{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 D¥ ×C¥
{R,R1,Y} R2 = R12 = E YR = RY YR1 = R1Y (R1 = RX)
- pg
-
{X,Y,P} XY = YX P2 = E XPX = P < 2,2,¥ >
{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 D¥ ×D¥
{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 (¥,¥| 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:
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:
Polarity of rotations: polar rotations - p2, pmg,
pgg, p3, p4,
p6;
Polarity of generating translation axes:
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):
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 D¥ ×D¥
T1, T2, T3 = RT1R generate p2
R1, R3, Y = R2R4 generate pm D¥ ×C¥
Y, X = R1R3 generate p1 C¥ ×C¥
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+
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.
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.
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.
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