# Conformal Symmetry Rosettes   and Ornamental Art

An inversion transforms a point A of the plane E2\{O} onto the point A' of the same plane, where the vector relationship (O,A)°(O,A') = r2 holds, and r is the length of a radius of the inversion circle. An important characteristic of the inversion RI is the property of equiangularity - the maintenance of the angle between two arbitrary vectors in the plane E2\{O}, transformed by the inversion RI. The non-metric construction of the inverse point RI(A) of a point A is based on the fact that the base point of the hypotenuse altitude and a vertex that belongs to the hypotenuse are the homologous points of the inversion in the circle with the center in the other vertex of the hypotenuse and with the radius equal to the cathete to which this vertex belongs (Figure 4.2). All the lines containing point O and points of the inversion circle mI are, respectively, the invariant lines and points of the inversion RI. Circle lines containing the point O are transformed onto the lines that do not contain the point O, circle lines that do not contain the point O are transformed onto the circle lines that do not contain the point O, while to all the lines not containing the point O correspond the circle lines containing the point O. Hence, the following relationships hold: RI(A) = A iff A Î mI, RI(l) = l iff O Î l, RI(l) = c iff O not Î l and O Î c, RI(c) = c1 iff O not Î c and O not Î c1.

Every circle perpendicular to the inversion circle mI is transformed by the inversion RI onto itself and represents its invariant, so the relationship RI(c) = c iff c^ mI holds. All the constructions in which an inversion takes part can be considerably simplified by using those invariance relations - the invariance of all the points of the inversion circle mI, of lines containing the singular point O and of circles perpendicular to the inversion circle mI. Very important for the simplification of constructions is the fact that every circle line containing the point O and touching the inversion circle mI, is transformed onto the tangent line of the circle mI in the touch point and vice versa, and also every secant of the inversion circle mI is transformed onto the circle line containing the singular point O and the intersection points of the secant and inversion circle. The reverse transformations also hold. Owing to those characteristics of the non-metric construction of inverse figures, according to the criterion of maximal constructional simplicity, the metric construction was rarely used.

Figure 4.2
 The construction of homologous points of the inversion RI.

Every conformal symmetry group of the type CnRI is the direct product of the symmetry groups Cn (n) and RI. Hence, visual interpretations of conformal symmetry groups of the type CnRI (Figure 4.3, 4.4) can be constructed multiplying by the inversion RI a rosette with the symmetry group Cn (n), belonging to a fundamental region of the group RI, or multiplying by the n-fold rotation a figure with the symmetry group RI, belonging to the fundamental region of the group Cn (n). A fundamental region of the group CnRI is the section of the fundamental regions of the groups Cn (n) and RI. Owing to the presence of the indirect symmetry transformation - inversion RI - in groups of the type CnRI enantiomorphic modifications do not occur. The visual effect and degree of visual dynamism in rosettes with the conformal symmetry group CnRI depend exclusively on the choice of the form of a fundamental region, or on the position and form of an elementary asymmetric figure within a fundamental region. A degree of visual dynamism goes from conformal symmetry rosettes alike to rosettes with the symmetry group Cn (n) (Figure 4.3), to the conformal symmetry rosettes with a fully expressed stationary visual component resulting from the visual effect of the inversion RI, similar to the visual effect of a reflection. Those "static" conformal symmetry rosettes with the group CnRI can be constructed by using an asymmetric figure, with its shape very close to the inversion circle, or by using a fundamental region of a similar form (Figure 4.4). A fundamental region of the group CnRI offers a change of non-inversional boundaries, i.e. boundaries that do not belong to the inversion circle mI. This is the only restriction to the choice of a fundamental region, since the invariance of all the points of the inversion circle must be preserved.

Figure 4.3
 Conformal symmetry rosettes with the symmetry groups of the type CnRI, which satisfy the principle of maximal constructional simplicity.

Figure 4.4
 Examples of conformal symmetry rosettes with the symmetry groups of the type CnRI, with dominant static visual component produced by the inversion RI.

Owing to their low degree of symmetry and visual dynamism conditioned by the polarity of rotations, conformal symmetry rosettes with the symmetry group CnRI are very rare in ornamental art. Such examples as exist are constructed mostly by using half-circles containing the singular point O, touching the inversion circle and forming a rosette with the symmetry group Cn (n). They are transformed by the inversion RI onto the corresponding half-tangents in the touch points (Figure 4.3). In ornamental art, the frequent use of that construction is dictated by the principle of maximal constructional and visual simplicity, while other aspects of conformal symmetry rosettes with the symmetry group CnRI are rarely found in ornamental art. Conformal symmetry groups of the type CnRI can be obtained by desymmetrizations of groups of the type DnRI, the most frequent discrete conformal symmetry groups of the category C21 in ornamental art. Besides classical-symmetry desymmetrizations, frequently occurring is the antisymmetry desymmetrization resulting in the conformal antisymmetry group DnRI/CnRI, which in the classical theory of symmetry can be discussed as the group CnRI.

Every group of the type DnRI (Figure 4.5-4.7) is the direct product of the symmetry groups Dn (nm) and RI. Hence, visual interpretations of the group DnRI can be constructed multiplying by the inversion RI a rosette with the symmetry group Dn (nm), belonging to a fundamental region of the group RI or, less frequently, multiplying by symmetry transformations of the group Dn (nm) a figure with the symmetry group RI, belonging to a fundamental region of the group Dn (nm). A fundamental region of the group DnRI is the section of fundamental regions of the groups Dn (nm) and RI. In groups of the type DnRI, there are no enantiomorphic modifications.

Conformal symmetry rosettes with the symmetry group DnRI have visual characteristics similar to that of generating rosettes with the symmetry group Dn (nm). Owing to the presence of reflections and inversions, these conformal symmetry groups belong to the family of visually static symmetry groups with non-polar rotations.

Since groups of the type DnRI are generated by reflections (reflections and inversions), there is no possibility for changing boundaries of a fundamental region. Owing to the fixed shape of a fundamental region, tilings corresponding to the group DnRI, for fixed n, are reduced to only one figure (Figure 4.5). In ornamental art, the variety, richness and visual interest of conformal symmetry rosettes with the symmetry group DnRI, is achieved by applying different elementary asymmetric figures within a fundamental region. The visual effect of the inversion RI, within the group DnRI depends on the shape of that elementary asymmetric figure and its position within a fundamental region. The static visual function of the inversion RI comes to its full expression for an elementary asymmetric figure, by the shape and position being very close to the inversion circle.

Figure 4.5
 Examples of conformal symmetry rosettes with the symmetry groups of the type DnRI, constructed according to the principle of maximal constructional simplicity.

In ornamental art, there are many examples of conformal symmetry rosettes with symmetry groups of the type DnRI. The frequency of occurrence of a particular group depends on the frequency of occurrence of its generating group Dn (nm). Therefore, the groups DnRI, for n - an even natural number, especially for n = 2,4,6,8,12..., occur most often. These groups satisfy the principle of visual entropy and offer the possibility of choosing the position of a corresponding conformal symmetry rosette, such that its reflection lines coincide with the fundamental natural directions - the vertical and horizontal line.

Owing to maximal constructional simplicity, groups of the type DnRI have a special role in ornamental art. Very interesting visual interpretations of these groups are obtained, reproducing by the inversion RI a rosette with the symmetry group Dn (nm), constructed by circles (or their arcs) containing the singular point O and touching the inversion circle mI. The inversion RI transforms these circles (arcs) onto the tangent lines (parts of tangent lines) of the inversion circle in the touch points (Figure 4.6, 4.7). These conformal symmetry rosettes are used in ornamental art by almost all cultures. They have a special place in Romanesque and Gothic art, within rosettes used in architecture.

Figure 4.6
 Examples of conformal symmetry rosettes with the symmetry groups of the type DnRI, which are used in ornamental art.

The continuous conformal symmetry group D¥ RI possesses adequate visual interpretations. One of them is a circle. Regarded from the point of view of the isometric theory of symmetry, a circle possesses the continuous symmetry group of rosettes D¥ , but for conformal symmetry, its symmetry group is D¥ RI. Such a possibility for different symmetry treatments of the same figure occurs in all situations when a certain theory (e.g., the isometric theory of symmetry) is extended to a larger, more general theory (e.g., the theory of conformal symmetry).

Figure 4.7
 Examples of conformal symmetry rosettes with the symmetry groups of the type DnRI, which are used in ornamental art.

The simplest group of the type CnZI is the group ZI (n = 1), generated by the inversional reflection ZI, the commutative composition of a reflection and an inversion. The inversional reflection ZI is an equivalent of a two-fold rotation with the axis belonging to the invariant plane of the symmetry groups of tablets G320. Those isomorphism between conformal symmetry groups of the category C21 and the symmetry groups of tablets G320, in which the inversional reflection ZI corresponds to this two-fold rotation, indicates the properties of the inversional reflection ZI - its involutionality and relations to the other conformal symmetry transformations.

Since the inversional reflection ZI can be represented in the form ZI = RRI = RIR, as the commutative product of the reflection R with the reflection line containing the singular point O and the inversion RI, many properties of the inversion RI (e.g., the property of equiangularity, etc.) and the construction methods given analyzing the symmetry group RI, hold and can be transferred to the inversional reflection ZI. A fundamental region of the group ZI can coincide with that of the group RI, but offers a change of the shape of all its boundaries.

Every group of the type CnZI (Figure 4.8, 4.9) is a dihedral group derived as a superposition of the groups Cn (n) and ZI. Their visual interpretations can be constructed multiplying by the inversional reflection ZI a rosette with the symmetry group Cn (n), belonging to a fundamental region of the group ZI or, less frequently, multiplying by the n-fold rotation a conformal symmetry rosette with the symmetry group ZI, belonging to a fundamental region of the symmetry group Cn (n). According to the relationship ZI = RRI = RIR, conformal symmetry rosettes with the symmetry group CnZI can be directly derived from conformal symmetry rosettes with the symmetry group CnRI, by reproducing by the reflection R with the reflection line defined by the singular point O and by the section of the boundaries of the fundamental region of the group CnRI with the inversion circle mI, one class of fundamental regions of the group CnRI (the internal or external fundamental regions) (Figure 4.8). Conformal symmetry groups of the type CnZI offer the possibility for the enantiomorphism. All the other properties of groups of the type CnRI can be attributed to groups of the type CnZI.

Figure 4.8
 Examples of conformal symmetry rosettes with the symmetry groups of the type CnZI, constructed according to the principle of maximal constructional simplicity.

Figure 4.9
 Examples of conformal symmetry rosettes with the symmetry groups of the type CnZI.

Figure 4.10
 Examples of conformal symmetry rosettes with the symmetry groups of the type NI, which satisfy the principle of maximal constructional simplicity.

Figure 4.11
 Examples of conformal symmetry rosettes with the symmetry groups of the type NI.

Like the symmetry groups of the type CnRI, the groups of the type CnZI can be derived by a desymmetrization of the symmetry groups of the type DnRI. Besides the classical-symmetry desymmetrizations, antisymmetry desymmetrizations resulting in conformal antisymmetry groups of the type DnRI/CnZI, discussed in the classical theory of symmetry within the type CnZI, can be obtained.

Groups of the type NI (Figure 4.10, 4.11) are generated by the inversional rotation SI, the commutative composition of the rotation S of the order 2n and the inversion RI, so that the relationship SI = SRI = RIS holds. Hence, conformal symmetry rosettes with the symmetry group NI can be directly derived from conformal symmetry rosettes with the symmetry group CnRI, by transforming by the rotation S one class of fundamental regions (internal or external fundamental regions) of the group CnRI. A fundamental region of the group NI may coincide with that of the group CnRI, but it allows the varying of all the boundaries. Conformal symmetry rosettes with the symmetry group NI can also be constructed multiplying by the inversional rotation SI a conformal symmetry rosette with the symmetry group Cn ( n), belonging to a fundamental region of the group RI. Groups of the type NI are an equivalent of the symmetry groups of tablets G320 of the type ([]), generated by n-fold rotational reflection, so that the relationship S1 = SI2 holds, where by S1 is denoted the n-fold rotation. Despite the choice of a fundamental region or an elementary asymmetric figure within a fundamental region, among all the conformal symmetry groups of the category C21, conformal rosettes with groups of the type NI produce the maximal degree of visual dynamism. Consequently, corresponding conformal symmetry rosettes are very rare in ornamental art. Owing to their maximal constructional simplicity, the most frequent are examples of conformal symmetry rosettes with the symmetry group NI, which consist of half-circles containing the singular point O and touching the inversion circle mI, and of corresponding half-tangent lines onto which these half-circles are transformed by the inversional rotation SI (Figure 4.10). The enantiomorphism does not occur in groups of the type NI.

Groups of the type NI can also be obtained by desymmetrizations of groups of the type C2nRI. Besides the classical-symmetry desymmetrizations, the antisymmetry desymmetrizations resulting in conformal antisymmetry groups of the type C2nRI/NI, discussed in the classical theory of symmetry within the type NI, are very frequent.

According to the relationship SI = SRI = RIS, where by S is denoted the rotation of the order 2n, a connection, analogous to that existing between the types CnRI and NI, exists between the types DnRI and D1NI. Hence, conformal symmetry rosettes with the symmetry group D1NI (Figure 4.12, 4.13) can be directly derived from conformal symmetry rosettes with the symmetry group DnRI, by reproducing by the rotation S one class of the fundamental regions (the internal or external fundamental regions) of the group DnRI. The same result can be obtained multiplying by the inversional rotation SI a conformal symmetry rosette with the symmetry group D1 (m), or multiplying by the reflection a conformal symmetry rosette with the symmetry group NI. All the other properties of groups of the type D1NI - the absence of the enantiomorphism, visual characteristics of corresponding conformal symmetry rosettes, etc. - are similar to the properties of groups of the type DnRI.

Groups of the type D1NI are isomorphic to the symmetry groups of tablets G320 of the type ([])m, generated by the n-fold rotational reflection and the reflection in the plane containing the tablet axis.

The variety of conformal symmetry rosettes with the symmetry group D1NI can be accomplished by the choice of the form of non-reflectional boundaries of a fundamental region or by using different elementary asymmetric figures belonging to a fundamental region. In that way, a large spectrum of the possible degree of visual dynamism can be achieved (Figure 4.12). It goes from static conformal symmetry rosettes with the symmetry group D1NI, without the application of an elementary asymmetric figure within a fundamental region, to conformal symmetry rosettes of a higher degree of visual dynamism, which can be, for instance, formed by circle and line segments alternating along a radial line.

By applying the desymmetrization method on groups of the type D2nRI, groups of the type D1NI can be obtained. Besides classical-symmetry desymmetrizations, the antisymmetry desymmetrizations resulting in the conformal antisymmetry groups of the type D2nRI/D1NI, discussed in the classical theory of symmetry within the type D1NI, can also be used.

The symmetry groups of polar rods G31, isomorphic to conformal symmetry groups of the category C2, can be derived by extending by the translation, twist and glide reflection, the symmetry groups of tablets G320, isomorphic to conformal symmetry groups of the category C21. Hence, conformal symmetry groups of the category C2 can be derived by extending by the similarity transformations K, L, M, conformal symmetry groups of the category C21.

Figure 4.12
 Examples of conformal symmetry rosettes with the symmetry groups of the type D1NI.

Figure 4.13
 Examples of conformal symmetry rosettes with the symmetry groups of the type D1NI, which are used in ornamental art.

According to the theorem that the product of two reflections with parallel reflection lines is a translation, the modulus of the translation vector of which is twice the distance between the reflection lines, in the field of conformal symmetry a dual theorem holds: the product of two circle inversions with concentric inversion circles is a dilatation with the dilatation coefficient k = r2/r12, where r, r1 are the lengths of the radii of the inversion circles. It indicates the possibility of deriving conformal symmetry groups of the category C2 that contain a dilatation, by using inversions in concentric inversion circles.

Figure 4.14
 The conformal symmetry rosette with the symmetry group K4I.

Conformal symmetry rosettes with the symmetry group KNI (Figure 4.14) can be constructed multiplying by the dilatation K (with k > 0) a conformal symmetry rosette with the symmetry group NI, belonging to a fundamental region of the group K. A fundamental region of the group KNI is the section of fundamental regions of the groups K and NI, and it allows a varying of the form of all the boundaries. Since the dilatation K is the element of every group of the type KNI, to efficiently construct the corresponding visual interpretations one applies two inversions in the concentric inversion circles. All the visual properties of a generating conformal symmetry rosette with the symmetry group NI, after the introduction of the dilatation K, are maintained in the derived conformal symmetry rosette with the symmetry group KNI. The dilatation K stimulates their visual dynamism, producing the suggestion of centrifugal expansion. Owing to their low degree of symmetry and to their visual properties mentioned, the corresponding conformal symmetry rosettes are very rare in ornamental art. Those examples that exist in ornamental art, in the first place respect the principle of visual entropy.

For a derivation of groups of the type KNI, desymmetrizations of groups of the type KC2nRI, somewhat more frequent in ornamental art, are also used. Besides the classical-symmetry desymmetrizations, the antisymmetry desymmetrizations resulting in the conformal antisymmetry groups of the type KC2nRI/KNI, discussed in the classical theory of symmetry within the type KNI, are frequent.

Figure 4.15
 The conformal symmetry rosette with the symmetry group M6I.

The type MNI consists of the groups derived by the superposition of the groups M and NI. Hence, corresponding conformal symmetry rosettes with the group MNI can be constructed multiplying by the dilative reflection M (with k > 0) a conformal symmetry rosette with the group NI, belonging to a fundamental region of the group M (Figure 4.15). A fundamental region of the group MNI is the section of fundamental regions of the groups M and NI, and allows a varying of all the boundaries.

After the introduction of the dilative reflection M, the visual properties of a generating conformal symmetry rosette with the symmetry group NI, remain unchanged. In the conformal symmetry rosette with the symmetry group MNI obtained, the presence of the dilative reflection M increases the visual dynamism, suggesting alternating centrifugal expansion. Since conformal symmetry rosettes with the symmetry group MNI belong to a family of dynamic, complicated conformal symmetry rosettes, the construction and symmetry of which is not comprehensible by empirical methods, they are very rare in ornamental art.

In aiming to obtain groups of the type MNI, the desymmetrization method can be applied on groups of the type KD1NI. Besides the classical-symmetry desymmetrizations, also the antisymmetry desymmetrizations resulting in antisymmetry groups of the type KD1NI/MNI, discussed in the classical theory of symmetry within the type MNI, can also be derived.

Figure 4.16
 The conformal symmetry rosette with the symmetry group KD12I.

Conformal symmetry rosettes with the symmetry group KD1NI (Figure 4.16) can be constructed multiplying by the dilatation K (with k > 0) a conformal symmetry rosette with the symmetry group D1NI, belonging to a fundamental region of the symmetry group K. A fundamental region of the group KD1NI is the section of fundamental regions of the groups K and D1NI, and it allows a varying of non-reflectional boundaries.

The visual properties of generating conformal symmetry rosettes with the symmetry group D1NI are maintained in derived conformal symmetry rosettes with the symmetry group KD1NI. The introduction of the dilatation K results in the appearance of a new dynamic visual component - the suggestion of centrifugal expansion. As in all the other cases of conformal symmetry groups of the category C2 containing a dilatation, for the construction of conformal symmetry rosettes with the symmetry group KD1NI, it is very efficient to make the use of two circle inversions with the concentric inversion circles. Owing to the static visual component produced by reflections and the non-polarity of rotations, examples of conformal symmetry rosettes with the symmetry group KD1NI are more frequent in ornamental art, than conformal symmetry rosettes with the symmetry group KNI or MNI.

By using the desymmetrization method, besides classical-symmetry desymmetrizations, the antisymmetry desymmetrizations of groups of the type KD2nRI, resulting in conformal antisymmetry groups of the type KD2nRI/KD1NI, discussed in the classical theory of symmetry within the type KD1NI, can be obtained.

The type KCnZI consists of conformal symmetry groups formed by the superposition of the groups K and CnZI. Corresponding conformal symmetry rosettes can be constructed multiplying by the dilatation K (with k > 0) a conformal symmetry rosette with the symmetry group CnZI, belonging to the fundamental region of the group K, or by applying an inversion with the inversion circle concentric to a conformal symmetry rosette with the symmetry group CnZI. A fundamental region of the group KCnZI is the section of fundamental regions of the groups K and CnZI, and allows a varying of all boundaries. Visual properties of a derived conformal symmetry rosette with the symmetry group KCnZI are similar to that of the generating conformal symmetry rosette with the symmetry group CnZI. By introducing a new visual dynamic component - an impression of centrifugal expansion - the dilatation K contributes to the increase in the visual dynamism of the conformal symmetry rosette derived, with respect to the generating conformal symmetry rosette. Due to their visual dynamism conditioned by the bipolarity of rotations, enantiomorphism, visual function of the dilatation K, etc., examples of conformal symmetry rosettes with the symmetry group KCnZI, are very rare in ornamental art (Figure 4.17).

By desymmetrizations of groups of the type KDnRI - the most frequent conformal symmetry groups of the category C2 in ornamental art - it is possible to derive groups of the type KCnZI. Besides classical-symmetry desymmetrizations, the antisymmetry desymmetrizations resulting in conformal antisymmetry groups of the type KDnRI/KCnZI, discussed in the classical theory of symmetry within the type KCnZI, can be obtained.

Figure 4.17
 The conformal symmetry rosette with the symmetry group KC4ZI.

The type LCnZI consists of conformal symmetry groups that are the result of the superposition of the groups L and CnZI. Visual examples of conformal symmetry groups of the type LCnZI can be constructed multiplying by the dilative rotation L (with k > 0) a conformal symmetry rosette with the symmetry group CnZI, belonging to a fundamental region of the group L (Figure 4.18). The visual effect of the dilative rotation L is the appearance of a dynamic spiral-motion component. For a rational angle q of the dilative rotation L, q = pp/q, (p,q) = 1, p,q Î Z, it is possible to divide a conformal symmetry rosette with the symmetry group LCnZI into sectors of the dilatation K((-1)pkq). A fundamental region of the group LCnZI is the section of fundamental regions of the groups L and CnZI. Hence, a varying of the shape of all the boundaries of a fundamental region is allowed. Since the dilative rotation L is a composite transformation, the relationship L = KS = SK holds, where a rotation with the rotation angle q is denoted by S. Conformal symmetry rosettes with the symmetry group KCnZI can be constructed by using a generating conformal symmetry rosette with the symmetry group CnZI, but such a construction is, in a certain degree, complicated. The conformal rosette mentioned, must be first transformed by an inversion with the inversion circle mI concentric to this rosette. After that, the image obtained must be transformed by the reflection R with the reflection line containing the singular point O, and finally, by the rotation S.

Figure 4.18
 The conformal symmetry rosette with the symmetry group LC3ZI.

Groups of the type LCnZI belong to a family of visually dynamic conformal symmetry groups with bipolar rotations, and with the possibility for the enantiomorphism. The impression of visual dynamism, suggested by the corresponding conformal symmetry rosettes, is greater than that suggested by the generating conformal symmetry rosettes with a symmetry group of the type CnZI. It is the result of the presence of the dilative rotation L, producing the visual impression of spiral-motion rotational expansion, and representing by itself a visual interpretation of a twist within the plane. The desired intensity of a visual dynamic impression can be achieved by varying the form of a fundamental region, applying different elementary asymmetric figures within a fundamental region and choosing the parameters k, q. In ornamental art, the variety of conformal symmetry rosettes with symmetry groups of the type LCnZI is restricted by the principle of visual entropy. Therefore, the most frequent conformal symmetry rosettes with the symmetry group LCnZI are constructed multiplying by a dilative rotation, the simplest conformal symmetry rosettes with the symmetry group CnZI (Figure 4.18). For the same reason, more frequent are conformal symmetry rosettes with symmetry groups of the type LCnZI, with a rational angle of the dilative rotation L. Very important in ornamental art are conformal symmetry rosettes with a symmetry group of the subtype L2nCnZI (L2n = L(k,p/n)). The symmetry group mentioned is the subgroup of the index 2 of the group LDnRI. According to the relationship K = L(k,0) = L0, a consequent application of the criterion of subordination requires also that the type KCnZI must be considered as the subtype of the type LCnZI. This would result in the complete elimination of the type KCnZI. A similar situation occurs in all the cases of overlapping types or individual conformal symmetry groups of the category C2. When solving such a problem, an approach analogous to that already discussed with the similarity symmetry groups of rosettes S20, can be used.

By desymmetrizations of groups of the type LDnRI, the corresponding groups of the subtype L2nCnZI, can be derived. Adequate antisymmetry desymmetrizations of groups of the type LDnRI result in conformal antisymmetry groups of the type LDnRI/ L2nCnZI, in the classical theory of symmetry included in the type LCnZI, as groups of the subtype L2nCnZI.

Conformal symmetry rosettes with the symmetry group KCnRI can be constructed multiplying by the dilatation K (with k > 0) a conformal symmetry rosette with the symmetry group CnRI, belonging to a fundamental region of the group K, or multiplying the same conformal symmetry rosette by an inversion with the inversion circle concentric with it (Figure 4.19). A fundamental region of the group KCnRI is the section of fundamental regions of the groups K and CnRI, and allows a varying of boundaries that do not belong to the inversion circles, i.e. a varying of radial boundaries. The visual effect of the conformal symmetry rosettes derived is very similar to that produced by the generating conformal symmetry rosette with the symmetry group CnRI. The introduction of the dilatation K representing the new dynamic visual component - the suggestion of centrifugal expansion - results in an increase in the visual dynamism. Owing to their dynamic visual qualities, we would expect that conformal symmetry rosettes with symmetry groups of the type KCnRI are not so frequent in ornamental art. However, the possibility of deriving conformal symmetry groups of the type KCnRI by desymmetrizations of groups of the type KDnRI, the most frequently used conformal symmetry groups of the category C2 in ornamental art, caused their more frequent occurrence. Besides classical-symmetry desymmetrizations, the antisymmetry desymmetrizations, resulting in the conformal antisymmetry groups of the type KDnRI/KCnRI, discussed in the classical theory of symmetry within the type KCnRI, can be obtained.

Figure 4.19
 The conformal symmetry rosette with the symmetry group KC4RI.

The type MCnRI consists of conformal symmetry groups derived by extending by the dilative reflection M (with k > 0) conformal symmetry groups of the type CnRI. Corresponding conformal symmetry rosettes can be constructed multiplying by the dilative reflection M a conformal symmetry rosette with the symmetry group CnRI, belonging to a fundamental region of the group M. The same conformal symmetry rosettes can be constructed transforming by an inversion with the inversion circle mI concentric to it, a generating conformal symmetry rosette with the symmetry group CnRI. Afterward, that image obtained we must to copy by a reflection in the reflection line containing the singular point O (Figure 4.20). The extension of the symmetry group CnRI by the dilative reflection M will result, in the visual sense, in the appearance of a new dynamic visual component - centrifugal alternating expansion. The dominance of dynamic components caused the relatively rare occurrence of conformal symmetry groups of the type MCnRI in ornamental art.

Figure 4.20
 The conformal symmetry rosette with the symmetry group MC8RI.

By desymmetrizations of groups of the type KDnRI, conformal symmetry groups of the type MCnRI can be derived. In particular, antisymmetry desymmetrizations, resulting in antisymmetry groups of the type KDnRI/MCnRI, discussed by the classical theory of symmetry within the type MCnRI, can be obtained.

The group KDnRI can be derived extending by the dilatation K the conformal symmetry group DnRI. Corresponding conformal symmetry rosettes can be constructed multiplying by the dilatation K a conformal symmetry rosette with the symmetry group DnRI, belonging to a fundamental region of the group K (with k > 0), or multiplying the same conformal symmetry rosette by an inversion with the inversion circle mI concentric to it (Figure 4.21). A fundamental region of the group KDnRI is the section of fundamental regions of the groups K and DnRI. Since the group KDnRI is generated by reflections (reflections and inversions), its fundamental region is fixed. Therefore, a fundamental region of the group KDnRI is defined by two successive reflection lines and two successive inversion circles, corresponding to this conformal symmetry group. The varying of conformal symmetry rosettes with the symmetry group KDnRI is reduced to the use of different elementary asymmetric figures belonging to a fundamental region and to a change in the value of the parameter k.

Figure 4.21
 The conformal symmetry rosette with the symmetry group KD6RI.

The effect of the dilatation K on a generating conformal symmetry rosette with the symmetry group DnRI is reduced, in the visual sense, to the increase in the visual dynamism and the suggestion of centrifugal expansion. Since there are a large number of models in nature with the symmetry group of rosettes Dn, with a high degree of constructional and visual simplicity and symmetry, and with the dominance of the static visual impression, conformal symmetry groups of the type KDnRI are the most frequent discrete conformal symmetry groups of the category C2, in ornamental art. Besides their individual use, groups of the type KDnRI form the basis for applying the desymmetrization method, aiming to derive the other types of conformal symmetry groups of the category C21.

Figure 4.22
 The conformal symmetry rosette with the symmetry group L12C6RI.

The group LCnRI can be derived extending by the "centering" dilative rotation L = L2n = L(k,p/n) (with k > 0) the conformal symmetry group CnRI. Hence, conformal symmetry rosettes with the symmetry group LCnRI can be constructed multiplying by the dilative rotation mentioned, a generating conformal symmetry rosette with the symmetry group CnRI, belonging to a fundamental region of the group L (Figure 4.22). The same conformal symmetry rosettes can be constructed by using an inversion with the inversion circle concentric to the generating rosette. In that case, after transforming it by the inversion, the image obtained must be rotated through the angle q = p/n. A fundamental region of the group LCnRI is the section of fundamental regions of the groups L2n and CnRI, and allows a varying of the form of non-inversional boundaries, while the remaining boundaries are defined by the concentric inversion circles (their corresponding arcs).

The influence of the dilative rotation L = L2n on a generating conformal symmetry rosette with the symmetry group CnRI results in the formation of the visual impression of a spiral motion. All the other visual properties of the generating conformal symmetry rosette remain unchanged. Because of the rational dilative rotation angle q = p/n, there are sectors of the dilatation K(kn).

In ornamental art, apart from by those construction methods, groups of the type LCnRI can be obtained by desymmetrizations of groups of the type LDnRI. Since the group LCnRI is the subgroup of the index 2 of the group LDnRI, besides classical-symmetry desymmetrizations, very frequent are antisymmetry desymmetrizations, resulting in antisymmetry groups of the type LDnRI/LCnRI, discussed in the classical theory of symmetry within the type LCnRI.

Figure 4.23
 The conformal symmetry rosette with the symmetry group LD4RI.

The group LDnRI can be derived extending the group DnRI by the "centering" dilative rotation L = L2n = L(k,p/n) (with k > 0). In ornamental art, conformal symmetry rosettes with the symmetry group LDnRI are very frequent, since they can be derived multiplying by the dilative rotation L = L2n a conformal symmetry rosette with a symmetry group of the type DnRI, the most frequent conformal symmetry group of the category C2, which belongs to a fundamental region of the group L (Figure 4.23). The same conformal symmetry rosettes can be constructed transforming by a circle inversion with the concentric inversion circle mI, the generating rosette mentioned. Afterward, the image obtained must be rotated through the angle q = p/n. A fundamental region of the group LDnRI is the section of fundamental regions of the groups DnRI and L. Hence, a varying of the form of a fundamental region is restricted to a change in the shape of boundaries that do not belong to the reflection lines or inversion circles.

The dilative rotation L produces, in the visual sense, a dynamic effect and gives the impression of a spiral motion. Although, since the "centered dihedral" similarity symmetry group LDn is the subgroup of the index 2 of the group LDnRI, there exists a specific balance between static and dynamic visual components in conformal symmetry rosettes with the symmetry group LDnRI, and even a dominance of the static ones. Because of the rational dilative rotation angle q = p/n , there are sectors of the dilatation K(kn).

In ornamental art, conformal symmetry rosettes with the symmetry group LDnRI are very frequently used as individual ones, or as a basis for applications of the desymmetrization method. Various examples are obtained by applying a different elementary asymmetric figure within a fundamental region or by varying the form of a fundamental region and the value of the parameter k.

The complete survey of continuous conformal symmetry groups of the categories C21 and C2 can be derived directly from the data on the continuous symmetry groups of tablets G320 and non-polar rods G31, respectively (A.V. Shubnikov, V.A. Koptsik, 1974). According to the restrictions imposed by ornamental art, if textures are not applied, visually presentable are continuous conformal symmetry groups of the category C21 of the type D¥RI, and continuous conformal symmetry groups of the category C2 of the types KD¥RI, K1CnRI, K1DnRI, L1D ¥RI and L1CnZI, where a continuous dilatation group and continuous dilative rotation group is denoted by K1, L1, respectively. In terms of ornamental art, the most interesting are continuous conformal symmetry rosettes with the symmetry group L1CnZI, which can be constructed multiplying by the n-fold rotation a logarithmic spiral - the invariant line of the continuous conformal symmetry group L1ZI. All the other visually non-presentable continuous conformal symmetry groups can be visually interpreted by using textures. Regarding the physical interpretations, all the continuous conformal symmetry groups of the categories C21 and C2 can be modeled in the plane E2\{O} , by means of physical desymmetrization factors (e.g., by the uniform rotation around the singular point O, by adequate physical fields, etc.), as was done with the continuous symmetry groups of tablets G320 and rods G31, isomorphic to them (A.V. Shubnikov, V.A. Koptsik, 1974).

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As with all the previously discussed symmetry groups occurring in ornamental art, a significant prerequisite for their early appearance and frequent use in ornamental art is that they satisfy the principle of visual entropy - maximal constructional and visual simplicity and maximal symmetry. For many ornamental art motifs, their origin and use is not directly a function of the existence of models in nature with the corresponding symmetry. This especially refers to conformal symmetry rosettes. Hence, probably the most significant of the three mentioned criteria conditioning the time of origin and frequency of occurrence of different conformal symmetry groups in ornamental art, is the principle of maximal constructional simplicity.

An inversion is a constituent part of all the conformal symmetry transformations, as an independent symmetry transformation, or as a component of the composite transformations - inversional reflection ZI or inversional rotation SI, the commutative compositions of an inversion with a reflection or with n-fold rotation. All the other conformal symmetry transformations of the plane E2\{O} belong to isometries or similarity symmetry transformations. Therefore, all the construction problems in conformal symmetry rosettes with a symmetry group of the category C21 or C2, can be solved, in principle, by using the non-metric construction to obtain homologous points of the inversion RI. Since a non-metric construction fully satisfies the criterion of maximal constructional simplicity, for the conformal symmetry transformations RI, ZI, SI, there is no reason to use the metric construction method.

The property of equiangularity is satisfied by every inversion and by all the other isometries and similarity symmetry transformations consisting of conformal symmetry groups. Hence, this property is an invariant of all the conformal symmetry groups. When constructing conformal symmetry rosettes with an elementary asymmetric figure of an arbitrary form, belonging to a fundamental region, as with the similarity symmetry groups of rosettes S2, a construction of the type "point by point" is unavoidable. Since such a construction is very complicated, the invariance of the points of the inversion circle mI and the fact that circles and lines are homologous figures of an inversion, expressed by the relationships: RI(A) = A iff A Î mI, RI(l) = l iff O Î l, RI(l) = c iff O not Î l and O Î c, RI(c) = c1 iff O not Î c and O not Î c1, RI(c) = c iff c^mI are a basis upon which we can simplify constructions of all conformal symmetry rosettes.

For conformal symmetry groups of the category C21, this is sufficient for the construction of their visual interpretations. Besides the fact that they can be constructed multiplying by similarity transformations K, L, M a generating conformal symmetry rosette with a symmetry group of the category C21, for a construction of the conformal symmetry groups of the category C2 it is possible to use an inversion with the inversion circle concentric to the generating rosette mentioned. Due to the simplicity of constructions by circles and lines, in many cases this is the most suitable construction method. Whatever the approach is, the construction of visual interpretations of conformal symmetry rosettes with a symmetry group of the category C2 is reduced to a multiplication of a conformal symmetry rosette with the conformal symmetry group of the category C21. Since conformal symmetry groups of the category C21 are the extensions of the symmetry groups of rosettes Cn (n), Dn (nm), every construction of conformal symmetry rosettes can be reduced to the following procedure: the transformation of a generating rosette with the symmetry group Cn (n) or Dn (nm) by two circle inversions with the concentric inversion circles, by rotations with the singular point O and by reflections with the reflection line containing the point O.

For constructions of conformal symmetry rosettes with the desired symmetry, the desymmetrization method is also used. This indirect construction method is mainly applied to the conformal symmetry rosettes of the types DnRI, KDnRI, LDnRI. Such conformal symmetry rosettes, possessing a high degree of symmetry and representing the most frequent conformal symmetry groups in ornamental art, are a suitable medium for constructions of other conformal symmetry rosettes of a lower degree of symmetry.

Using the desymmetrization method, besides classical-symmetry desymmetrizations, antisymmetry desymmetrizations can be used in all the cases when the desired symmetry group is the subgroup of the index 2 of a certain larger group. With conformal symmetry groups of the category C21, a complete survey of them is given in the table of the antisymmetry desymmetrizations, i.e. of the corresponding conformal antisymmetry groups of the category C21'. In this table, symbols of antisymmetry groups are given in the group/subgroup notation G/H.

The table of antisymmetry desymmetrizations of conformal symmetry groups of category C21:

 NI/Cn CnRI/Cn DnRI/Dn C2nRI/CnRI D2nRI/DnRI D1NI/Dn C2nRI/CnZI DnRI/CnRI D1NI/NI D2nRI/D1NI D1NI/CnZI CnZI/Cn DnRI/CnZI C2nZI/CnZI

Besides being the basis for applications of the antisymmetry desymmetrization method, this table is an indicator of all the subgroups of the index 2 of conformal symmetry groups of the category C21.

The complete derivation and catalogue of conformal antisymmetry groups of the categories C21' and C2' is given by S.V. Jablan (1985).

Information on some possible color-symmetry desymmetrizations of crystallographic conformal symmetry groups of the categories C21 and C2, can be obtained from the work of A.M. Zamorzaev, E.I. Galyarski, A.F. Palistrant (1978), A.F. Palistrant (1980c), and E.I. Galyarski (1986), who discuss the color-symmetry groups of tablets G320 and non-polar rods G31.

With conformal symmetry groups of the category C21, it is possible to establish a connection between these and the corresponding symmetry groups of friezes G21. The following relationships hold: CnRI @ m1, DnRI @ mm, CnZI @ 12, D1NI @ mg, NI @ 1g. In this way, the problem of color-symmetry groups derived from conformal symmetry groups of the category C21 can be reduced to the color-symmetry groups of friezes (J.D. Jarratt, R.L.E. Schwarzenberger, 1981), i.e. to the use of the table of color-symmetry desymmetrizations of the corresponding symmetry groupsof friezes. In doing so, it is necessary to be aware on the identification pxn = E.

Conformal symmetry tilings of the plane E2\{O} , are discussed by E.A. Zamorzaeva (1985). In this work, a connection is established between the different types of conformal symmetry groups of the category C2, the symmetry groups of non-polar rods G31 and the symmetry groups of ornaments G2. The following relationships hold : KCnZI, LCnZI @ p2, KNI @ pg, KCnRI @ pm, LCnRI @ cm, MNI @ pgg, KD1NI, MCnRI @ pmg, LDnRI @ cmm, KDnRI @ pmm. Using such an approach, the problem of isogonal tilings, which correspond to conformal symmetry groups of the category C2, is solved.

The discussion on the visual properties of conformal symmetry groups of the category C21, can be reduced to an analysis of the effects of the conformal transformations RI, ZI, SI on generating rosettes with the symmetry groups Cn (n), Dn (nm).

In groups of the type CnRI, the inversion RI causes the absence of the enantiomorphism, existing in the symmetry group Cn (n). The intensity of the static visual impression produced by an inversion depends on the position and form of an elementary asymmetric figure belonging to a fundamental region of the conformal symmetry group containing this inversion. It comes to its full expression only for figures that are, by their shape, close to the inversion circle. In the geometric sense, the inversion RI causes the constancy of the shape of the boundary of a fundamental region, which coincides with the inversion circle mI (its arc), and the non-polarity of radial rays.

The inversional rotation SI mainly keeps the properties of generating rosettes with symmetry groups of the category G20 and somewhat intensifies their dynamic visual properties. The inversional reflection ZI causes the bipolarity of rotations and of radial rays, and preserves the property of the enantiomorphism. The dynamic or static visual properties of conformal symmetry rosettes with symmetry groups of the category C21 will depend on the analogous properties of generating rosettes with symmetry groups of the category G20.

Conformal symmetry groups of the category C2 are derived extending by the similarity transformations K, L, M, conformal symmetry groups of the category C21. The dilatation K and dilative rotation L maintain all the geometric-visual properties of the generating conformal symmetry groups of the category C21 and introduce a new dynamic visual component - a suggestion of centrifugal expansion, or of rotational centrifugal expansion. The dilative reflection M, in the visual sense, produces the impression of centrifugal alternating expansion. In the geometric sense, it eliminates the possibility for the enantiomorphism.

The form of a fundamental region of conformal symmetry groups is defined by the invariance of all the points of inversion circles and reflection lines. In this way, the conformal symmetry groups of the types NI, CnZI, KNI, KCnZI, LCnZI, MNI, offer the possibility to change the shape of all the boundaries of a fundamental region; groups of the types CnRI, MCnRI, LCnRI offer the possibility to change the shape of non-inversional boundaries; groups of the types D1NI, KD1NI offer the possibility to change the shape of non-reflectional boundaries; groups of the type LDnRI type offer the possibility to change the shape of non-reflectional and non-inversional boundaries, while groups of the types DnRI, KDnRI, generated by reflections and inversions, do not offer the possibility to change the shape of boundaries of a fundamental region.

In conformal symmetry groups that do not require the constancy of the form of a fundamental region, a variety of corresponding conformal symmetry rosettes in ornamental art is achieved by varying the boundaries of a fundamental region or the form of an elementary asymmetric figure belonging to a fundamental region. In the remaining conformal rosettes the variety is achieved exclusively by the second of these possibilities.

In the geometric-visual sense, the inversion RI represents an adequate interpretation of "two-sideness" in the "one-sided" plane, i.e. the interpretation of the symmetry transformation of the space E3 - the plane reflection in the invariant plane of the symmetry groups of tablets G320, in the plane E2\{O}. In the same way, because of the isomorphism between the symmetry groups of tablets G320 and conformal symmetry groups of the category C21, and the isomorphism between the symmetry groups of non-polar rods G31 and conformal symmetry groups of the category C2, apart from the schematic visual interpretations - Cayley diagrams and tables of the graphic symbols of symmetry elements - conformal symmetry rosettes represent a completely adequate visual model of the symmetry groups of tablets G320 and non-polar rods G31. The symmetry groups of polar rods G31 possess a similar visual interpretation in the plane E2 - similarity symmetry rosettes. On the basis of those isomorphisms, the presentations, the geometric and visual properties of conformal symmetry groups of the categories C21, C2 can be fully transferred, respectively, to the symmetry groups of tablets G320 and non-polar rods G31.

In the table of the group-subgroup relations (Figure 4.24), a survey is given of all the group-subgroup relations between the visually-presentable continuous conformal symmetry groups, the discrete conformal symmetry groups of the categories C21, C2 and group-subgroup relations between conformal symmetry groups of the category C2 and the similarity symmetry groups of the category S20. Although incomplete, as they do not include all the group-subgroup relations but only the most important ones, the tables can serve as a basis on which to apply the desymmetrization method for obtaining conformal symmetry groups or similarity symmetry groups and also for the geometric-visual evidence of symmetry substructures of conformal symmetry groups. Aiming for a more complete consideration of those problems, the given tables can be used with the analogous tables corresponding to similarity symmetry groups of the category S20.

Figure 4.24

The time and frequency of occurrence of different conformal symmetry groups in ornamental art are related to the periods when various constructional problems were solved. According to the criterion of maximal constructional and visual simplicity, constructions of conformal symmetry rosettes are mostly based on the use of circles and lines as homologous elements of conformal symmetry transformations. Combinations of elementary geometric figures (regular polygons, circles) with a common singular point, found in the earliest periods of ornamental art, gave as a result the first examples of conformal symmetry rosettes with symmetry groups of the category C21, mostly of the type DnRI (n =1,2,3,4,6,8,...). In the further development of ornamental art, examples of all other conformal symmetry groups of the category C21 appeared. The dominance of visually static conformal symmetry rosettes with a higher degree of symmetry, is expressed throughout the history of ornamental art.

Very important in the formation of conformal symmetry rosettes was the existence of certain models in nature - the flowers of different plants, forms of growth, etc., possessing or suggesting different kinds of conformal symmetry.

Conformal symmetry rosettes with a symmetry group of the category C2 are constructed multiplying by similarity transformations K, L, M, a generating conformal symmetry rosette with the symmetry group of the category C21, or multiplying a generating rosette with the same symmetry group, by an inversion with the inversion circle concentric to it. The second construction, based on the non-metric construction method, using invariance of all the points of inversion circles and reflection lines, and circles and lines as homologous figures of conformal symmetry transformations, offers better possibilities, in the sense of maximal constructional simplicity. It came to its fullest expression in the corresponding elementary geometric constructions by means of circles and lines used in ornamental art. In ornamental art, conformal symmetry rosettes came to their peak in the work of Romanesque and Gothic architects, artisans and artists. Examples of almost all the conformal symmetry groups date from these periods. When calculating a building proportions and other architectural elements, especially when drawing-up the plans of the decorative architectural elements - window and floor rosettes - Medieval architects used those constructions.

Directly linked to these problems, and covered by the theory of similarity symmetry and conformal symmetry, are the questions of the theory of proportions, the roots of which date from Greek geometry. It held a special place in Medieval and Renaissance architectural planning and it reached its fullest expression in applications of the "aurea sectio" (or the "golden section") and musical harmonies used in architecture and in the visual arts. In more recent periods, examples of conformal symmetry rosettes with symmetry groups of the category C2 can be found in the work of M.C. Escher (1971a, b; 1986), who, besides classical-symmetry, often used conformal antisymmetry and color-symmetry rosettes, and greatly contributed to the analysis of different conformal color-symmetry groups and conformal tilings.

The problems of visual perception, referring here to conformal symmetry rosettes, can be solved analogously to the same problems previously discussed in the other categories of symmetry groups, through the analysis of the symmetrization and desymmetrization factors caused by the visual effects of the physiological-psychological elements of visual perception.

The approach to ornamental art from the theory of symmetry, makes possible the recognition, classification and the exact analysis of all the various kinds of conformal symmetry rosettes occurring in ornamental art, and also highlights the different possibilities for constructing these conformal symmetry rosettes possessing the symmetry and geometric-visual properties already anticipated.