 Chapter 1.3

Classification of Symmetry  Transformations and Groups

As the basis for the classification of the symmetry groups G three elements were taken into consideration: the types of symmetries (isometries, similarity symmetries, conformal symmetries) that occur in G, the space on which the group G acts, and the sequence of maximal included proper subspaces, invariant with respect to the group G. According to this, the Bohm symbols (J. Bohm, K. Dornberger-Schiff, 1966) are used for the categorization of the groups of isometries. Symbols of the same type are applied to the similarity symmetry and conformal symmetry groups. For example, the symmetry group of square D4 acts in plane and possesses only one invariant point, so it belongs to the category G20 - the symmetry groups of rosettes. Figure 1.6
 (a) Direct and (b) indirect plane isometry.

The classification of isometric transformations and corresponding symmetry groups is common for spaces En, Sn, Ln for n < 2, while for n ł 2 different possibilities of relations of disjoint lines, which are defined by the axiom of parallelism, condition specific differences. This work exclusively discusses Euclidean spaces.

In the space E2 (plane) we distinguish the following isometric transformations (Figure 1.7): Figure 1.7
 (a) Identity transformation; (b) reflection; (c) rotation; (d) translation; (e) glide reflection.

1) identity transformation E, with the minimal reflectional representation of the length 2 (R2 = E);

2) reflection R;

With respect to the invariant figures, all the points of the plane E2 are invariant points of the identity transformation E, reflection R maintains the invariance of all the points of the reflection line, rotation S possesses a single invariant point - the center of rotation, while translation and glide reflection have no invariant points. A glide reflection possesses a single invariant line - the axis, and a translation keeps invariant all the lines parallel to the translation axis. Figure 1.8
 Central reflection Z.

When orientation is considered we distinguish direct transformations (or sense preserving transformations): identity transformation E, translation, rotation, and indirect transformations (sense reversing transformations): reflection and glide reflection. Since direct transformations are the product of an even, and the indirect ones of an odd number of reflections, we can call them respectively even and odd transformations.

{S,R}     S4 = R2 = (RS)2 = E:    ER = E,    RR = R,    (RS)R = SR,
(SR)R = RS,    SR = S3,     (S2)R = S2,     (S3)R = S.

In the same way, it is defined an external automorphism of the rotational group of square H, given by presentation

{S}     S4 = E,    SR = S3,     (S2)R = S2,     (S3)R = S,

where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient tool for extending symmetry groups.

Since the product of direct transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct transformations of the group G. For example, the rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the space En can be identified as movements of a material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not represent motion in E3).

Since reflections have a role of elementary isometric transformations, while all other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of which consists exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow variations and will have rectilinear edges. All symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9
 "Left" and "right" rosette with the symmetry group C4, consisting of direct symmetries. Figure 1.10
 Five plane Bravais lattices.

Because the symmetry groups of friezes G21 are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. Figure 1.11
 (a) Dilatation; (b) dilative rotation; (c) dilative reflection.

Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes.

Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,Ľ) will be discussed analogously to the case of isometry groups. Figure 1.12
 Circle inversion.

Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: Figure 1.13
 (a) Inversional reflection; (b) inversional rotation.