure 2.29) with the non-polar translation axis, adequate visual interpretations of the other continuous symmetry groups of friezes with the polar or bipolar translation axis, can be realized, in the sense of the objective symmetry, exclusively by textures. Textures can be realized by equal average density of the same asymmetric elementary figures arranged along the singular direction in accordance with the corresponding continuous symmetry group. For a schematic visual interpretation of these groups by using the graphic symbols of symmetry elements, certain supplementary symbols (e.g., arrows) are indispensable (B. Grünbaum, G.C. Shephard, 1987).

The oldest examples of friezes are found in the art of the Paleolithic (Magdalenian, 12000-10000 B.C.) and Neolithic. In fact, examples of all seven discrete and two visually presentable symmetry groups of friezes are known from the Magdalenian period.

Despite a relative variety of motifs in ornamental art, there is also a repetition of ornamental motifs - basic, elementary patterns occurring in different parts of Europe, Asia and Africa, where the late Paleolithic and Neolithic cultures were formed. Since the possibilities for communication between distant areas were remote, we can assume that common ornamental motifs result from similar or the same models found in nature by prehistoric peoples and from the laws of symmetry.

Most frequently, friezes are the result of a translational repetition of different motifs, where the symmetry of the original motif - a rosette - determines the symmetry of the frieze itself, in the sense of composition of the translational group 11 and the symmetry group of the rosette or by an artistic schematization of natural objects possessing by themselves the symmetry of friezes.

The origins of friezes are visible in cave drawings and engravings on stones or bones from the earliest period - late Paleolithic.

Representing a herd of deer (Figure 2.31), prehistoric man had abstracted a motif, almost reducing it to a translational repetition of a pair of horns, i.e. to the frieze with the symmetry group 11. A similar process, from the motif of dance to a frieze with the symmetry group m1, from the motif of harpoon to a frieze with the symmetry group 1m, from the motif of waves to a frieze with the symmetry group 12 or mg, shows the original symbolic meanings friezes carried. Thus, friezes became one of the first visual communication means.

Also, the symmetry of friezes based on repetition, have made possible a symbolic representation of certain periodic natural phenomena - the turn of day and night, the daily and annual revolving of the Sun - where friezes played the role of a calendar. This is witnessed by friezes originating from ethnical art and having even today precisely defined symbolic meanings and adequate names (Figure 2.54).

The symmetry group of friezes 11 is the result of a periodic, translational repetition of an asymmetric figure. The form of the fundamental region can be arbitrary. Owing to the polarity of the frieze axis and the sheer repetition of the motif, besides a possibility for figurative representation, there is also a possibility for a geometric-symbolic representation of directed phenomena. Hence, directed tension resulting from the polarity of the axis creates an impression of "motionless motion", thus forming the time component of painting. In combination with overlapping which makes possible the suggestion of the perspective in the sense "in front-behind", friezes have been frequently used in Egyptian art, both decorative and painting, and in the art of cultures tending to the "objective", natural axonometric presentation of spatial groups in motion (Figure 2.32-2.35).

A similar dynamic effect is produced by friezes with the symmetry groups 1g (Figure 2.36, 2.37) and 1m (Figure 2.38-2.40) which have a polar axis. By using different elementary asymmetric figures belonging to the fundamental region and different directed forms with acute angles oriented toward the direction of the axis, it is possible to intensify the already existing impression of motion. Owing to their relation to the growth of certain plants, friezes with the symmetry group 1g hold an important place in plant ornaments (Figure 2.37). Since they contain a glide reflection as the symmetry element, such friezes are suitable for representing all directed alternating phenomena or objects by means of geometric ornaments.

Owing to the existence of a central reflection as the symmetry transformation of the group 12, the frieze axis is bipolar, so friezes with the symmetry group 12 offer the possibility for registering oppositely directed elementary asymmetric figures along the singular direction, i.e. two oppositely directed friezes with the symmetry group 11. Friezes with the symmetry group 12 occur in many cultures (in the Neolithic, Egyptian, Aegean, etc.), with the application of spiral motifs (Figure 2.41-2.45).

All the other discrete symmetry groups of friezes m1, mg, mm and the continuous symmetry groups of friezes m₀1 and m₀m with the non-polar axis, which contain reflections with reflection lines perpendicular to the frieze axis, create an impression of stationariness and balance.

Friezes with the symmetry group m1 were frequently used in the prehistoric period with the motif of the cult dance "kolo". The origin and development of this motif can be seen in the stone drawings from the period of cave painting (Figure 2.46). This motif, very characteristic and suitable for the analysis of the history of ornamental art, underwent a considerable stylization in the Neolithic period (Figure 2.47). During the Neolithic, by losing its symbolic meaning and by being enriched by new elements, the motif of dance and the corresponding friezes were reduced to sheer decorativeness. All types of ornaments underwent this process (Figure 2.48).

Owing to a glide reflection and reflections in reflection lines perpendicular to the frieze axis, friezes with the symmetry group mg, among geometric ornaments occur as symbols of regular alternating phenomena (Figure 2.49-2.51). Different variations of these friezes have in ethnical art the following meanings (Figure 2.54a, b, c, d): "Up and down", "The daily motion of the Sun", "The Sun above and below the water level (the horizon)", "Breathing", "Water", "The rhythm of water".

Friezes with the symmetry group mm (Figure 2.52, 2.53) often symbolize an even flow of time, years and similar phenomena with a high degree of symmetry (Figure 2.54f, g, h). It is interesting that in such cases time is considered as non-polar. Owing to the maximal degree of symmetry and to the fact that it contains all the other discrete symmetry groups of friezes as subgroups, besides having a significant independent function, the symmetry group mm will be a basis from which all the other discrete symmetry groups of friezes can be derived by means of the desymmetrization method. Having in mind the fixed shape of the fundamental region - rectilinear, perpendicular, unbounded fundamental region - a variety of friezes with the symmetry group mm can be achieved only by using different elementary asymmetric figures belonging to the fundamental region. This holds for all the symmetry groups of friezes generated by reflections (m1, mm).

Already in ornamental art of the Neolithic and of the first great cultures - the Egyptian, Mesopotamian, and Aegean cultures, and the pre-Columbian culture in America, etc. - by introducing new ornamental motifs and by enriching existing ones, the variety of friezes is achieved (Figure 2.55). Like for rosettes, superpositions of friezes are frequent. Some primary symbolic meanings have been gradually replaced by new ones. The application of different motifs unavoidably leads to decorativeness. The empirical perception of the properties of friezes and the regularities they are based on, resulted in new friezes constructed by using the construction rules comprehended, thus opening the way to artistic imagination and creation-play.

Examples of all seven discrete symmetry groups of friezes 11, 1g, 12, m1, 1m, mg, mm and two continuous visually presentable symmetry groups of friezes m₀1 and m₀m, represented in bone engravings or cave drawings, date from Paleolithic art. In ornamental art, friezes are a way to express regularity, repetition and periodicity, while with polar friezes, where there occurs a singular oriented direction, they represent a way to express motion or dynamic tendencies.

In the theory of symmetry, although derived relatively late by registering the one-dimensional subgroups of the symmetry groups of ornaments G₂ (G. Pòlya, 1924; P. Niggli, 1924; A. Speiser, 1927), the symmetry groups of friezes are a suitable ground for different research. For example, the development of the theory of antisymmetry is greatly stimulated by studies of the symmetry groups of friezes. Namely, its first accomplishment - the antisymmetry groups of friezes (H. Heesch, 1929; H.J. Woods, 1935; A.V. Shubnikov, 1951) - resulted from the Weber diagrams of the symmetry groups of bands G₃₂₁. Owing to their simplicity, the symmetry groups of friezes G₂₁ - the first category of infinite isometry groups - were used as a suitable medium for constructing and analyzing new theories, e.g., the theory of antisymmetry, colored symmetry, etc.

In the same way as with rosettes, according to the principle of visual entropy - maximal visual and constructional simplicity and maximal symmetry - there may be established a relation between geometric properties of the symmetry groups of friezes and their visual interpretations - friezes, their frequency of occurrence, period of origin and variety. The earlier appearance and dominance of friezes satisfying this principle, is evident.

The table of the group-subgroup relations points to the possibility to apply the desymmetrization method for the derivation of the symmetry groups of friezes. According to the relations (Figure 2.56) and the tables of the (minimal) indexes of subgroups in groups, the classical-symmetry, antisymmetry (for subgroups of the index 2) and color-symmetry desymmetrizations or their combinations can be used aiming to obtain the symmetry groups of friezes of a lower degree of symmetry.

A survey of the antisymmetry desymmetrizations of friezes is given in the corresponding table. Symbols of antisymmetry groups G' are given in the group/subgroup notation G/H, offering information on the generating symmetry group G and its subgroup H of the index 2 - symmetry subgroup H of the group G', which is the final result of the antisymmetry desymmetrization. So that, by antisymmetry desymmetrizations it is possible to obtain all subgroups of the index 2 of the given symmetry group (H.S.M. Coxeter, 1985).

The table of antisymmetry desymmetrizations of symmetry groups of friezes G₂₁:

The table of the color-symmetry desymmetrizations relies on the works of J.D. Jarratt, R.L.E. Schwarzenberger (1980) and H.S.M. Coxeter (1987). By color-symmetry desymmetrizations, the symmetry groups of friezes 11, 1g, 1m, m1 and 12, may be obtained. Complete information on the color-symmetry desymmetrizations of friezes is given in the corresponding table. Each of the infinite classes of colored symmetry groups is denoted by a symbol G/H/H₁; its first datum represents the generating symmetry group G, the second its stationary subgroup H consisting of transformations of the colored symmetry group G^* that maintain each individual index (color) unchanged, and the third the symmetry subgroup H₁ of the group G^* . The symmetry group H₁ is the final result of the color-symmetry desymmetrization. A number N (N ³ 3) is the number of "colors" used to obtain the particular color-symmetry group. For H = H₁, i.e. iff H is a normal subgroup of the group G, the symbol G/H/H₁ is reduced to the symbol G/H.

The table of color-symmetry desymmetrizations of symmetry groups of friezes G₂₁:

Due to the objective stationariness of works of ornamental art, continuous friezes with a polar or bipolar axis do not have adequate visual interpretations. The visual models of these groups can be constructed by using textures (A.V. Shubnikov, N.V. Belov et al., 1964).

According to the principle of maximal symmetry, a survey of group-subgroup relations may, in a way, serve as an indicator of the frequency of occurrence of the particular symmetry groups of friezes in ornamental art, and also for recognition and evidence of symmetry substructures. The influence of the principle of maximal symmetry on the frequency of occurrence of certain symmetry groups of friezes in ornamental art comes to its full expression for the most common frieze - the straight line, which represents a visual illustration of the maximal continuous symmetry group of friezes m₀m and in the frequent occurrence of friezes with the symmetry groups mm, mg, etc.

The origin of friezes arose also out of the existence of natural models, so that friezes with the symmetry group 12 or mg may be considered as stylized forms of waves; motifs with the symmetry 1g or 1m, that are found in arrangements of leaves in many plants, served as a source of many friezes; while the importance of mirror symmetry in nature caused friezes with the symmetry group m1, 1m or mm. This especially refers to the maximal discrete symmetry group of friezes mm, which contains as subgroups all the other discrete symmetry groups of friezes and the symmetry group of rosettes D₂ (2m) and expresses, in the visual sense, the relation "vertical-horizontal" and the quality of perpendicularity. Besides natural objects, the origin of friezes is also to be found in the periodic character of many natural phenomena (the turn of day and night, the turn of the seasons, the phases of the moon, the tides etc.). Therefore, friezes represent a record of the first human attempts to register periodic natural phenomena, i.e. the first calendars. In time, with their symbolic meanings clearly defined, friezes became a visual communication means: each frieze contains a message, i.e. its meaning is harmonized with its visual form. This can be proved by the preserved names of friezes in ethnical art. Later, with the development of all other communication means, friezes lose their original symbolic function, to be partially or completely replaced by a decorative one.

The visual impression produced by a real frieze results from the interaction between its symmetry group, human mirror symmetry and binocularity, the symmetry group of a limited part of the plane to which the frieze belongs, and the symmetry group D₂ (2m) caused by the fundamental natural directions - the vertical and horizontal line. Regarding the first, the other symmetry groups mentioned have a role of desymmetrization or symmetrization factors. Besides the objective elements of symmetry, the visual impression is influenced by the subjective elements referring to the physiological-psychological properties of the visual perception (e.g., perception of the "right" diagonal as "ascending" and the "left" as "descending", etc.), so that this dependence may be very complex. Aiming to perceive and recognize the objective, geometric symmetry, the observer must eliminate these secondary, subjective visual factors.

A fundamental region of the symmetry group of friezes m1 or mm is rectilinear, with boundaries incident withthe reflection lines. All the other friezes with partly or completely curved boundaries of the fundamental region, offer a change of its shape. Aiming to increase the variety of friezes with the symmetry group m1 or mm possessing a rectilinear fundamental region, these possibilities are reduced to the use of different elementary asymmetric figures belonging to the fundamental region.

Data on the polarity of friezes and enantiomorphism, in the visual sense refer to the dynamic or static impression created by them. Friezes with the symmetry groups 11, 1g, 1m with a polar, oriented singular direction will produce a dynamic effect. Since all friezes with the polar axis may have a curvilinear fundamental region, their visual dynamism may be emphasized by choosing an acuteangular fundamental region with the acute angle oriented toward the direction of the axis or by choosing an acuteangular elementary asymmetric figure belonging to the fundamental region and directed in the same way. The symmetry group of friezes 12 that contains a central reflection and possesses the bipolar axis, offers a similar possibility: recognition of two oppositely oriented polar friezes with the symmetry group 11, which produce the visual impression of two-way motion.

Since the discrete symmetry groups m1, mg and mm and the continuous groups m₀1 and m₀m contain reflections with reflection lines perpendicular to the frieze axis, friezes corresponding to them belong to a family of friezes with a non-polar axis. Enantiomorphic modifications of friezes with the discrete symmetry groups 1g, m1, 1m, mg and mm and continuous visually presentable symmetry groups m₀1 and mm, which contain indirect isometries - reflections or glide reflections - does not occur.

Data on the polarity of friezes and enantiomorphism may be a basic indicator of the static or dynamic visual properties of friezes. Non-polar friezes produce static, while polar friezes produce a dynamic visual impression. Another component of a dynamic visual impression produced by a certain frieze, may be the presence of a glide reflection, suggesting the impression of alternating motion. The enantiomorphism is the "left" or "right" orientation of a frieze - the existence of only "left" or "right " homologous asymmetric elementary figures or fundamental regions. According to the principle of maximal symmetry, those symmetry groups of friezes with a high degree of symmetry, mg and mm, prevail. Among them, more frequent are static friezes with the symmetry group mm. Their distinctive stationariness results from the fact that each of them may be placed in such a position that the reflection lines corresponding to the rosettal subgroup D₂ (2m) coincide with the fundamental natural directions - the vertical and horizontal line. For the maximal continuous symmetry group of friezes m₀m that may be visually modeled by a straight (horizontal) line, a similar argument hold.

A table survey of subgroups of the symmetry groups of friezes and their decompositions (reducibility) offers complete evidence of their symmetry substructures. Through its use, a visual recognition of friezes and rosettes that a particular frieze contains, will be simplified. Certainly, static substructures with a higher degree of symmetry may be easily perceived and visually recognized, but low-symmetry substructures demand its use. Besides the visual simplicity of substructures, possibilities for their visual recognition will be caused by all the other elements taking part in the formation of a visual impression: the visual qualities of suprastructure, visual simplicity, stationariness or dynamism, the relation of substructures to vertical and horizontal line, to the surrounding, to the observer, etc.

Cayley diagrams of the symmetry groups of friezes are another suitable visual interpretation. Besides pointing out characteristics of generators, relations that consist of the presentation of the corresponding symmetry group and its structure, they indicate the visual qualities of the corresponding friezes. In the tables of the graphic symbols of symmetry elements, similar such information may be given.

The symmetry groups of friezes G₂₁ are the simplest category of the infinite groups of isometries. In the development of the generalizations of the theory of symmetry - antisymmetry and colored symmetry, they had a significant role. Since visual models are the most obvious interpretation of abstract geometric-algebraic structures, among infinite discrete symmetry groups, friezes are the simplest and most suitable medium for analyzing such generalizations.

Through knowledge of the geometric-algebraic properties of the symmetry groups of friezes, the visual qualities of the corresponding friezes may be anticipated directly from the presentations and structures of their symmetry groups. This opens a large field for ornamental design; for the planning of visual effects produced by friezes; and for aesthetic analyses based on exact grounds. The presence of generators of infinite order in the symmetry groups of friezes and their possible identification with time, results in the occurrence of the time component, representing for ornamental art the possibility to suggest motion.

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