The continuous symmetry group D_{¥} (¥m), the symmetry group of a circle, has a relative priority in ornamental art, both in the frequency of occurrence and in chronology. Among the earliest art forms (such as bone engravings, stone carvings and drawings from the late period of the Paleolithic, the Magdalenian, 12000-10000 B.C.), it is possible to find examples of the oldest rosettes. Among them, the most important is the circle - a rosette with maximal symmetry.
If we accept the idea that a man, in his early childhood, is repeating the phases of the development of humans, then the first drawings of a child from the "scribbling" phase, which often show a distinct circular structure, suggest the primariness of the circle as a geometric and visual form. In his discussion of the appearance of circles and circular forms in the drawings of children, Rudolph Arnheim said that a child discovered the shape of a circle lead by "the tendency toward a simple form in the visual and motor behavior", and not by imitating round objects. Having maximal symmetry, the circle satisfies the "principle of uniformity", which is in the basis of the entire activity of nature, and according to which "asymmetry is reduced in isolated systems".
This principle can be applied also to the visual field as the principle of visual entropy - maximal symmetry, visual and constructional simplicity. By the principle of visual entropy we understand an affirmation of the universal natural principle of economy. In the visual arts it will be expressed through domination of "well-behaved forms" (R. Arnheim, 1965). In ornamental art, the afore mentioned "good behavior" mostly corresponds to the notion of symmetry. Thanks to their uniform structure, such forms offer a possibility for their visual perception as integral entities ("gestalts"). In the process of visual thinking they ask for a minimum of effort, thus satisfying the principle of economy, i.e. the principle of maximal visual simplicity. Although ornamental art does not make use of exact geometric constructions, it uses its own artistic construction methods - the artistic laws of composition. The process of the artistic "construction" of an ornament consists of the creative anticipation of what it will look like, by choosing an elementary asymmetric figure and rules for its regular multiplication. Having in mind the complexity of the process of creating ornamental motifs, in the early phases of ornamental art the motifs satisfying the principle of maximal constructional simplicity, based on the simplest geometric regularities, were used first. Forms with maximal symmetry often satisfy the principles of maximal visual and constructional simplicity as well. This is the reason the principle of visual entropy combines three notions: the principles of maximal visual simplicity, maximal constructional simplicity and maximal symmetry. According to their connections, mutual dependence and the inseparability of these notions, figures satisfying one of these requirements usually satisfy the rest as well.
Owing to the completeness, compactness, boundness and uniformity of its structural segments, the circle in its primary sense does not only designate "roundness" but offers a possibility to designate any other abstract form - a unit. Associated to different possible meanings, the circle becomes the symbol of the Sun, completeness and perfection, and remains that throughout history (Figure 2.2).
Variations of the Sun symbol (Paleolithic, Neolithic and Bronze Age). |
In ornamental art, the circle a leading element, either as independent or in combination with concentric circles, or as the basis upon which some concentric rosette of a lower degree of symmetry can be added. In such a desymmetrization, the newly formed rosette - the result of the superposition - possesses the symmetry of the rosette that has caused the desymmetrization. In this process, the circle plays the role of the neutral element.
Among the symmetry groups of the type D_{n} ( nm), the group D_{1} (m) is most frequent in ornamental art. Usually, it is presented by a vertical or horizontal line segment, to which in the geometric sense corresponds symmetry group D_{2} (2m), and in the visual sense group D_{1} (m). Discussing a vertical line segment, S. Langfeld (R. Arnheim, 1965, pp. 20) says: "If one is asked to bisect a perpendicular line without measuring it, one almost invariably places the mark too high. If a line is actually bisected, it is with difficulty that one can convince itself that the upper half is not longer then the lower half.". Therefore, from the visual point of view, a vertical or horizontal line segment possesses the symmetry group D_{1} (m). The primariness of these two directions is governed by their meaning in the physical world. The symbolic meanings of a vertical line are: a man walking upright, growth, upright trees, firmness, activity, while the meanings of a horizontal line are: earth, matter, passiveness, rest, sleep, death,¼ Most of them result from the physiological organization of man, its perpendicular attitude toward the base and its plane symmetry. This is the reason, in drawings, reflections are usually vertical, and very seldom horizontal (Figure 2.3-2.8).
The rosette with symmetry D_{1} (m) (Paleolithic, El Pendo, Spain). |
Formation of rosettes with the symmetry group D_{2} (2m) by a superposition of rosettes with the symmetry group D_{1} (m) (Paleolithic, France). |
Examples of rosettes with the symmetry group D_{1} (m) (Paleolithic, France and Spain). |
Stylized human faces with the symmetry D_{1} (m) (Paleolithic and Neolithic of Europe). |
Stylized human figures with the symmetry D_{1} (m) (Paleolithic and Neolithic of Italy and Spain). |
Stylized vehicle motifs with the symmetry D_{1} (m). |
Rosettes with the symmetry group D_{1} (m) have a dominant position in ornamental art. In the visual sense, the best illustrations of the importance of mirror symmetry are different forms of two-headed mythological animals: the two-headed snake of the Maya Indians, Amphisbaena, the two-headed eagle or other motifs found in heraldry, or rosettes with motifs showing only one head and a "development" of the body (Figure 2.9, 2.24c).
Examples of the symmetry group D_{1} (m): (a) the two-headed winged lion (Tell Hallaf, about 5000 B.C.); (b) formation of motifs with the symmetry D_{1} (m) (cave paintings, La Pileta, Spain); (c) art of the pre-dynastic period of Egypt; (d) motif from the Ionian amphora; (e) two-headed Mayans snake; (f) primitive art of the American Indians. |
Very frequent is the substitution of a reflection by a visually suggested 2-rotation axis belonging to the plane of drawing (e.g. H. Weyl, 1952, pp. 9, Nermer's Palette). For the symmetry of rosettes in E^{2} such motifs are asymmetrical. In the case of a presentation of two identical figures arranged according to the symmetry group D_{1} (m), they are in the same vertical plane, at the same distance from the observer, giving the impression of balance and stationariness, without suggesting a space component. A visual suggestion of the 2-rotation axis in the plane of drawing implies the relation ïn front-behind", so there is another element of painting - space (Figure 2.10). This phenomenon is especially present in interlaced rosettes and similar layered ornamental motifs (B. Grünbaum, G.C. Shephard, 1983; B. Grünbaum, Z. Grünbaum, G.C. Shephard, 1986; B. Grünbaum, G.C. Shephard, 1987).
The example of the substitution of the reflection m by a suggestion of the 2-rotation axis in the plane of the figure (Egypt). |
By combining a vertical and horizontal line we get the sign of the cross, which possesses another fundamental characteristic - perpendicularity. Depending upon its construction, this symbol has three different symmetry aspects: D_{1} (m), D_{2} (2m) and D_{4} (4m) (Figure 2.11).
The cross motif - rosettal symbol with the symmetry group D_{1} (m), D_{2} (2m) or D_{4} (4m). |
Rosettes with the symmetry groups D_{2} (2m) (Figure 2.13), D_{3} (3m) (the equilateral triangle), D_{4} (4m) (the square) (Figure 2.12, 2.13), D_{6} (6m) (the regular hexagon) etc., based on the symmetry of regular polygons, are very frequent. In the later periods of ornamental art they occur mostly with plant or geometric ornamental motifs.
The rosette with the symmetry group D_{4} (4m) occurring in Paleolithic art. |
The rosettes with the symmetry groups D_{2} (2m), D_{4} (4m) (Paleolithic, Maz d'azil). |
Although the principle of crystallographic restriction (n=1,2,3,4,6) is not respected in ornamental art, rosettes with the symmetry groups 1(m), 2(m), 3(m), 4(m), 6(m) prevail over rosettes with the symmetry group 5(m), 7( m), 9(m), etc., probably because of a simpler construction of the corresponding regular polygons. For practical reasons, rosettes with rotations of a higher order occur very seldom (Figure 2.14-2.20).
The rosette with the symmetry group C_{2} (2) (Paleolithic, Magdalenian, around 10000 B.C., Laugerie Basse, France). |
The spiral rosette with the symmetry group C_{2} (2) (Paleolithic, Magdalenian, around 10000 B.C., Mal'ta, USSR). |
Examples of rosettes of the type C_{n} (n) and D_{n} (nm) in the Neolithic ceramics of Middle Asia: (a) C_{4} (4); (b) C_{6} (6); (c) C_{5} (5); (d) D_{4} (4m); (e) C_{4} (4); (f) D_{4} (4m); (g) C_{4} (4) ((a)-(d), (g) Samara; (e), (f) Susa; around 5500-5000 B.C.). |
Examples of rosettes of the type C_{n} (n) and D_{n} (nm) in the Neolithic ceramics of Middle Asia (Susa, Hacilar, Catal Hüjük, Hallaf, Eridu culture), around 6000-4500 B.C. (7500-5000 B.C.?). |
Rosettes with the symmetry group C_{2} (2) and D_{4} (4m): (a) Hacilar, about 6000 B.C.; (b) Aznabegovo-Vrshnik, Yugoslavia, around 5000 B.C. |
Examples of rosettes of the type C_{n} (n) and D_{n} (nm) in European ancient art: (a) C_{4} (4), Neolithic, Rumania; (b) C_{4} (4), Knossos, Crete, around 2500 B.C.; (c) C_{10} (10), Mycenae; (d) C_{4} (4), Crete; (e) D_{3} (3m), Mycenae. |
The rosettes of the type C_{n} (n) and D_{n} (nm) in the art of Egypt: (a) pre-dynastic; (b) dynastic period. |
In deriving rosettes with the symmetry group C_{n}
(n) from rosettes with the symmetry group D_{n}
(nm), the desymmetrization method can be used. The relationship
[D_{n}:C_{n}] = 2 holds. Even from the Neolithic, for
obtaining rosettes with the symmetry group C_{n}
(n), the antisymmetry group D_{n}/C_{n} has been used
(Figure 2.21, 2.27).
Examples of antisymmetry rosettes with the antisymmetry group D_{8}/C_{8}, that in the classical theory of symmetry are treated as rosettes with the symmetry group C_{8} (8) (Hajji Mohammed, around 5000 B.C.). |
Variations of the (a) Chinese symbol "yang-yin" with the symmetry group C_{2} (2) and (b) the triquetra motif with the symmetry group C_{3} (3). |
The rosette with the symmetry group C_{4} (4) (ceramics of the Neolithic of Middle Asia, around 6000 B.C.). |
Examples of rosettes with the symmetry group (a) C_{3} (3); (b) C_{4} (4); (c) D_{1} (m) in the art of American Indians. The rosette (c) represents an example of the "development" of the body, that results in the symmetry D_{1} (m). |
Examples of rosettes with the symmetry group (a) C_{4} (4), 15th century; (b) C_{3} (3), with the dominant decorative component. |
Owing to the reflections, rosettes with the symmetry group D_{n} (nm) produce a visual impression of balance and stationariness, where one of the reflections is supposed to be vertical. According to the principle of visual entropy - the tendency toward a high degree of symmetry - rosettes with the symmetry group D_{n} (nm) will be more frequent than rosettes with the symmetry group C_{n} (n). They have no enantiomorphic modifications - the oriented "left" and "right" rosettes with a singular direction of rotations.
Distinct from the previous type, type C_{n} (n) consists of the symmetry groups to which correspond visually dynamic rosettes with a polar singular direction of rotation, occurring in two enantiomorphic modifications. For ornamental art, enantiomorphic ("left" and "right") modifications of rosettes, which produce an impression of the "left" and "right" rotations, introduce the possibility to suggest motion. Depending on the orientation, the same enantiomorphic rosettes may even have different symbolic meanings. The suggestion of motion can be stressed by using forms with acute angles oriented toward the direction of rotation.
Typical examples of rosettes with the symmetry group C_{n} (n) are the triquetra (C_{3} (3)) (Figure 2.22b), the swastika (C_{4} (4)) (Figure 2.23) and similar motifs represented in different civilizations (e.g. Babylonian, Chinese, Aegean and Mayan) a symbol of the Sun, and lastly the Chinese symbol "yang-yin" (C_{2} (2)) symbolizing a dynamic balance between the male and female principle (Figure 2.22a), etc.
In the early phases of ornamental art, these symbols were used mostly in their simplest form. Further development led toward more complexity, and the enrichment and variation of basic elementary asymmetric figures belonging to a fundamental region, which were multiplied by symmetry transformations. For the symmetry groups of the type C_{n} (n), a variety of rosettes can be achieved by using a curvilinear fundamental region, while with the symmetry groups of the type D_{n} (nm), generated by reflections, the fundamental region must be rectilinear.
Even in early ornamental art (e.g., in the ceramics of the American Indians before Columbus) with its very complicated geometric ornaments, there are no deviations from the strict principle of symmetry (Figure 2.24a, b).
Interesting examples are derived by superposing concentric rosettes with different symmetry groups. In such a case, the symmetry of the system is the symmetry of the least symmetrical rosette belonging to the composition or some of its subgroups, usually non-trivial (Figure 2.19e, 2.23).
In time, the symbolic meanings of rosettes, which in the beginning of ornamental art had played a role as a specific means of communication, were also lost. This way, rosettes became only decorations, and they remained so till modern times (Figure 2.25).
In the modern age, aiming for the simplest possible means of communication - mainly visual - the modern designer has developed a whole system of signs (traffic signs, trade marks, etc.) which have the symmetry of rosettes. Also, by a multidisciplinary approach uniting ornamental art, the theory of symmetry, and the sciences which need for the visual modeling of rosettal symmetry structures (Crystallography, Physics, Chemistry,¼), rosettes gained new meanings.
Rosettes date back to the period of Paleolithic art and represent one of the oldest examples of the human aim to express regularity and symmetry. For the mathematical theory of symmetry, they are the simplest basis for an adequate mathematical treatment of ornaments, based on the theory of symmetry, and the record of the historic development going from visuality to the mathematical abstraction. Even to those acquainted with the theory of symmetry, rosettes remain the most evident visual illustration of the visually presentable symmetry groups C_{n} (n), D_{n} (nm), D_{¥} (¥m). By analyzing rosettes from the point of view of the theory of symmetry, it is possible to note the common characteristics of rosettes and create a link between presentations of the symmetry groups of rosettes, their structures and the visual properties of corresponding rosettes. By using the principle of visual entropy, it is possible to establish relations between the maximal constructional and visual simplicity and maximal symmetry on the one side, and the period of origin, frequency of occurrence and variety of rosettes on the other. Because of this, we have the early appearance and dominance of rosettes that satisfy this principle.
A survey on the symmetry groups of rosettes and the group-subgroup relations can serve as a basis for the construction of rosettes by the desymmetrization method. These relations are schematically shown in Figure 2.26, where an arrow designates the group-subgroup relation, and an attached symbol the index of the subgroup in the group. These relations determine the possibilities available to the classical-symmetry, antisymmetry (for subgroups of the index 2) and color-symmetry desymmetrizations or their combinations, aiming to obtain rosettes of a lower degree of symmetry.
In the table of antisymmetry desymmetrizations, symbols of antisymmetry groups are given in the group/subgroup notation G/H (V.A. Koptsik, 1966; A.M. Zamorzaev, 1976; H.S.M. Coxeter, 1985). The group/subgroup notation G/H gives information on the generating symmetry group G and its (normal) subgroup H of the index 2, derived by the antisymmetry desymmetrization (Figure 2.27). The relation G/H @ C_{2} holds.
Antisymmetry rosettes in Neolithic ornamental art: (a) D_{4}/D_{2}, Danilo, Yugoslavia, about 3500 B.C.; (b) D_{6}/D_{3}, Near East; (c) D_{4}/D_{2}, Near East; (d) D_{4}/C_{4}, Middle East; (e) C_{8}/C_{4}, Middle East; (f) C_{4}/C_{2}, Dimini, Greece. |
The table of antisymmetry desymmetrizations of symmetry groups of rosettes G_{20}:
D_{2n} / D_{n} |
D_{n} / C_{n} |
C_{2n} / C_{n} |
In the table of color-symmetry desymmetrizations the symbols of colored symmetry groups G^{*} are given in the notation G/H/H_{1} (A.M. Zamorzaev, E.I. Galyarski, A.F. Palistrant, 1978; V.A. Koptsik, J.N. Kotzev, 1974). In the symbol G/H/H_{1}, the first datum denotes the generating symmetry group G, the second gives the stationary subgroup H of the colored symmetry group G^{*} , which consists of transformations maintaining an individual index (color) unchanged, while the third denotes the symmetry subgroup H_{1} of the colored symmetry group G^{*} . The subgroup H_{1} is the result of the color-symmetry desymmetrization. A number N (N ³ 3) is the number of "colors" used to derive the colored symmetry group. For H = H_{1}, i.e. iff H is a normal subgroup of the group G, the symbol G/H/H_{1} is reduced to the symbol G/H.
The table of color-symmetry desymmetrizations of symmetry groups of rosettes G_{20}:
N | |
D_{an} / D_{n} / C_{n} | a |
D_{an} / C_{n} | 2a |
C_{an} / C_{n} | a |
So that, by color-symmetry desymmetrizations of the symmetry groups of rosettes, it is possible to obtain exclusively symmetry groups of the type C_{n} (n).
Group-subgroup relations can also serve as an indicator of the frequency of occurrence of certain rosettes in ornamental art, in line with the tendency toward maximal visual simplicity and symmetry. The full expression of the principle of visual entropy is the very frequent use of circles - visual models of the maximal continuous symmetry group of rosettes D _{¥} (¥m). The same is a reason for the domination of the dihedral symmetry groups of rosettes D_{n} (nm) over the corresponding cyclic groups C_{n} (n), resulting from the relationship [D_{n}:C_{n}] = 2. The priority of rosettes with the symmetry group D_{2n} (2nm) over rosettes with the symmetry group D_{n} (nm), or rosettes with the symmetry group C_{2n} (2n) over rosettes with the symmetry group C_{n} (n) ( [D_{2n}:D_{n}] = 2, [ C_{2n}:C_{n}] = 2 ), may be reduced to the question of the existence of a central reflection as the element of the symmetry group. A similar situation holds for all other group-subgroup relations.
For larger values of n, rosettes with the symmetry group D_{n} (nm) or C_{n} (n) are rare. However, since the process of visual perception often results in a visual, subjective symmetrization of these rosettes, perceived by the observer as circles, even in such a case the principle of maximal symmetry is respected, but only in the sense of visual perception.
The causes of the very early appearance and frequent occurrence of rosettes with the symmetry groups D_{1} ( m) and D_{2} (2m) are mainly of a physical-physiological nature: D_{1} (m) - human symmetry and binocularity, D_{2} (2m) - the relation between a vertical and horizontal line and perpendicularity of the reflections. The origins of rosettes with the symmetry groups D_{4} (4m), D_{6} (6m), D_{8} (8m), D_{12} (12m) can be found in the relation vertical-horizontal, perpendicularity of the reflections (because the symmetry group D_{2} ( 2m) is the subgroup of all the symmetry groups mentioned) and constructional simplicity, while the frequent use of rosettes with the symmetry groups 3(m), 5( m),¼ results from their constructional simplicity. A considerable influence is the existence of models in nature: D_{1} (m ) - the symmetry of almost all living beings, D_{2} (2m ) - vertical and horizontal line, D_{5} (5m) - a starfish, D_{6} (6m) - a honeycomb, D_{¥} (¥m) - all circular forms found in nature, etc.
The geometric basis of rosettes with the symmetry group D_{n} (nm) or C_{n} (n) is the construction of regular polygons, which is possible iff the number of the sides of the polygon is of the form: 2^{m}p_{1}p_{2}¼p_{n}, where p_{1}, p_{2},¼, p_{n} are prime Fermat numbers, i.e. prime numbers of the form 2^{2n}+1, m Î N È{0} and n Î N.
The visual impression produced by a certain rosette will be influenced also by the enantiomorphism - the "left" and "right" orientation of rosettes, representing an important part of the general problem of orientation in nature (H. Weyl, 1952; R. Arnheim, 1965).
From the point of view of ornamental art, of special interest are the symmetry groups D_{n} (nm) with n - an even natural number. Since one of their subgroups is the symmetry group D_{2} (2m), there is the possibility to place the rosette with the symmetry group D_{n} (nm) in such a position that the perpendicular reflections of the subgroup D_{2} (2m) coincide with the fundamental natural perpendiculars - vertical and horizontal lines.
A complete table survey of subgroups of the given symmetry group presents, in the geometric and also in the visual sense, the evidence of their symmetry substructures. Possibilities for their visual recognition depend on the nature of the substructures themselves, their impressiveness, visual simplicity, dynamism or stationariness, the nature of the rosette to which they belong and on the position of the substructure regarding the visual dominants - reflections, vertical and horizontal lines, etc. Necessary data can be also supplied by a survey giving decompositions of symmetry groups of rosettes.
Cayley diagrams are visual interpretations of symmetry groups of rosettes, giving complete information on symmetry groups. According to the established connection between the geometric-algebraic properties of the symmetry groups of rosettes and their visual models, many important visual characteristics of a rosette are implied by the structure of their symmetry group.
In the development of the theory of symmetry very important have been the visual interpretations of symmetry groups: ornaments, graphic symbols of symmetry elements and Cayley diagrams. Visual examples have been the motives for further analysis and discussion on the corresponding symmetry groups. In a modern science, instances of a reversed process - from abstract groups to their visual models - are frequent, especially in those cases where theory precedes the practice.
The discussion of the visual characteristics of rosettes given in this work, based on the theory of symmetry, can be used also in ornamental design, for the construction of new rosettes through anticipating their visual properties, and as a basis for exact aesthetic analyses. Also, they can be applied in all scientific fields in needing of visual interpretations of symmetry groups.