Conformal Symmetry Rosettes
  and Ornamental Art

Every conformal symmetry group of the type C_nR_I is the direct product of the symmetry groups C_n (n) and R_I. Hence, visual interpretations of conformal symmetry groups of the type C_nR_I (Figure 4.3, 4.4) can be constructed multiplying by the inversion R_I a rosette with the symmetry group C_n (n), belonging to a fundamental region of the group R_I, or multiplying by the n-fold rotation a figure with the symmetry group R_I, belonging to the fundamental region of the group C_n (n). A fundamental region of the group C_nR_I is the section of the fundamental regions of the groups C_n (n) and R_I. Owing to the presence of the indirect symmetry transformation - inversion R_I - in groups of the type C_nR_I enantiomorphic modifications do not occur. The visual effect and degree of visual dynamism in rosettes with the conformal symmetry group C_nR_I 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 C_n (n) (Figure 4.3), to the conformal symmetry rosettes with a fully expressed stationary visual component resulting from the visual effect of the inversion R_I, similar to the visual effect of a reflection. Those "static" conformal symmetry rosettes with the group C_nR_I 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 C_nR_I offers a change of non-inversional boundaries, i.e. boundaries that do not belong to the inversion circle m_I. 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.

Owing to their low degree of symmetry and visual dynamism conditioned by the polarity of rotations, conformal symmetry rosettes with the symmetry group C_nR_I 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 C_n (n). They are transformed by the inversion R_I 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 C_nR_I are rarely found in ornamental art. Conformal symmetry groups of the type C_nR_I can be obtained by desymmetrizations of groups of the type D_nR_I, 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 D_nR_I/C_nR_I, which in the classical theory of symmetry can be discussed as the group C_nR_I.

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

Conformal symmetry rosettes with the symmetry group D_nR_I have visual characteristics similar to that of generating rosettes with the symmetry group D_n (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 D_nR_I 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 D_nR_I, 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 D_nR_I, is achieved by applying different elementary asymmetric figures within a fundamental region. The visual effect of the inversion R_I, within the group D_nR_I depends on the shape of that elementary asymmetric figure and its position within a fundamental region. The static visual function of the inversion R_I comes to its full expression for an elementary asymmetric figure, by the shape and position being very close to the inversion circle.

In ornamental art, there are many examples of conformal symmetry rosettes with symmetry groups of the type D_nR_I. The frequency of occurrence of a particular group depends on the frequency of occurrence of its generating group D_n (nm). Therefore, the groups D_nR_I, 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 D_nR_I have a special role in ornamental art. Very interesting visual interpretations of these groups are obtained, reproducing by the inversion R_I a rosette with the symmetry group D_n (nm), constructed by circles (or their arcs) containing the singular point O and touching the inversion circle m_I. The inversion R_I 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.

The continuous conformal symmetry group D_¥ R_I 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_¥ R_I. 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).

The simplest group of the type C_nZ_I is the group Z_I (n = 1), generated by the inversional reflection Z_I, the commutative composition of a reflection and an inversion. The inversional reflection Z_I 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 Z_I corresponds to this two-fold rotation, indicates the properties of the inversional reflection Z_I - its involutionality and relations to the other conformal symmetry transformations.

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

Every group of the type C_nZ_I (Figure 4.8, 4.9) is a dihedral group derived as a superposition of the groups C_n (n) and Z_I. Their visual interpretations can be constructed multiplying by the inversional reflection Z_I a rosette with the symmetry group C_n (n), belonging to a fundamental region of the group Z_I or, less frequently, multiplying by the n-fold rotation a conformal symmetry rosette with the symmetry group Z_I, belonging to a fundamental region of the symmetry group C_n (n). According to the relationship Z_I = RR_I = R_IR, conformal symmetry rosettes with the symmetry group C_nZ_I can be directly derived from conformal symmetry rosettes with the symmetry group C_nR_I, 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 C_nR_I with the inversion circle m_I, one class of fundamental regions of the group C_nR_I (the internal or external fundamental regions) (Figure 4.8). Conformal symmetry groups of the type C_nZ_I offer the possibility for the enantiomorphism. All the other properties of groups of the type C_nR_I can be attributed to groups of the type C_nZ_I.

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

Groups of the type N_I (Figure 4.10, 4.11) are generated by the inversional rotation S_I, the commutative composition of the rotation S of the order 2n and the inversion R_I, so that the relationship S_I = SR_I = R_IS holds. Hence, conformal symmetry rosettes with the symmetry group N_I can be directly derived from conformal symmetry rosettes with the symmetry group C_nR_I, by transforming by the rotation S one class of fundamental regions (internal or external fundamental regions) of the group C_nR_I. A fundamental region of the group N_I may coincide with that of the group C_nR_I, but it allows the varying of all the boundaries. Conformal symmetry rosettes with the symmetry group N_I can also be constructed multiplying by the inversional rotation S_I a conformal symmetry rosette with the symmetry group C_n ( n), belonging to a fundamental region of the group R_I. Groups of the type N_I are an equivalent of the symmetry groups of tablets G₃₂₀ of the type ([2ñ]), generated by n-fold rotational reflection, so that the relationship S₁ = S_I² 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 N_I 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 N_I, which consist of half-circles containing the singular point O and touching the inversion circle m_I, and of corresponding half-tangent lines onto which these half-circles are transformed by the inversional rotation S_I (Figure 4.10). The enantiomorphism does not occur in groups of the type N_I.

Groups of the type N_I can also be obtained by desymmetrizations of groups of the type C_2nR_I. Besides the classical-symmetry desymmetrizations, the antisymmetry desymmetrizations resulting in conformal antisymmetry groups of the type C_2nR_I/N_I, discussed in the classical theory of symmetry within the type N_I, are very frequent.

According to the relationship S_I = SR_I = R_IS, where by S is denoted the rotation of the order 2n, a connection, analogous to that existing between the types C_nR_I and N_I, exists between the types D_nR_I and D₁N_I. Hence, conformal symmetry rosettes with the symmetry group D₁N_I (Figure 4.12, 4.13) can be directly derived from conformal symmetry rosettes with the symmetry group D_nR_I, by reproducing by the rotation S one class of the fundamental regions (the internal or external fundamental regions) of the group D_nR_I. The same result can be obtained multiplying by the inversional rotation S_I a conformal symmetry rosette with the symmetry group D₁ (m), or multiplying by the reflection a conformal symmetry rosette with the symmetry group N_I. All the other properties of groups of the type D₁N_I - the absence of the enantiomorphism, visual characteristics of corresponding conformal symmetry rosettes, etc. - are similar to the properties of groups of the type D_nR_I.

Groups of the type D₁N_I are isomorphic to the symmetry groups of tablets G₃₂₀ of the type ([2ñ])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 D₁N_I 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 D₁N_I, 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 D_2nR_I, groups of the type D₁N_I can be obtained. Besides classical-symmetry desymmetrizations, the antisymmetry desymmetrizations resulting in the conformal antisymmetry groups of the type D_2nR_I/D₁N_I, discussed in the classical theory of symmetry within the type D₁N_I, 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.

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.

Conformal symmetry rosettes with the symmetry group KN_I (Figure 4.14) can be constructed multiplying by the dilatation K (with k > 0) a conformal symmetry rosette with the symmetry group N_I, belonging to a fundamental region of the group K. A fundamental region of the group KN_I is the section of fundamental regions of the groups K and N_I, 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 KN_I, 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 N_I, after the introduction of the dilatation K, are maintained in the derived conformal symmetry rosette with the symmetry group KN_I. 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 KN_I, desymmetrizations of groups of the type KC_2nR_I, 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 KC_2nR_I/KN_I, discussed in the classical theory of symmetry within the type KN_I, are frequent.

The type MN_I consists of the groups derived by the superposition of the groups M and N_I. Hence, corresponding conformal symmetry rosettes with the group MN_I can be constructed multiplying by the dilative reflection M (with k > 0) a conformal symmetry rosette with the group N_I, belonging to a fundamental region of the group M (Figure 4.15). A fundamental region of the group MN_I is the section of fundamental regions of the groups M and N_I, 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 N_I, remain unchanged. In the conformal symmetry rosette with the symmetry group MN_I obtained, the presence of the dilative reflection M increases the visual dynamism, suggesting alternating centrifugal expansion. Since conformal symmetry rosettes with the symmetry group MN_I 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 MN_I, the desymmetrization method can be applied on groups of the type KD₁N_I. Besides the classical-symmetry desymmetrizations, also the antisymmetry desymmetrizations resulting in antisymmetry groups of the type KD₁N_I/MN_I, discussed in the classical theory of symmetry within the type MN_I, can also be derived.

Conformal symmetry rosettes with the symmetry group KD₁N_I (Figure 4.16) can be constructed multiplying by the dilatation K (with k > 0) a conformal symmetry rosette with the symmetry group D₁N_I, belonging to a fundamental region of the symmetry group K. A fundamental region of the group KD₁N_I is the section of fundamental regions of the groups K and D₁N_I, and it allows a varying of non-reflectional boundaries.

The visual properties of generating conformal symmetry rosettes with the symmetry group D₁N_I are maintained in derived conformal symmetry rosettes with the symmetry group KD₁N_I. 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 KD₁N_I, 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 KD₁N_I are more frequent in ornamental art, than conformal symmetry rosettes with the symmetry group KN_I or MN_I.

By using the desymmetrization method, besides classical-symmetry desymmetrizations, the antisymmetry desymmetrizations of groups of the type KD_2nR_I, resulting in conformal antisymmetry groups of the type KD_2nR_I/KD₁N_I, discussed in the classical theory of symmetry within the type KD₁N_I, can be obtained.

The type KC_nZ_I consists of conformal symmetry groups formed by the superposition of the groups K and C_nZ_I. Corresponding conformal symmetry rosettes can be constructed multiplying by the dilatation K (with k > 0) a conformal symmetry rosette with the symmetry group C_nZ_I, 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 C_nZ_I. A fundamental region of the group KC_nZ_I is the section of fundamental regions of the groups K and C_nZ_I, and allows a varying of all boundaries. Visual properties of a derived conformal symmetry rosette with the symmetry group KC_nZ_I are similar to that of the generating conformal symmetry rosette with the symmetry group C_nZ_I. 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 KC_nZ_I, are very rare in ornamental art (Figure 4.17).

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

The type LC_nZ_I consists of conformal symmetry groups that are the result of the superposition of the groups L and C_nZ_I. Visual examples of conformal symmetry groups of the type LC_nZ_I can be constructed multiplying by the dilative rotation L (with k > 0) a conformal symmetry rosette with the symmetry group C_nZ_I, 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 LC_nZ_I into sectors of the dilatation K((-1)^pk^q). A fundamental region of the group LC_nZ_I is the section of fundamental regions of the groups L and C_nZ_I. 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 KC_nZ_I can be constructed by using a generating conformal symmetry rosette with the symmetry group C_nZ_I, 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 m_I 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.

Groups of the type LC_nZ_I 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 C_nZ_I. 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 LC_nZ_I is restricted by the principle of visual entropy. Therefore, the most frequent conformal symmetry rosettes with the symmetry group LC_nZ_I are constructed multiplying by a dilative rotation, the simplest conformal symmetry rosettes with the symmetry group C_nZ_I (Figure 4.18). For the same reason, more frequent are conformal symmetry rosettes with symmetry groups of the type LC_nZ_I, 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 L_2nC_nZ_I (L_2n = L(k,p/n)). The symmetry group mentioned is the subgroup of the index 2 of the group LD_nR_I. According to the relationship K = L(k,0) = L₀, a consequent application of the criterion of subordination requires also that the type KC_nZ_I must be considered as the subtype of the type LC_nZ_I. This would result in the complete elimination of the type KC_nZ_I. 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 LD_nR_I, the corresponding groups of the subtype L_2nC_nZ_I, can be derived. Adequate antisymmetry desymmetrizations of groups of the type LD_nR_I result in conformal antisymmetry groups of the type LD_nR_I/ L_2nC_nZ_I, in the classical theory of symmetry included in the type LC_nZ_I, as groups of the subtype L_2nC_nZ_I.

Conformal symmetry rosettes with the symmetry group KC_nR_I can be constructed multiplying by the dilatation K (with k > 0) a conformal symmetry rosette with the symmetry group C_nR_I, 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 KC_nR_I is the section of fundamental regions of the groups K and C_nR_I, 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 C_nR_I. 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 KC_nR_I are not so frequent in ornamental art. However, the possibility of deriving conformal symmetry groups of the type KC_nR_I by desymmetrizations of groups of the type KD_nR_I, 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 KD_nR_I/KC_nR_I, discussed in the classical theory of symmetry within the type KC_nR_I, can be obtained.

The type MC_nR_I consists of conformal symmetry groups derived by extending by the dilative reflection M (with k > 0) conformal symmetry groups of the type C_nR_I. Corresponding conformal symmetry rosettes can be constructed multiplying by the dilative reflection M a conformal symmetry rosette with the symmetry group C_nR_I, 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 m_I concentric to it, a generating conformal symmetry rosette with the symmetry group C_nR_I. 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 C_nR_I 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 MC_nR_I in ornamental art.

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

The group KD_nR_I can be derived extending by the dilatation K the conformal symmetry group D_nR_I. Corresponding conformal symmetry rosettes can be constructed multiplying by the dilatation K a conformal symmetry rosette with the symmetry group D_nR_I, 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 m_I concentric to it (Figure 4.21). A fundamental region of the group KD_nR_I is the section of fundamental regions of the groups K and D_nR_I. Since the group KD_nR_I is generated by reflections (reflections and inversions), its fundamental region is fixed. Therefore, a fundamental region of the group KD_nR_I 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 KD_nR_I 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.

The effect of the dilatation K on a generating conformal symmetry rosette with the symmetry group D_nR_I 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 D_n, 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 KD_nR_I are the most frequent discrete conformal symmetry groups of the category C₂, in ornamental art. Besides their individual use, groups of the type KD_nR_I form the basis for applying the desymmetrization method, aiming to derive the other types of conformal symmetry groups of the category C₂₁.

The group LC_nR_I can be derived extending by the "centering" dilative rotation L = L_2n = L(k,p/n) (with k > 0) the conformal symmetry group C_nR_I. Hence, conformal symmetry rosettes with the symmetry group LC_nR_I can be constructed multiplying by the dilative rotation mentioned, a generating conformal symmetry rosette with the symmetry group C_nR_I, 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 LC_nR_I is the section of fundamental regions of the groups L_2n and C_nR_I, 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 = L_2n on a generating conformal symmetry rosette with the symmetry group C_nR_I 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 LC_nR_I can be obtained by desymmetrizations of groups of the type LD_nR_I. Since the group LC_nR_I is the subgroup of the index 2 of the group LD_nR_I, besides classical-symmetry desymmetrizations, very frequent are antisymmetry desymmetrizations, resulting in antisymmetry groups of the type LD_nR_I/LC_nR_I, discussed in the classical theory of symmetry within the type LC_nR_I.

The group LD_nR_I can be derived extending the group D_nR_I by the "centering" dilative rotation L = L_2n = L(k,p/n) (with k > 0). In ornamental art, conformal symmetry rosettes with the symmetry group LD_nR_I are very frequent, since they can be derived multiplying by the dilative rotation L = L_2n a conformal symmetry rosette with a symmetry group of the type D_nR_I, 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 m_I, the generating rosette mentioned. Afterward, the image obtained must be rotated through the angle q = p/n. A fundamental region of the group LD_nR_I is the section of fundamental regions of the groups D_nR_I 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 LD_n is the subgroup of the index 2 of the group LD_nR_I, there exists a specific balance between static and dynamic visual components in conformal symmetry rosettes with the symmetry group LD_nR_I, 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 LD_nR_I 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.