The idea of conformal symmetry was given in the monograph Colored Symmetry, its Generalizations and Applications by A.M. Zamorzaev, E.I. Galyarski and A.F. Palistrant (1978) as a generalization of similarity symmetry. Along with it, there was established the isomorphism between the symmetry groups of non-polar rods G_{31} and the conformal symmetry groups of the category C_{2}. The isomorphism between the similarity symmetry groups S_{20} and the symmetry groups of polar rods G_{31} is pointed out within the analysis of the similarity symmetry groups in E^{2} (Chapter 3). According to the same isomorphism, the category C_{2} consists of the conformal symmetry groups isomorphic to the symmetry groups of non-polar rods G_{31}.
In conformal symmetry groups of the category C_{2}, besides the isometries and similarity transformations, which are the elements of the similarity symmetry groups S_{20}, there will be, as basic transformations, three more conformal symmetry transformations: inversion R_{I} in a circle with the center in the singular point O of the plane E^{2}, inversional reflection Z_{I} and inversional rotation S_{I}. An inversion circle can be denoted by m_{I}, inversional reflection center by 2_{I} and inversional rotation center by n_{I}.
According to the isomorphism existing between the types of the discrete symmetry groups of non-polar rods G_{31} (A.V. Shubnikov, V.A. Koptsik, 1974) and the corresponding types of discrete conformal symmetry groups of the category C_{2}, it is possible to conclude that there will be ten types of discrete conformal symmetry groups of the category C_{2}.
Isomorphism between discrete symmetry groups of non-polar rods G_{31} and discrete conformal symmetry groups of category C_{2}:
KN_{I} | (a)([2ñ]) | MN_{I} | (a)([2ñ])ã |
KC_{n}R_{I} | (a)n:m | KD_{n}R_{I} | (a)mn:m |
KC_{n}Z_{I} | (a)n:2 | MC_{n}R_{I} | (a)ãn:m |
LC_{n}Z_{I} | (a_{t})n:2 | LC_{n}R_{I} | (a)(2n)_{n}:m |
KD_{1}N_{I} | (a)([2ñ])m | LD_{n}R_{I} | (a)m(2n)_{n}:m |
Besides conformal symmetry groups of the category C_{2}, there exist five types of discrete conformal symmetry groups of the category C_{21}, isomorphic to the five existing types of the discrete symmetry groups of tablets G_{320} (A.V. Shubnikov, V.A. Koptsik, 1974): C_{n}Z _{I} @ n:2, C_{n}R_{I} @ n:m, N_{I} @ ([2ñ]), D_{1}N_{I} @ ([2ñ])m and D_{n}R_{I} @ mn:m.
Those isomorphisms make possible the defining of the continuous conformal symmetry groups of the categories C_{2} and C_{21}, which in these isomorphisms correspond, respectively, to the continuous symmetry groups of non-polar rods G_{31} and tablets G_{320}.
Owing to the existence of a singular point, any figure the symmetry group of which is a conformal symmetry group is called a conformal symmetry rosette. All the conformal symmetry groups of the category C_{2} can be derived as the extensions of the conformal symmetry groups of the category C_{21} by the similarity transformations K, L, M. Therefore, every conformal symmetry rosette with a conformal symmetry group of the category C_{2} can be constructed, multiplying by those similarity transformations a conformal symmetry rosette with the corresponding conformal symmetry group of the category C_{21}. Developing our analysis further we will consider first the conformal symmetry groups of the category C_{21}. They will be used to generate conformal symmetry groups of the category C_{2}. An analogous approach to the symmetry groups of rods G_{31} treated as extensions of the symmetry groups of tablets G_{320}, is given by A.V. Shubnikov and V.A. Koptsik (1974).
According to the criterion of subordination, for the conformal symmetry groups of the category C_{2}, it is possible to consider a certain type as the subtype of the more general type (e.g., the type KC_{n}Z_{I} as the subtype of the type LC_{n}Z_{I}, according to the relationship K = L(k,0) = L_{0}). That and similar problems can be solved in the same way as with the similarity symmetry groups S_{20}.
Presentations and structures of conformal symmetry groups of category C_{21}:
N_{I} {S_{I}} S_{I}^{2n} = E C_{2n}
D_{1}N_{I} {S_{I},R_{I}} S_{I}^{2n} = E R_{I}^{2} = E R^{2} = E RS_{I} = S_{I}R C_{2n}×D_{1}
C_{n}R_{I} {S,R_{I}} S^{n} = E SR_{I} = R_{I}S C_{n}×D_{1}
C_{n}Z_{I}
{S,Z_{I}}
S^{n} = Z_{I}^{2} = (SZ_{I})^{2} = E
D_{n}
{Z_{I},Z_{I}'} Z_{I}^{2} = Z_{I}^{'2} = (Z_{I}Z_{I}')^{n} = E
D_{n}R_{I}
{S,R,R_{I}}
S^{n} = R^{2} = (SR)^{2} = E R_{I}^{2} = E
SR_{I} = R_{I}S
RR_{I} = R_{I}R D_{n}×D_{1}
{R,R_{1},R_{I}} R^{2} = R_{1}^{2} = (RR_{1})^{n} = E R_{I}^{2} = E
RR_{I} = R_{I}R R_{1}R_{I} = R_{I}R_{1}
Presentations of conformal symmetry groups of category C_{2}:
KN_{I} {K,S_{I}} S_{I}^{2n} = E KS_{I}K = S_{I}
MN_{I} {M,S_{I}} S_{I}^{2n} = E (MS_{I})^{2} = E
KD_{1}N_{I} {K,S_{I},R} S_{I}^{2n} = R^{2} = (RS_{I})^{2} = E KR = RK (KS_{I})^{2n} = E
KC_{n}R_{I} {K,S,R_{I}} S^{n} = R_{I}^{2} = E SR_{I} = R_{I}S KS = SK (KR_{I})^{2} = E
KC_{n}Z_{I} {K,S,Z_{I}} S^{n} = Z_{I}^{2} = (SZ_{I})^{2} = E KS = SK (KZ_{I})^{2} = E
LC_{n}R_{I} {L,S,R_{I}}
S^{n} = R_{I}^{2} = E
SR_{I} = R_{I}S SL = LS
LR_{I}LR_{I} = R_{I}LR_{I}L = S (L = L_{2n} = L(k,p/n))
LC_{n}Z_{I} {L,S,Z_{I}} S^{n} = Z_{I}^{2} = (SZ_{I})^{2} = E SL = LS (LZ_{I})^{2} = E
MC_{n}R_{I} {M,S,R_{I}} S^{n} = R_{I}^{2} = E SR_{I} = R_{I}S SMS = M (MR_{I})^{2} = E
KD_{n}R_{I} {K,S,R,R_{I}} S^{n} = R^{2} = (SR)^{2} = E
R_{I}^{2} = E
SR_{I} = R_{I}S
RR_{I} = R_{I}R KS = SK KR = RK (KR_{I})^{2} = E
LD_{n}R_{I}
{L,S,R,R_{I}} S^{n} = R^{2} = (SR)^{2} = E R_{I}^{2} = E
SR_{I} = R_{I}S
RR_{I} = R_{I}R LS = LS LRLR = RLRL
LR_{I}LR_{I} = R_{I}LR_{I}L = S RLR = LS
(L = L_{2n} = L(k,p/n))
Besides these presentations of conformal symmetry C_{2}, it is possible to have presentations with a different choice of generators. For example, groups of the types KC_{n}R_{I}, KC_{n}Z_{I}, KD_{n}R_{I} offer the following possibilities:
KC_{n}R_{I} {S,R_{I},R_{I}'}
S^{n} = R_{I}^{2} = R_{I}^{'2} = E SR_{I} = R_{I}S
SR_{I}' = R_{I}'S
(R_{I}' = R_{I}K)
KC_{n}Z_{I} {S,Z_{I},Z_{I}'} S^{n} = Z_{I}^{2} = Z_{I}^{'2} = E
(SZ_{I})^{2} = (SZ_{I}')^{2} = E
(Z_{I}' = Z_{I}K)
KD_{n}R_{I} {S,R,R_{I},R_{I}'} S^{n} = R^{2} = (SR)^{2} = E
SR_{I} = R_{I}S SR_{I}' = R_{I}'S
R_{I}^{2} = R_{I}^{'2} = E RR_{I} = R_{I}R RR_{I}' = R_{I}'R
(R_{I}' = R_{I}K)
Similar possibilities exist in all the other conformal symmetry groups. Besides indicating the different presentation possibilities, certain choices of generators and corresponding presentations make possible a direct recognition of the structure of a group considered (e.g., the structure of groups of the type KC_{n}R_{I} is D_{¥}×C_{n}, of groups of the type KD_{n}R_{I} is D_{¥}×D_{n}, etc.).
Enantiomorphism:
the enantiomorphism exists in the conformal symmetry
groups C_{21}
of the type C_{n}Z_{I} type and
conformal symmetry groups of the
category C_{2} of
the types KC_{n}Z_{I} and LC_{n}R_{I}.
Polarity of rotations:
polar rotations - N_{I}, BC_{n}R_{I},
KN_{I}, KC_{n}R_{I}, LC_{n}R_{I};
bipolar rotations - C_{n}Z_{I}, MN_{I},
MC_{n}R_{I},
KC_{n}Z_{I}, LC_{n}Z_{I};
non-polar rotations - D_{1}N_{I}, D_{n}R_{I},
KD_{1}N_{I},
KD_{n}R_{I}, L_{n}R_{I}.
Polarity of radial rays:
polar - KN_{I};
bipolar - MN_{I},
KD_{1}N_{I}, KC_{n}Z_{I}, LC_{n}Z_{I} (if they
exist, i.e. if the dilative
rotation angle is a rational one);
non-polar - KC_{n}R_{I}, KD_{n}R_{I},
MC_{n}R_{I}, LC_{n}R_{I}, LD_{n}R_{I}.
Form of the fundamental region:
unbounded in the conformal symmetry groups
C_{21}; bounded in conformal
symmetry groups of the category C_{2}. There exists
the possibility of varying
the shape of boundaries that do not belong to
reflection lines or inversion
circles. In conformal symmetry groups of the
category C_{21}, variation of
all the boundaries is possible in groups of the
types N_{I}, C_{n}Z_{I}, of
non-reflectional boundaries in groups
of the type D_{1}N_{I}, of non-inversional
boundaries in groups of the type
C_{n}R_{I}, while in groups of the type D_{n}R_{I},
it is not possible at all.
Regaring the changes of the form of a fundamental
region, conformal symmetry
groups of the category C_{2} offer similar
possibilities as generating
conformal symmetry groups of the category C_{21}.
Number of edges of the fundamental region:
KC_{n}R_{I},
KD_{n}R_{I} - 4;
KN_{I} - 4,6;
MN_{I}, KD_{1}N_{I},
KC_{n}Z_{I},
LC_{n}R_{I},
LC_{n}Z_{I}, MC_{n}R_{I},
LD_{n}R_{I} - 3,4,5,6.
Among the continuous conformal symmetry groups, visually presentable are groups D_{¥} R_{I} of the category C_{21} and groups KD_{¥} R_{I}, K_{1}C_{n}R_{I}, K_{1}D_{n}R_{I}, K_{1}D_{¥} R_{I}, L_{1}C_{n}Z_{I} of the category C_{2}.
Cayley diagrams (Figure 4.1):