 ## Chapter 1.2

### Transformations and  Symmetry Groups

a1) (closure): for all A1,A2 Î G, A1A2 Î G is satisfied;

a2) (associativity): for all A1,A2,A3 Î G, (A1A2)A3 = A1(A2A3) is satisfied;

a4) (existence of inverse element): for each A1 Î G there exists A1-1 Î G so that A1-1A1 = E is satisfied.

If besides a1-a4) also holds Figure 1.1
 (a) Symmetric figure (square) consisting of equaly arranged congruent parts (1-8) and its symmetry transformations: identity transformation E ( 1 « 1, 2 « 2, 3« 3, 4 « 4, 5 « 5, 6« 6, 7 « 7, 8 « 8), reflections R ( 1 « 2, 3 « 8, 4« 7, 5 « 6), R1 ( 1« 4, 2 « 3, 5 « 8, 6« 7), R1RR1 ( 1 « 6, 2« 5, 3 « 4, 7 « 8), RR1R ( 1 « 8, 2 « 7, 3« 6, 4 « 5), rotations R1R ( 1® 7, 2 ® 8, 3 ® 1, 4 ®2, 5 ® 3, 6 ® 4, 7 ® 5, 8® 6), RR1 ( 1 ® 3, 2 ® 4, 3® 5, 4 ® 6, 5 ® 7, 6 ®8, 7 ® 1, 8 ® 2) and half-turn (RR1)2 ( 1 « 5, 2 « 6, 3 « 7,4 « 8). The order of the symmetry group of square D4 is equal to the number of congruent parts (8); (b) plane tiling with the same symmetry.

The symmetry group of square is given by Cayley table:

 E R R1 R1RR1 RR1R RR1 (RR1)2 R1R E E R R1 R1RR1 RR1R RR1 (RR1)2 R1R R R E RR1 (RR1)2 R1R R1 R1RR1 RR1R R1 R1 R1R E RR1 (RR1)2 R1RR1 RR1R R R1RR1 R1RR1 (RR1)2 R1R E RR1 RR1R R R1 RR1R RR1R RR1 (RR1)2 R1R E R R1 R1RR1 RR1 RR1 RR1R R R1 R1RR1 (RR1)2 R1R E (RR1)2 (RR1)2 R1RR1 RR1R R R1 R1R E RR1 R1R R1R R1 R1RR1 RR1R R E RR1 (RR1)2

and by the presentation:

 { R,R1 }     R2 = R12 = (RR1)4 = E,

or by Cayley table:

 E R RS RS2 SR S S2 S3 E E R RS RS2 SR S S2 S3 R R E S S2 S3 RS RS2 SR RS RS S3 E S S2 RS2 SR R RS2 RS2 S2 S3 E S SR R RS SR SR S S2 S3 E R RS RS2 S S SR R RS RS2 S2 S3 E S2 S2 RS2 SR R RS S3 E S S3 S3 RS RS2 SR R E S S2

and by the presentation:

 { S,R }     S4 = R2 = (RS)2 = E.
Two groups G1 and G2 which are given with their presentations:
 G  { S1,S2,Ľ,Sm }     gk(S1,S2,Ľ,Sm) = E    k = 1,2,Ľ,s     (1)
 G1  { S1',S2',Ľ,Sn' }     hl(S1',S2',Ľ,Sn') = E    l = 1,2,Ľ,t     (2)
are isomorphic iff there exist relations:
 Sj' = Sj(S1,S2,Ľ,Sm)     j = 1,2,Ľ,n     (1')
 Si = Si(S1',S2',Ľ,Sn')     i = 1,2,Ľ,m     (2')
such that the systems of relations (1), (1') are algebraically equivalent to (2), (2') (H.S.M. Coxeter, W.O.J. Moser, 1980). This means, that the second presentation can be obtained from the first by the substitutions (2'), and the first can be obtained from the second by the substitutions (1'). For example, the groups G1 and G2, given by the presentations:
 G1     { R,R1}     R2 = R12 = (RR1)4 = E      (1)
 G2     { S,R}     S4 = R2 = (RS)2 = E     (2)
are isomorphic, because there exist the relations:
 S = RR1      (1')
 R1 = RS      (2')
so that the systems of relations (1), (1') are algebraically equivalent to (2), (2'). Namely, by the substitution (2') R1 = RS, the relations (1) R2 = R12 = (RR1)4 = E are transformed into algebraically equivalent relations
 R2 = (RS)2 = (RRS)4 = E    S4 = R2 = (RS)2 = E     (2)
and by the substitution (1') S = RR1, the relations (2) are transformed into algebraically equivalent relations
 (RR1)4 = R2 = (RRR1)2 = E   R2 = R12 = (RR1)4 = E     (1).
Their isomorphism, defined by the mapping i(R) = R, i(R1) = RS is also simply visible from the corresponding Cayley tables.

The term "decomposition" can be used in another sense. Each group can be decomposed according to its subgroup H:

 G = g1 H Čg2H ČĽČgnH ČĽ
where giH = {gih | gi Î G, h Î H}. The expression giH is called the left coset which corresponds to element gi with respect to subgroup H. Analogously, there is the possibility of the right decomposition of group G according to subgroup H. If the above decompositions are finite, the number of cosets is called the index of the subgroup H in the group G; in the case of infinite decomposition we say that H is a subgroup of infinite index. We should also note the property that every two cosets are either disjoint or identical, and that the order of the group is equal to the product of the order of the subgroup H and its index. From this results the statement that the order of a subgroup is a divisor of the order of the group. A subgroup H of a group G is called a normal subgroup if gH = Hg holds for every element g Î G. For example, for the symmetry group of square G and its rotatational subgroup H holds the decomposition G = H ČRH, and gH = Hg holds for every element g Î G, so H is the normal subgroup of index 2 in G. The order of H is 4, and order of G (8) is the product of the order of H (4) and index of H in G (2).

According to those basic geometric-algebraic assumptions, we can consider as the subject of this study the analysis of plane figures - ornamental motifs and their invariance with respect to symmetry groups. Figure 1.2
 (a) Graph of the group C4 given by the presentation {S}    S4 = E; (b) the complete graph of the same group.

a) for each point P of the space where the group of transformations G acts, there exists S Î G that P Î S(F);

b) for each S Î G\{E} holds int(F)Çint(S(F)) = Ć. If Cl(F) is the closure of F, the orbit G(Cl(F)) represents a tiling of the space on which the group G acts. A space tiling or tessellation is a countable family of closed sets T = {T1,T2,Ľ} covering space without gaps or overlaps. More explicitly, the union of the sets T1, T2,Ľ, which are known as the tiles of T, is to be the whole space, and the interiors of the sets Ti are to be pairwise disjoint (B. Grünbaum, G.C. Shephard, 1987). Since a fundamental region F has no points which are equivalent under any transformation of the group G, unless they are on the boundary, each internal point of F is a point in general position with respect to the group G. Regarding the extent of the fundamental region we distinguish between groups with bounded and unbounded fundamental regions. A discrete group of transformations G usually does not determine uniquely the fundamental region, or the induced tiling G(Cl(F)). Therefore, it is of interest to inquire about the different possible shapes of the fundamental region. In the tiling G(Cl(F)) the intersection of tiles of any finite set of tiles (containing at least two distinct tiles) may be empty or may consist of a set of isolated points (vertices) and arcs (edges). When discussing variations of the form of the fundamental region F we distinguish between two aspects of change: the change in the number of vertices and edges of the fundamental region F, and the change of the form of the edges (arcs) themselves in which the number of vertices and edges remains unchanged. As the result of the action of the symmetry groups we have tile-transitive or isohedral tilings. Their tiles belong to the same class of transitivity G(Cl(F)), since for every two tiles of G(Cl(F)) there exists a transformation of group G which maps one tile onto the other (Figure 1.3). Figure 1.3
 (a) Isohedral plane tiling corresponding to the symmetry group D4; (b) two isohedral plane tilings with different shape of the fundamental region, corresponding to its rotational symmetry subgroup C4. Figure 1.4
 Regular tilings {4,4} , {3,6} and {6,3}. Figure 1.5
 Archimedean tilings.

a) every tile of T is a topological disk;

b) the intersection of every two tiles of T is a connected set, i.e. does not consist of two closed and disjoint subsets;

c) the tiles of T are uniformly bounded, i.e. there exist circles c and C, with fixed radiuses, such that every tile Ti of tiling T contains a translate of c and is contained in a translate of C.

(i) G(P) is the complete space on which G acts; or