Chapter 2

  Symmetry Groups
  of Friezes G21

In the plane S2 and E2 there are seven discrete, one-dimensional, line groups of isometries, the symmetry groups of friezes G21: p11 (11), p1g (1g), p12 (12), pm1 (m1), p1m (1m), pmg (mg), pmm (mm) and two visually presentable continuous symmetry groups of friezes: p0m1=pm01 (m01) and p0mm= pm0m (m0m).

To denote them, we have used the simplified version of International Two-dimensional Symbols (M. Senechal, 1975; H.S.M. Coxeter, 1985). Here, the first symbol represents an element of symmetry perpendicular to the direction of the translation, while the second denotes an element of symmetry parallel or perpendicular (exclusively for 2-rotations) to the direction of the translation.

Presentations and structures:

11 {X} C
1g {P} C
12 {X,T} T2 = (TX)2 = E D
{T,T1} T2 = T12 = E (T1 = TX)
m1 {X,R1} R12 = (R1X)2 = E D
{R1,R2} R12 = R22 = E (R2 = R1X)
1m {X,R} R2 = E RX = XR C ×D1
mg {P,R1} R12 = (R1P)2 = E D
{R1,T} R12 = T2 = E (T = R1P)
mm {X,R,R1} R2 = R12 = (R1X)2 = E RX = XR RR1 = R1R D ×D1
{R,R1,R2} R2 = R12 = R22 = E RR1 = R1R RR2 = R2R (R2 = R1X)

Form of the fundamental region: unbounded, allows variation of the boundaries that do not belong to reflection lines.

Enantiomorphism: 11, 12 possess enantiomorphic modifications, while in other cases the enantiomorphism does not occur.

Polarity of rotations: polar rotations - 12, mg;
                                  non-polar rotations - mm, m0m.

Polarity of translations: polar translations - 11, 1g, 1m;
                                      bipolar translations - 12;
                                      non-polar translations - m1, mg, mm, m01, m0m.

The table of minimal indexes of subgroups in groups:

11 1g 12 m1 1m mg mm
11 2
1g 2 3
12 2 2
m1 2 2
1m 2 2 2
mg 4 2 2 2 3
mm 4 4 2 2 2 2 2

All the discrete symmetry groups of friezes are subgroups of the group mm generated by reflections and given by the presentation:

{R,R1,R2}     R2 = R12 = R22 = E     RR1 = R1R     RR2 = R2R     D ×D1

R1, T = RR2 generate mg D
R, X = R1R2 generate 1m C ×D1
R1, X generate m1 D
X, T = RR1 generate 12 D
P = RR1R2 generates 1g C
X generates 11 C

The survey of the characteristics of the symmetry groups of friezes relies on the work of A.V. Shubnikov, V.A. Koptsik, 1974; H.S.M. Coxeter, W.O.J. Moser, 1980.

The first derivation of the symmetry groups of friezes as the line subgroups of the symmetry groups of ornaments G2 and their complete list, was given by G. Pòlya (1924), P. Niggli (1926) and A. Speiser (1927).

Cayley diagrams (Figure 2.28):

Figure 2.28