Chapter 1.4

 Visual Interpretations
 of Symmetry Groups

All discrete symmetry groups can be visually modeled by adequate ornamental motifs (patterns, tilings) which for centuries have been an important part of applied art. Besides different symbolic or schematic visual interpretations of symmetry groups (such as Cayley diagrams, tables of graphic symbols of symmetry elements, where the different symmetry transformations are denoted by graphic symbols: rotations by oriented regular polygons, reflections by full lines, glide reflections by dotted ones, etc.), they are an important aspect of "imaginative geometry" ("anshauliche geometrie" of D. Hilbert) - geometry of everyday life, and its relations with art (Figure 1.14).

When trying to translate the meanings of geometric properties of symmetry transformations and of symmetry groups into the visual sphere, one can note the links between the geometric-algebraic properties of transformations and the different visual parameters (stationariness, dynamism,). A survey of geometric characteristics and their visual interpretations, which are relevant for such a study, is given in each chapter of this book. Alongside the elements already mentioned: presentations of symmetry groups, Cayley diagrams, data on enantiomorphism, also the orientability: polarity, non-polarity and bipolarity of different lines, invariants of symmetry transformations will be discussed.

Figure 1.14

Graphic notation of the symmetry groups (a) D4; (b) C4.

In addition to the orientability of invariant lines - axes, also the orientability of radial rays - half-lines invariant to some dilatation K, of circles - invariants of rotations, or of equiangular, logarithmic spirals - invariants of corresponding similarity symmetry or conformal symmetry groups will be discussed. The (curved) line l is a polar invariant line of the symmetry S if the relation S(l) = l holds, where l is an orientation of the line l. A line l is a polar invariant line of the group G if this relation is satisfied for all the elements of the group G. A (curved) invariant line l of the symmetry S is non-polar if the relation S(l) = -l is satisfied, where -l denotes the oppositely oriented line l. A line invariant with respect to the group G is non-polar if there exists at least one indirect symmetry S G which satisfies the condition S(l) = -l. A non-polar (curved) line l, invariant of the symmetry S is bipolar if S is a direct symmetry. A non-polar (curved) invariant line of the group G is bipolar if the set Si = {S | S(l) = -l, S G} contains only direct transformations.

In the visual sense, the term "polarity" can be connected with the dynamism of ornamental motifs corresponding to discrete symmetry groups. For continuous symmetry groups it is immediately linked with the term visual presentability. As opposed to the discrete symmetry groups which can always be visually interpreted by means of ornamental motifs, the continuous symmetry groups will not always offer an adequate visualization. So, for example, for the continuous line group of translations, its visualization is not possible without introducing a supplementary symbol (e.g., the symbol --------------------------> a suitable symbol of such a translation). Not having a previous agreement or convention about its meaning, it is not possible to give to the observer an adequate visual interpretation of this symmetry group, comprehensible without further explanation. Under the term "visual presentability" we understand the visual modeling of symmetry groups which offers to observer the complete visual information on the observed symmetry (in the sense of objective, geometric symmetry), without needing for an additional explanation. In visual arts, apart from the objective, geometric symmetry, very important are the effects of "visual forces" (R. Arnheim, 1965, 1969). They are, for example, the upward tendency of a vertical line, the visual effect of the "ascending" and "descending" diagonal, the "left" and "right" orientation. These subjective visual factors, having a great influence on the visual perception of symmetry and representing a subject of study in the psychology of visual perception, are not discussed under the term "visual presentability", which refers only to objective, geometric symmetry and its visual perception. Although a detailed analysis of the subjective, visual factors of symmetry is omitted, mainly because of the complexity of the problems of visual perception, this work offers a potential approach to such problems. Continuous symmetry groups with continuous non-polar elements of symmetry allow an immediate visual interpretation, while for representing groups with polar or bipolar continuous elements of symmetry we can apply textures - an equal, average density of the asymmetric figures arranged along the invariant polar or bipolar line, in accordance with the given continuous symmetry group (A.V. Shubnikov, N.V. Belov et al., 1964). So, for example, continuous line group of translations can be interpreted by means of textures as the series
,, , ,,,, ,,, ,,, , ,, In contrast to physical interpretations of all continuous symmetry groups which can be obtained by motion or some other physical effect, the domain of the visual presentability of continuous symmetry groups, if textures are not applied, is limited by the objective stationariness of ornamental art works to the continuous groups with non-polar continuous elements of symmetry.