Chapter 1.2


 Transformations and
 Symmetry Groups




A function m is a mapping of a set A to a set B if for every element a Î A there exists exactly one element b Î B such that m(a) = b. The mapping m is one-to-one if m(a) = m(a') implies a = a', and it is onto if m(A) = B, where m(A) = {m(a) | a Î A}. A transformation is a mapping of a space to itself that is both one-to-one and onto, i.e. it is a one-to-one correspondence from the set of points in the space onto itself (H.S.M. Coxeter, 1969; G.E. Martin, 1982). If we denote a transformation of space by t, then for each point P which we call the original there exists exactly one point Q, the image of point P derived by transformation t and we write t(P) = Q. Each point Q of the space is the image of some point P derived by transformation t, where to equal images correspond equal originals. Points P, Q in the given order are called homologous points of transformation t.

A figure f is any non-empty subset of points of space. A figure f is called invariant with respect to a transformation S if S(f) = f; in this case the transformation S is called a symmetry of the figure f. The set of points invariant with regard to all the powers of a given symmetry S is called the element of symmetry of the figure f. The identity transformation of space is the transformation E under which every point of space is invariant, i.e. E(P) = P holds for each point P of the space. The identity transformation is a symmetry of any given figure. Any figure whoose set of symmetries consists only of the identity transformation E is called asymmetric; any other figure is called symmetric. For example, the capital letters A,B,C,D,E,K,M,T,U,V,W,Y are mirror-symmetric, H,I,O,X doubly mirror-symmetric and point-symmetric, N,S,Z point-symmetric, and F,G,J,P,Q,R asymmetric. The letters b d or p q form the mirror symetric pairs, and b q or p d the point-symmetric pairs.

For every two transformations S1, S2 of the same space we define the product S1S2, as the composition of the transformations: S1S2(P) = S2(S1(P)). In other words, by product we mean the successive action of transformations S1, S2. As a symbol for the composition S¼S, where S occurs n times, we use Sn, i.e. the n-th power of the transformation S. The order of the transformation S is the minimal n (n Î N) for which Sn = E holds. If there is no finite number n which satisfies the given relation, then the transformation S is called a transformation of infinite order. If n = 2, then the transformation S is called an involution. If transformations S1 and S2 are such that S1S2 = E, then S1 is called the inverse of S2, and vice versa. We denote this relationship as S1 = S2-1 and S2 = S1-1. For an involution S we have S = S-1, and for the product of two transformations (S1S2)-1 = S2-1S1-1 holds.

A transformation t which maps every line l onto a line t(l) is a collineation. An affine transformation (or linear transformation) is a collineation of the plane that preserves parallels.

As a binary operation * we understand any rule which assigns to each ordered pair (A,B) a certain element C written as A *B = C, or in the short form, AB = C. A structure (G,*) formed by a set G and a binary operation * is a group if it satisfies the axioms:

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;

a3) (existence of neutral element): there exists E Î G that for each A1 Î G the equality A1E = A1 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

a5) (commutativity): for all A1,A2 Î G, A1A2 = A2A1 is satisfied, the group is commutative or abelian.

The order of a group G is the number of elements of the group; we distinguish finite and infinite groups. The power and the order of a group element are defined analogously to the definition of the power and the order of a transformation.

A figure f is said to be an invariant of the group of transformations G if it is invariant with respect to all its transformations, i.e. if A1(f) = f for every A1 Î G. All symmetries of a figure f form a group, that we call the group of symmetries of f and denote by Gf. For example, all the symmetries of a square (Figure 1.1a) form the non-abelian group, consisting of identity transformation E, reflections R, R1, R1RR1, RR1R, and rotations RR1, (RR1)2, R1R - the symmetry group of square D4. The order of reflections is 2, the order of rotations RR1, R1R is 4, and the order of half-turn (RR1)2 is 2. This group consists of 8 elements, so it is of order 8. The elements of the same group, expressed as products of reflection R and rotation S of order 4 are: identity E, reflections R, RS, RS2, SR, and rotations S, S2 and S3. Instead of a square, we may consider the plane tiling having the same symmetry (Figure 1.1b).

A subset H of group G, which by itself constitutes a group with the same binary operation, is called a subgroup of group G if and only if (iff) for all A1, A2 Î H, A1A2-1 Î H. Subgroups H = G and H = {E} of each group G are called trivial, while the other subgroups are nontrivial subgroups of the group G. In the symmetry group of square, identity transformation E and rotations S, S2, S3 form the subgroup of the order 4 - the rotational subgroup of square C4.

Groups (G1, *) and (G2, °) are called isomorphic if there exists a one-to-one and onto mapping i of elements of the group G1 onto elements of the group G2, so that for all A1,A2 Î G1, i(A1 *A2) = i(A1) °i(A2) holds; the mapping i is called an isomorphism. For example, by the mapping i(R) = R, i(R1) = RS is defined the isomorphism of the symmetry group of square generated by reflections R,R1, with the same group generated by reflection R and rotation S. Any isomorphism of a group G with itself is called an automorphism.




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.

Instead of representing the group in the traditional way, by means of its Cayley table, which offers a listing of all the elements of the group and their compositions (products), complete information about the group is given more effectively and concisely by a group presentation (i.e. abstract, generating definition): a set of generators and defining relations. The group of transformations G is discrete if for each point P of the space in which the group G acts there is a positive distance d = d(P) such that no image of P (distinct from P) under an element of G is at distance less than d from P. The set { S1,S2,¼,Sm } of elements of a discrete group G is called a set of generators of G if every element of the group can be expressed as a finite product of their powers (including negative powers). Relations gk(S1,S2,¼,Sm) = E, k = 1,2,¼,s, are called defining relations if all other relations which S1, S2,¼, Sm satisfy are algebraic consequences of the defining relations (H.S.M. Coxeter, W.O.J. Moser, 1980). So that, in further discussions each discrete group will be given by a set of generators and defining relations, i.e. by a presentation.

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,S2,¼,Sm }     gk(S1,S2,¼,Sm) = E    k = 1,2,¼,s     (1)
G1  { S1',S2',¼,Sn' }     hl(S1',S2',¼,Sn') = E    l = 1,2,¼,t     (2)
are isomorphic iff there exist relations:
Sj' = Sj(S1,S2,¼,Sm)     j = 1,2,¼,n     (1')
Si = Si(S1',S2',¼,Sn')     i = 1,2,¼,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.

By "structure of the group" we understand its isomorphism with some of the basic, well known groups (e.g., cyclic group Cn, dihedral group Dn,¼) or with a direct product of such groups. The cyclic group Cn is given by the presentation: {S}     Sn = E, and the dihedral group Dn can be given by two isomorphic presentations: {R,R1}     R2 = R12 = (RR1)n = E or {S,R}    Sn = R2 = (RS)2 = E. Hence, the structure of the symmetry group of square is D4, and the structure of its rotational subgroup is C4.

For groups G and G1, G ÇG1 = {E}, given by presentations (1), (2) we define the direct product G×G1 as the group with the set of generators {S1,S2,¼,Sm,S1',S2',¼,Sn'}, the set of defining relations of which is, besides the relations (1), (2), made up of relations SiSj' = Sj'Si, i = 1,2,¼,m, j = 1,2,¼,n. For each group G we can discuss the possibility of it being decomposed, i.e. represented as the direct product of its nontrivial subgroups. A group which allows such a decomposition we call reducible, otherwise it is called irreducible. For example, the direct product of two cyclic groups, C3 given by the presentation {S}     S3 = E and C2 given by {T}  T2 = E is the group {S,T}    S3 = T2 = E  ST = TS. By the substitution U = ST, this results in the presentation {U}     U6 = E, so C3×C2 @ C6, showing that the group C6 is reducible.

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

G = g1 H Èg2H ȼÈgnH ȼ
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 ÈRH, 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.

The set of points G(P) = {g(P) | g Î G}, obtained from a point P by all transformations of the group G, is called the orbit of P with respect to G; it is the set of points equivalent to point P (or the transitivity class of P) with respect to the group G. Analogously we can also define the orbit (or transitivity class) of any figure f with respect to the group G and denote it by G(f). A point P which is invariant with respect to a transformation S, i.e. a point for which S(P) = , is also called singular. A figure f is invariant with respect to a transformation S if S(f) = f. A point P is a singular (invariant) point of a group G if it is a singular (invariant) point of all transformations of G. A point which is not an invariant point of a transformation S is also called a point in general position with respect to the transformation S. A point is said to be a point in general position with respect to a group of transformations G if it is in general position with respect to all the transformations of the group G, i.e. if it is not an invariant point of any transformation of the group G. For example, the singular (invariant) point of the symmetry group of square is the center of square. The points belonging to the mirror-reflection lines are the invariant points of the corresponding reflections. All other plane points, are the points in general position with respect to the symmetry group of square (Figure 1.1).

The orbit of some point P in general position with respect to the discrete group of transformations G makes possible a schematic interpretation of the group G: a Cayley diagram or a graph of the group G - a visual model of discrete group of transformations G. To each vertex of the graph corresponds exactly one element of the group, and to each edge corresponds one transformation. The edges which connect the homologous points of the same transformation are denoted by the same type of line (full, broken, dotted). The non-oriented edges correspond to the involutions. For any other, oriented edge, the motion in the direction of the arrow indicates the multiplication by the corresponding transformation from the right, and the motion in the opposite direction of the arrow corresponds to multiplication by the inverse of the corresponding transformation on the right. A Cayley diagram is a connected graph, i.e. there exists a path which connects every two vertexes of the graph. It represents the direct visual interpretation of the presentation of the group, since to every closed cycle there corresponds one defining relation (1). A complete graph is considered to be the graph in which every two vertexes are directly linked by the edge (Figure 1.2).




Figure 1.2

(a) Graph of the group C4 given by the presentation {S}    S4 = E; (b) the complete graph of the same group.

For a discrete group G it is possible to define a fundamental region of G. A fundamental region F is a figure which satisfies the following conditions:

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)) = Æ. 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,T2,¼} covering space without gaps or overlaps. More explicitly, the union of the sets T1, T2,¼, 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.

If the symmetry group GT contains also transformations which map any vertex of tiling T onto any other vertex, i.e. if the vertices make up one class of transitivity, the tiling is said to be isogonal. By a flag in a tiling we mean a triple (V,E,T) consisting of a vertex V, an edge E and a tile T which are mutually incident. A tiling is called regular if its symmetry group is transitive on the flags of the tiling. In particular, for the symmetry groups of ornaments there exist exactly three regular tilings (regular tessellations) by means of regular polygons. Each of them can be denoted by a Schläfli symbol {p,q} denoting regular p-gons, where q of them are incident with each vertex of the regular tessellation: {4,4} , {3,6} , {6,3} . A dual of regular tiling {p,q} is the regular tiling {q,p} (Figure 1.4).




Figure 1.4

Regular tilings {4,4} , {3,6} and {6,3}.

A uniform or Archimedean tiling is an isogonal plane tiling by regular polygons, which is edge-to-edge, i.e. in which every vertex and edge of a tile is a vertex and edge of the tiling. Each of the 11 types of uniform tilings can be denoted by the symbol (p1q1 p2q2 ¼pnqn) where p1,p2,¼,pn denote regular p-gons, and q1,q2,¼,qn the number of adjacent regular p-gons of the same type which are incident with one vertex. Besides regular tessellations (36) = {3,6} ,(63) = {6,3} and (44) = {4,4} the family of uniform tilings consists of (34.6), (33.42), (32.4.3.4), (3.4.6.4), (3.6.3.6), (3.122), (4.6.12) and (4.82) (J. Kepler, 1619) (Figure 1.5). The Archimedean tiling (34.6) occurs in two enantiomorphic forms - "left" and "right".




Figure 1.5

Archimedean tilings.

An open circle (or open circular disk) is the set of points X such that OX < r, where O is a fixed point and r is a positive number. For OX £ r, the circle (circular disk) is called a closed circle.

A transformation t is continuous if for any two points P, Q of the plane it is possible to make t(P) and t(Q) as close together as we wish, by taking P and Q sufficiently close, and bicontinuous if both t and t-1 are continuous. A homeomorphism or topological transformation is any bicontinuous transformation. The open (closed) topological disk is any plane set which is homeomorphic image of an open (closed) circle.

A tiling T is normal if:

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.

A tiling T is called homeohedral if it is normal and is such that for any two tiles T1, T2 of T there exists a homeomorphism of the plane that maps T onto T and T1 onto T2. A normal tiling is called two-homeohedral if its tiles form two transitivity classes under a homeomorphism mapping T onto itself. For example, all non-regular Archimedean tilings (Figure 1.5) are two-homeohedral.

A continuous set of points is any set of points which satisfies the axiom(s) of continuity. Every continuous set of points is a homeomorphic image of a line. Alongside the discrete groups of transformations, continuous symmetry groups may also be discussed. A symmetry group G of the space E2 or E2\{O} is called continuous if the orbit G(P) of a point in general position P with respect to the group G satisfies one of the following conditions:

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

(ii) G(P) can be divided into disjoint continuous sets of points, and for every point of each of these sets there is a positive distance d = d(P) such that the circle c(P,d) contains no points of any other of the sets mentioned. By the terms "continuous group of translations, rotations, central dilatations and dilative rotations" we mean that all translations along one line, all rotations around one center, all central dilatations with a common center, and all dilative rotations with a common center and with a fixed angle, are elements of such a group. In particular, the continuous symmetry groups of ornaments, depending on whether they satisfy condition (i) or (ii), are called the symmetry groups of continua or semicontinua.

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