Chapter 4.1

  Conformal Symmetry
  Groups in E²\{O}

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 G₃₁ and the conformal symmetry groups of the category C₂. The isomorphism between the similarity symmetry groups S₂₀ and the symmetry groups of polar rods G₃₁ is pointed out within the analysis of the similarity symmetry groups in E² (Chapter 3). According to the same isomorphism, the category C₂ consists of the conformal symmetry groups isomorphic to the symmetry groups of non-polar rods G₃₁.

In conformal symmetry groups of the category C₂, besides the isometries and similarity transformations, which are the elements of the similarity symmetry groups S₂₀, 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 E², 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 G₃₁ (A.V. Shubnikov, V.A. Koptsik, 1974) and the corresponding types of discrete conformal symmetry groups of the category C₂, it is possible to conclude that there will be ten types of discrete conformal symmetry groups of the category C₂.

Isomorphism between discrete symmetry groups of non-polar rods G₃₁ and discrete conformal symmetry groups of category C₂:

Besides conformal symmetry groups of the category C₂, there exist five types of discrete conformal symmetry groups of the category C₂₁, isomorphic to the five existing types of the discrete symmetry groups of tablets G₃₂₀ (A.V. Shubnikov, V.A. Koptsik, 1974): C_nZ _I @ n:2, C_nR_I @ n:m, N_I @ ([2ñ]), D₁N_I @ ([2ñ])m and D_nR_I @ mn:m.

Those isomorphisms make possible the defining of the continuous conformal symmetry groups of the categories C₂ and C₂₁, which in these isomorphisms correspond, respectively, to the continuous symmetry groups of non-polar rods G₃₁ and tablets G₃₂₀.

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 C₂ can be derived as the extensions of the conformal symmetry groups of the category C₂₁ by the similarity transformations K, L, M. Therefore, every conformal symmetry rosette with a conformal symmetry group of the category C₂ can be constructed, multiplying by those similarity transformations a conformal symmetry rosette with the corresponding conformal symmetry group of the category C₂₁. Developing our analysis further we will consider first the conformal symmetry groups of the category C₂₁. They will be used to generate conformal symmetry groups of the category C₂. An analogous approach to the symmetry groups of rods G₃₁ treated as extensions of the symmetry groups of tablets G₃₂₀, 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 C₂, it is possible to consider a certain type as the subtype of the more general type (e.g., the type KC_nZ_I as the subtype of the type LC_nZ_I, according to the relationship K = L(k,0) = L₀). That and similar problems can be solved in the same way as with the similarity symmetry groups S₂₀.

Presentations and structures of conformal symmetry groups of category C₂₁:

N_I         {S_I}    S_I²ⁿ = E          C_2n

D₁N_I     {S_I,R_I}    S_I²ⁿ = E    R_I² = E    R² = E   RS_I = S_IR         C_2n×D₁

C_nR_I     {S,R_I}   Sⁿ = E    SR_I = R_IS         C_n×D₁

C_nZ_I      {S,Z_I}   Sⁿ = Z_I² = (SZ_I)² = E         D_n
              {Z_I,Z_I'}    Z_I² = Z_I^'2 = (Z_IZ_I')ⁿ = E

D_nR_I     {S,R,R_I}   Sⁿ = R² = (SR)² = E    R_I² = E   SR_I = R_IS   
                              RR_I = R_IR          D_n×D₁
              {R,R₁,R_I}    R² = R₁² = (RR₁)ⁿ = E    R_I² = E    RR_I = R_IR    R₁R_I = R_IR₁

Presentations of conformal symmetry groups of category C₂:

KN_I           {K,S_I}    S_I²ⁿ = E    KS_IK = S_I

MN_I          {M,S_I}    S_I²ⁿ = E    (MS_I)² = E

KD₁N_I      {K,S_I,R}    S_I²ⁿ = R² = (RS_I)² = E    KR = RK    (KS_I)²ⁿ = E

KC_nR_I      {K,S,R_I}    Sⁿ = R_I² = E    SR_I = R_IS    KS = SK    (KR_I)² = E

KC_nZ_I      {K,S,Z_I}    Sⁿ = Z_I² = (SZ_I)² = E    KS = SK    (KZ_I)² = E

LC_nR_I       {L,S,R_I}    Sⁿ = R_I² = E    SR_I = R_IS    SL = LS
                                    LR_ILR_I = R_ILR_IL = S    (L = L_2n = L(k,p/n))

LC_nZ_I       {L,S,Z_I}    Sⁿ = Z_I² = (SZ_I)² = E    SL = LS    (LZ_I)² = E

MC_nR_I      {M,S,R_I}    Sⁿ = R_I² = E    SR_I = R_IS    SMS = M    (MR_I)² = E

KD_nR_I      {K,S,R,R_I}    Sⁿ = R² = (SR)² = E    R_I² = E   SR_I = R_IS
                                      RR_I = R_IR KS = SK KR = RK (KR_I)² = E

LD_nR_I       {L,S,R,R_I}    Sⁿ = R² = (SR)² = E    R_I² = E   SR_I = R_IS
                                      RR_I = R_IR    LS = LS    LRLR = RLRL   
                                      LR_ILR_I = R_ILR_IL = S    RLR = LS
                                      (L = L_2n = L(k,p/n))

Besides these presentations of conformal symmetry C₂, it is possible to have presentations with a different choice of generators. For example, groups of the types KC_nR_I, KC_nZ_I, KD_nR_I offer the following possibilities:

KC_nR_I        {S,R_I,R_I'}    Sⁿ = R_I² = R_I^'2 = E    SR_I = R_IS    SR_I' = R_I'S
                                      (R_I' = R_IK)

KC_nZ_I        {S,Z_I,Z_I'}    Sⁿ = Z_I² = Z_I^'2 = E    (SZ_I)² = (SZ_I')² = E
                                      (Z_I' = Z_IK)

KD_nR_I      {S,R,R_I,R_I'}    Sⁿ = R² = (SR)² = E    SR_I = R_IS    SR_I' = R_I'S
                                        R_I² = R_I^'2 = E    RR_I = R_IR    RR_I' = R_I'R
                                        (R_I' = R_IK)

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_nR_I is D_¥×C_n, of groups of the type KD_nR_I is D_¥×D_n, etc.).

Enantiomorphism:
         the enantiomorphism exists in the conformal symmetry groups C₂₁
         of the type C_nZ_I type and conformal symmetry groups of the
         category C₂ of the types KC_nZ_I and LC_nR_I.

Polarity of rotations:
         polar rotations - N_I, BC_nR_I, KN_I, KC_nR_I, LC_nR_I;
         bipolar rotations - C_nZ_I, MN_I, MC_nR_I, KC_nZ_I, LC_nZ_I;
         non-polar rotations - D₁N_I, D_nR_I, KD₁N_I, KD_nR_I, L_nR_I.

Polarity of radial rays:
         polar - KN_I;
         bipolar - MN_I, KD₁N_I, KC_nZ_I, LC_nZ_I (if they exist, i.e. if the dilative
         rotation angle is a rational one);
         non-polar - KC_nR_I, KD_nR_I, MC_nR_I, LC_nR_I, LD_nR_I.

Form of the fundamental region:
         unbounded in the conformal symmetry groups C₂₁; bounded in conformal
         symmetry groups of the category C₂. 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 C₂₁, variation of
         all the boundaries is possible in groups of the types N_I, C_nZ_I, of
         non-reflectional boundaries in groups of the type D₁N_I, of non-inversional
         boundaries in groups of the type C_nR_I, while in groups of the type D_nR_I,
         it is not possible at all. Regaring the changes of the form of a fundamental
         region, conformal symmetry groups of the category C₂ offer similar
         possibilities as generating conformal symmetry groups of the category C₂₁.

Number of edges of the fundamental region:
         KC_nR_I, KD_nR_I - 4;
         KN_I - 4,6;
         MN_I, KD₁N_I, KC_nZ_I, LC_nR_I, LC_nZ_I, MC_nR_I, LD_nR_I - 3,4,5,6.

Among the continuous conformal symmetry groups, visually presentable are groups D_¥ R_I of the category C₂₁ and groups KD_¥ R_I, K₁C_nR_I, K₁D_nR_I, K₁D_¥ R_I, L₁C_nZ_I of the category C₂.

Cayley diagrams (Figure 4.1):

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