Chapter 1.7

 Geometric-Visual Analysis
 of Symmetry Groups

As an illustration of the methodological approach used in this work, we will give the example of one symmetry group and its complete comparative analysis from the point of view of the theory of symmetry and ornamental art.

Let the discrete group of isometric transformations of the plane E2, generated by a glide reflection P and a reflection R1, be given by the presentation:

{P,R1}    R12 = (R1P)2 = E.
The first defining relation R12 = E shows that the reflection R1 is an involution, i.e. that R1 = R1-1 while from the other relation follows:
(R1P)2 = E R1PR1 = P-1 R1-1PR1 = P-1 PR = P-1.
The glide reflection P is a transformation without invariant points, with an invariant line- the axis l of the glide reflection. The conjugate of the transformation P derived by reflection R1 is also a glide reflection with the invariant line R1(l). Since the axis of the glide reflection P-1 is the line -l, from the previous relations we conclude that R1(l) = -l; consequently, it follows that the reflection line of the reflection R1 is perpendicular to the axis of the glide reflection P, and that its axis l is non-polar (since there exists an indirect transformation, the reflection R1, which transforms it onto the oppositely oriented line -l).

The group discussed possesses an invariant space E2 - the plane, an invariant subspace E1 - a line, and has no invariant points. Therefore it belongs to the category of symmetry groups of friezes G21 - the line groups of the plane E2 (S2) without invariant points. Distinguishing between the spaces E2, S2, L2 is not necessary because we are dealing with the line groups. Because this group is generated by a glide reflection P perpendicular to the reflection R1, its crystallographic symbol will be pmg, or in short form mg (M. Senechal, 1975). Within the crystallographic symbol pmg, p denotes the presence of a translation X = P2, i.e. the translational subgroup 11={X}; the symbol m denotes a reflection R1 perpendicular to this translation, and the symbol g denotes the glide reflection P. In the short symbol mg, the translation symbol p is omitted.

Since the set P,R1 is a generator set of the group mg, after concluding that the reflection line of R1 is perpendicular to the axis of the glide reflection P, we can construct an appropriate ornamental motif, the visual model of the frieze symmetry group mg. This is achieved by applying the transformations P and R1 to the chosen asymmetric figure, which belongs to a fundamental region of the symmetry group mg (Figure 1.19).

Figure 1.19

Representing the glide reflection P as the commutative composition X1R3 = R3X1 of a translation X1 = R1R2 (composition of reflections R1, R2 with parallel reflection lines) and a reflection R3 with the reflection line parallel to the axis of the translation X1, we come to the conclusion that the product R1P = R1R1R2R3 = R2R3 is the commutative composition of perpendicular reflections R2, R3, i.e. a half-turn T. The conjugates of reflection R1 and half-turn T, derived by the powers of the glide reflection P, define respectively the set of reflections with equidistant reflection lines parallel to the reflection line of R1, and the set of rotations of the order 2, where the distance between the neighboring reflection lines (rotation centers) is |X1|. So that, we come to the diagrammatic interpretation of the group mg - a table of graphic symbols of symmetry elements, where the axis of the glide reflection is indicated by the dotted line and by the vector of translation, reflection lines by solid lines, and centers of rotations of the order 2 by the symbol §  (Figure 1.20).

Figure 1.20

Using the substitution T = R1P we come to an algebraic equivalent of the previous presentation of the group mg - a new presentation of the same group:

{R1,T}  R12 = T2 = E.
Besides indicating another way to construct the frieze with symmetry group mg, this presentation directly shows that the group mg has structure D. Namely, group Dn has the presentation:
{S1,S2}    S12 = S22 = (S1S2)n = E.
If S1S2 is an element of infinite order, we obtain the group D having the following presentation:
{S1,S2}     S12 = S22 = E
which is isomorphic with the group mg.

Instead of the asymmetric figure, which under the action of the group mg gives the frieze pattern, by considering the orbit of the closure of a fundamental region of the group mg we obtain the corresponding frieze tiling. The fundamental region of the group mg and all other frieze symmetry groups, is unbounded and allows the variation of all boundaries which do not belong to reflection lines. Figure 1.21 shows two of these possibilities.

The Cayley diagram of the group mg is derived as the orbit of a point in general position with respect to the group mg. Instead of a direct mutual linking of all vertexes (i.e. orbit points) and obtaining the complete graph, we can, aiming for simplification, link only the homologous points of the group generators. By denoting with the broken oriented line the glide reflection P, and with the dotted non-oriented line the reflection R1, we get the Cayley diagram which corresponds to the first presentation of the group mg (Figure 1.22a).

Figure 1.21

By an analogous procedure we come to the graph which corresponds to its second presentation with the generator set {R1,T}, where a half-turn is indicated with the dot-dash line (Figure 1.22b).

Let us note also, that the defining relations can be read off directly from the graph of the group. Each cycle, i.e. closed path in which the beginning point coincides with the endpoint, corresponds to a relation between the elements of the group and vice versa. Cayley diagrams (graphs of the groups) may also very efficiently serve to determine the subgroups of the given symmetry group. Namely, every connected subgraph of the given graph satisfying the following condition determines a certain subgroup of the group discussed, and vice versa. The condition in question is: an element (transformation) is included in the subgraph either wherever it occurs, or not at all (i.e. it is deleted). Of course, to be able to determine all the subgroups of a given group, it is necessary to use its complete graph as the basis for defining the subgraphs.

Figure 1.22

Since in the group mg there are indirect isometries, this group does not give enantiomorphic modifications. For the groups consisting only of direct symmetries, the enantiomorphic modifications can be obtained by applying the "left" (e.g., b) and "right" (d) form of an elementary asymmetric figure. For example, for the group 11, generated by a translation X, this results in the enantiomorphic friezes: bbbbbbbbbbbbbbbbbbbbbbbbbbbbb and dddddddddddddddddddddddddddddd. The translation axis l of the group mg is non-polar, because there exists an indirect transformation, the reflection R1 for which the relation R1(l) = -l holds. Rotations of the order 2 in the group mg are polar because each circle c drawn around the center of rotation of the order 2 is invariant only with respect to this rotation and to the identity transformation E, so that the group C2 ( 2) (generated by the half-turn T) of transformations preserving the circle c invariant, a rosette subgroup C2 (2) of the group mg, consists of direct transformations. Besides the rosette subgroups C2 ( 2), the group mg has also the rosette subgroups D1 (m), namely the one generated by the reflection R1, or by its conjugates.

The group mg contains as subgroups the following symmetry groups of friezes: p1 (11) generated by the translation X = P2, p1g (1g) generated by the glide reflection P, pm1 (m1) generated by the translation X and the reflection R1, and itself. Besides the list of all frieze groups, subgroups of the group mg, the table of the minimal indexes of subgroups of the given group points out the possible desymmetrizations which lead to this subgroup. In particular, considering the use of antisymmetry and color- symmetry desymmetrizations, from this table we can see that antisymmetry desymmetrizations of group mg result in the subgroups of the index 2: 1g, 12 and m1. This can be achieved by a black-white coloring (or, e.g., 1-2 indexing) according to the laws of antisymmetry, using the following systems of (anti)generators: {P,e1R1} or {e1R1,e1T} for obtaining the antisymmetry desymmetrization mg/1g; {e1P,R1} or {R1,e1T} for obtaining the antisymmetry desymmetrization mg/m1; {e1P,e1R1} or {e1R1,T} for obtaining the antisymmetry desymmetrization mg/12, where e1 = (12), i.e. the group of color permutations PN = P2 = C2 (Figure 1.23).

Figure 1.23

The junior antisymmetry groups obtained can be understood also as adequate visual interpretations of the symmetry groups of bands G321 - as the Weber diagrams of the symmetry groups of bands p2111, pm11 and p112 respectively. In this case the alternation of colors white-black is understood in the sense "above-under" the invariant plane of the frieze, i.e. as the identification of the antiidentity transformation e1 with the plane reflection in the invariant plane of the group mg. The seven generating symmetry groups of friezes G21, seven senior antisymmetry groups and seventeen junior antisymmetry groups correspond to the 31 groups of symmetry of bands, offering complete information on their presentations and structures.

Using N = 4 colors and the system of colored generators {c1P,c2R1} or {c2R1,c1c2T}, we get the color-symmetry desymmetrization mg/11, where c1 = (12)(34) and c2 = (13)(24); hence, the group of color permutations is PN = P4 = C2×C2 = D2 (Figure 1.24).

Figure 1.24

In all the antisymmetry and color-symmetry desymmetrizations mentioned, for which the group PN is regular, the subgroup H derived by the desymmetrization is a normal subgroup of the group mg (1g, m1, 12, 12). Because of this, complete information on the antisymmetry or colored symmetry group, i.e. on the corresponding desymmetrization, is given by the number N and by the group/subgroup symbol G/H. The next case of coloring with N = 3 colors, the irregular group PN and the subgroup H which is not a normal subgroup of the group G, demands the symbols G/H/H1. In this case, besides the number N, the group of colored symmetry G*, i.e. the corresponding color-symmetry desymmetrization is uniquely defined by the generating group G, the stationary subgroup H of G*, which maintains every individual index (color) unchanged and its symmetry subgroup H1 which is the final result of the color-symmetry desymmetrization. The index of the subgroup H in the group G is equal to N and the product of the index of the subgroup H1 in group H and the number N is equal to the order of the group of color permutations PN, i.e. [G:H] = N, [H:H1] = N1, and the order of the group PN is NN1.

As an example of the irregular case we can use the color-symmetry desymmetrization of the group mg obtained by N = 3 colors, i.e. by the system of colored generators: {c1P,c2R1} or {c2R1,c1c2T}, which results in the color-symmetry desymmetrization mg/mg/1g, where c1 = (123), c2 = (23), PN = P3 = D3 and [mg: mg]=3, [mg:1g]=2. This color-symmetry desymmetrization mg/mg/1g, N = 3 is shown on Figure 1.25a, while the stationary subgroup H (mg) which maintains each individual index (color) unchanged is singled out on Figure 1.25b. All cases of subgroups which are not normal subgroups of the given group are denoted in the tables of (minimal) indexes of subgroups in groups by italic indexes (e.g., [mg:mg]=3).

Figure 1.25

In terms of construction, for frieze group mg we can also distinguish the rosettal method of construction - the multiplication of a rosette with the symmetry group C2 (2) (generated by the half-turn T) or D1 ( m) (generated by the reflection R1) by the glide reflection P (Figure 1.26a, b). Like all other symmetry groups of friezes, the group mg is the subgroup of the maximal symmetry group of friezes mm generated by reflections. Since it is the normal subgroup of the index 2, the antisymmetry desymmetrization of the generating group mm with a set of generators {X,R,R1} or {R,R1,R2} where X is the translation, R the reflection in translation axis line, and R1, R2 reflections with reflection lines perpendicular to the translation axis, can be used.

Figure 1.26

By means of the system of (anti)generators: {e1X,e1R,R1} = {e1X, e1R,e1R1} or {e1R,R1,e1R2} the antisymmetry desymmetrization mm/mg is obtained (Figure 1.27), where e1 = (12), PN = P2 = C2, [mm:mg] = 2.

Figure 1.27

Many visual properties of the group mg, e.g., a relative constructional and visual simplicity of corresponding friezes conditioned by a high degree of symmetry, specific balance of the stationariness conditioned by the presence of reflections, by the non-polarity of the glide reflection axis, by the absence of enantiomorphism, and the dynamism conditioned by the presence of glide reflection and by polar, oriented rotations, are the direct consequences of the algebraic-geometric characteristics mentioned. Also, the different possibilities that the group mg offers, e.g., the possibilities for antisymmetry and color-symmetry desymmetrizations, the ways of varying the form of the fundamental region, construction possibilities etc., become evident after the analysis of this symmetry group of friezes from the point of view of the theory of symmetry.

Even such a concise illustration of the connections between the theory of symmetry and ornamental art raises the question of the place that ornamental art has today, from the point of view of both the artist and the scientist. When analyzing works of art, the approach to ornamental art from the standpoint of the theory of symmetry offers the possibility for serious analysis, a more profound study of the complete historical development of ornamental art, the regularity and laws on which the constructions of ornaments are founded, and an efficient classification method of a large domain (isometric, non-isometric, antisymmetry, color-symmetry ornamental motifs). It opens for the artist a new field of exploration - a more exact planning of visual effects, based on the knowledge of the theory of symmetry and the psychology of visual perception. An example of successful creativity, artistic imagination and knowledge of exact geometric rules, is given by the work of M.C.Escher, which points to the future of ornamental art as a specific synthesis of science and art. On the other hand, to the scientists of different disciplines, the theory of symmetry offers various possibilities - to archaeologists an efficient and reliable method of classification and comparative analysis; to theorists of art the basis for working out exact aesthetic criteria; to crystallographers, physicians and chemists an obvious model of symmetry structures. Last, but not least, to mathematicians ornamental art, as the treasury of the implicit mathematical knowledge of humankind, represents an inspiring field, rich with questions seeking an answer.