Chapter 2.5


  Symmetry Groups
  of Ornaments G2




In the plane E2 there is 17 discrete symmetry groups without invariant lines or points, the crystallographic symmetry groups of ornaments: p1, p2, pm, pg, pmm, pmg, pgg, cm, cmm, p4, p4m, p4g, p3, p3m1, p31m, p6, p6m, two visually presentable symmetry groups of semicontinua p101m (s1m), p10mm (smm) and also one visually presentable symmetry group of continua p00Ąm (sĄĄ). The simplified International Symbols by Hermann and Maugin (H.S.M. Coxeter, W.O.J. Moser, 1980, pp. 40) are used to denote the discrete symmetry groups of ornaments, while the symbols introduced by A.V. Shubnikov, V.A. Koptsik (1974), B. Grünbaum, G.C. Shephard (1983) are used to denote the continuous symmetry groups of ornaments - symmetry groups of semicontinua and continua.

A complete survey of the presentations, structures, possible decompositions and Cayley diagrams of the 17 discrete symmetry groups of ornaments can be found in the monograph by H.S.M. Coxeter and W.O.J. Moser: Generators and Relations for Discrete Groups (1980, pp. 40-51). In the same book one can find discussion of all the symmetry groups of ornaments treated as subgroups of the maximal symmetry groups of ornaments p4m and p6m, generated by reflections (pp. 51-52), the table surveys of group-subgroup relations and minimal indexes of subgroups of the symmetry groups of ornaments (pp. 136, Table 4).

Presentations and structures:


p1
{X,Y}     XY = YX     CĄ×CĄ
{X,Y,Z}     XYZ = ZYX = E     (Z = X-1Y-1)

p2
{X,Y,T}     XY = YX     T2 = (TX)2 = (TY)2 = E
{T1,T2,T3}     T12 = T22 = T32 = (T1T2T3)2 = E    
(T1 = TY, T2 = XT, T3 = T)
{T1,T2,T3,T4}     T12 = T22 = T32 = T22 = T1T2T3T4 = E    
(T4 = T1T2T3 = T1X)

pm
{X,Y,R}     XY = YX     R2 = (RX)2 = E)     RYR = Y     DĄ ×CĄ
{R,R1,Y}     R2 = R12 = E     YR = RY     YR1 = R1Y     (R1 = RX)

pg
{X,Y,P}     XY = YX     P2 = E     XPX = P     < 2,2,Ą >
{P,Q} P2 = Q2     (Q = PX)

cm
{P,Q,R}     P2 = Q2     R2 = E     RPR = Q
{P,R}     R2 = E     RP2 = P2R
{R,S}     R2 = E     (RS)2 = (SR)2     (S = PR)

pmm
{R,R1,R2,Y}     R2 = R12 = R22 = (RR1)2 = (R1R2)2 = (R2Y)2 = E
YR = RY     YR1 = R1Y     DĄ ×DĄ
{R1,R2,R3,R4}
R12 = R22 = R32 = R42 = (R1R2)2 = (R2R3)2 = (R3R4)2 = (R4R1)2 = E
(R1 = R, R3 = R1, R4 = R2Y)

pmg
{P,Q,R}     P2 = Q2     R2 = (RP)2 = (RQ)2 = E
{R,T1,T2}     R2 = T12 = T22 = E     T1RT1 = T2RT2     (T1 = PR, T2 = QR)

pgg
{P,Q,T}     P2 = Q2     T2 = E     TPT = Q-1     (Ą,Ą| 2,2)
{P,O}     (PO)2 = (P-1O)2 = E     (O = PT)

cmm
{R1,R2,R3,R4,T}     T2 = E     TR1T = R3     TR2T = R4
R12 = R22 = R32 = R42 = (R1R2)2 = (R2R3)2 = (R3R4)2 = (R4R1)2 = E
{R1,R2,T}     R12 = R22 = T2 = (R1R2)2 = (R1TR2T)2 = E

p4
{T1,T2,T3,T4,S}     T12 = T22 = T32 = T42 = T1T2T3T4 = E
S4 = E     S-iT4Si = Ti     i = 1,2,3     [4,4]+
{S,T}     S4 = T2 = (ST)2 = E     (T = T4)

p4m
{R1,R2,R3,R4,R}     R2 = E     RR1R = R4     RR2R = R3
R12 = R22 = R32 = R42 = (R1R2)2 = (R2R3)2 = (R3R4)2 = (R4R1)2 = E     [4,4]
{R,R1,R2}     R2 = R12 = R22 = (RR1)4 = (R1R2)2 = (R2R)4 = E

p4g
{R1,R2,R3,R4,S}     S4 = E     S-iR4Si = Ri     i = 1,2,3
R12 = R22 = R32 = R42 = (R1R2)2 = (R2R3)2 = (R3R4)2 = (R4R1)2 = E     [4+,4]
{S,R}     R2 = S4 = (RS-1RS)2 = E     (R = R4)

p3
{X,Y,Z,S1}     XYZ = ZYX     S13 = E     S1-1XS1 = Y
S1-1YS1 = Z     S1-1ZS1 = X     D+
{S1,S2,S3}     S13 = S23 = S33 = S1S2S3 = E     (S2 = S1X, S3 = X-1S1)
{S1,S2}     S13 = S23 = (S1S2)3 = E

p31m
{S1,S2,R}     S13 = S23 = (S1S2)3 = E     R2 = E     RS1R = S2-1     [3+,6]
{R,S}     R2 = S3 = (RS-1RS)3 = E     (S = S1)

p3m1
{S1,S2,R}     S13 = S23 = (S1S2)3 = E     R2 = E
RS1R = S1-1     RS2R = S2-1     D
{R1,R2,R3}     R12 = R22 = R32 = (R1R2)3 = (R2R3)3 = (R3R1)3 = E
(R1 = RS2, R2 = S1R, R3 = R)

p6
{S1,S2,T}     S12 = S22 = (S1S2)2 = E     T2 = E     TS1T = S2     [3,6]+
{S,T}     S3 = T2 = (ST)6 = E     (S = S1)

p6m
{R1,R2,R3,R}     R12 = R22 = R32 = (R1R2)3 = (R2R3)3 = (R3R1)3 = E
R2 = E     RR1R = R3     RR2R = R2     [3,6]
{R,R1,R2}     R2 = R12 = R22 = (R1R2)3 = (R2R)2 = (RR1)6 = E


All the discrete symmetry groups of ornaments are subgroups of the groups generated by reflections p4m and p6m, given by the presentations:

p4m
{R,R1,R2}     R2 = R12 = R22 = (RR1)4 = (RR1)2 = (R2R)4 = E     [4,4]

R1, S = RR2 generate p4g [4+,4]
S, T1 = R1R2 generate p4 [4,4]+
T1, R2, R1 = R2S generate cmm
P = R1S, O = SR1 generate pgg
R, P generate cm
P, Q = RPR generate pg
T1, T2 = S2, R4 = RR1 generate pmg
R1, R2, R4, R3 = SR generate pmm DĄ ×DĄ
T1, T2, T3 = RT1R generate p2
R1, R3, Y = R2R4 generate pm DĄ ×CĄ
Y, X = R1R3 generate p1 CĄ ×CĄ

p6m
{R,R1,R2}     R2 = R12 = R32 = (R1R2)3 = (R2R)2 = (RR1)6 = E     [3,6]

R, S = R1R2 generate p31m [3+,6]
S, T = R2R generate p6 [3,6]+
R1, R2, R3 = RR1R generate p3m1 D
S1 = R1R2, S2 = R2R3 generate p3 D+

Form of the fundamental region: bounded, offers a change of boundaries that do not belong to reflection lines. The groups generated by reflections pmm, p3m1, p4m, p6m do not offer any change of the shape of a fundamental region.

Number of edges of the fundamental region:
                                    pm, pmm - 4;
                                    p4m - 3,4;
                                    p1, pg, p3 - 4,6;
                                    p4, p4g - 3,4,5;
                                    p31m, p6m - 3,4,6;
                                    p2, pmg, pgg, cm, cmm, p6 - 3,4,5,6.

Enantiomorphism: p1, p2, p3, p4, p6 possesses enantiomorphic modifications, while in the other cases the enantiomorphism does not occur.

Polarity of rotations: polar rotations - p2, pmg, pgg, p3, p4, p6;
                                  non-polar rotations - pmm, cmm, p3m1, p4m, p6m.
                                  The symmetry group p4g contains polar 4-rotations
                                  and non-polar 2-rotations, and the symmetry group p31m
                                  contains polar and non-polar 3-rotations.

Polarity of generating translation axes:
                                   both axes are polar - p1, pg,p3, p31m;
                                   both axesare bipolar -p2, pgg, p4, p4g, p6;
                                   one axis is polar, the other non-polar - pm, cm;
                                   one axisis non-polar, the other bipolar - pmg, cmm;
                                   both axes are non-polar - pmm, p4m, p3m, p6m.

The first studies on the symmetry groups of ornaments G2 were undertaken by C. Jordan (1868/69), but he did not succeed in discovering all the existing 17 symmetry groups. Namely, he omitted the group pgg, discovered by L. Sohncke (1874), who, on the other hand, omitted three other groups. The complete list of the discrete symmetry groups of ornaments was given by E.S. Fedorov (1891b).

Cayley diagrams (Figure 2.57):



Figure 2.57

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