Transformations and Groups

As the basis for the **classification of the symmetry groups***G* three elements were taken into consideration: the types of
symmetries (isometries, similarity symmetries, conformal
symmetries) that occur in *G*, the space on which the group *G*
acts, and the sequence of maximal included proper subspaces,
invariant with respect to the group *G*. According to this, the
* Bohm symbols*
(J. Bohm, K. Dornberger-Schiff, 1966)
are used for the categorization of the groups of isometries. Symbols
of the same type are applied to the similarity symmetry and
conformal symmetry groups. For example, the symmetry group of
square

A transformation *S* of *n*-dimensional space is called
* indirect* (or reflective, sense reversing, opposite, odd)
if it transforms any oriented (

(a) Direct and (b) indirect plane isometry. |

As an elementary isometric transformation we can take
the * reflection*, non-identical isometry of space

In the space *E*^{2} (plane) we distinguish the following
isometric transformations (Figure 1.7):

(a) Identity transformation; (b) reflection; (c) rotation; (d) translation; (e) glide reflection. |

1) **identity transformation***E*, with the minimal reflectional representation
of the length 2 (*R*^{2} = *E*);

2) **reflection***R*;

3) **rotation***S* = *R*_{1}*R*_{2}, the product of two reflections in
the reflection lines crossing in the invariant point (center of
rotation). The oriented angle of rotation is equal to twice the
angle between the reflection lines *R*_{1}, *R*_{2};

4) **translation***X* = *R*_{1}*R*_{2}, the product of two reflections
with parallel reflection lines, such that the translation vector
is perpendicular to them and equal to twice the oriented distance
between the reflection lines *R*_{1}, *R*_{2};

5) **glide reflection***P* =
*R*_{3}*X* = *XR*_{3} = *R*_{1}*R*_{2}*R*_{3}, the
commutative product of a translation *X* and a reflection *R*_{3}
with the reflection line parallel to the translation axis.

In the case of rotation, if the relation
*S* = *R*_{1}*R*_{2} =
*R*_{2}*R*_{1} holds, i.e.
if the reflection lines *R*_{1},
*R*_{2} are perpendicular, as a result we get the special
involutional rotation - **central reflection***Z* (two-fold
rotation, half-turn, point-reflection) (Figure 1.8).

Central reflection |

When orientation is considered we distinguish *
direct transformations* (or sense preserving transformations):
identity transformation

If a symmetry transformation *S* can be represented as a
composition *S* = *S*_{1}¼*S*_{n} such that *S*_{i}*S*_{j}
= *S*_{j}*S*_{i},
*i*,*j* = 1,¼,*n*, we can call it a * complex* or

An analogous procedure makes possible the classification
of isometries of the space *E*^{3}, where each isometry can be
represented as the composition of four plane reflections at the
most. Besides the transformations of the space *E*^{2} afore
mentioned with the line reflections substituted by plane
reflections, as the new transformations of the space *E*^{3} we
have two more transformations. They are a direct isometry -
* twist* (screw), the commutative composition of a rotation
and a translation, the canonic representation of which consists
of four plane reflections and indirect isometry -

For every element *S*_{1} of a transformation group *G* we
can define the * conjugate* of the element

{*S*} *S*^{4} = *E*,
*S*^{R} = *S*^{3},
(*S*^{2})^{R} = *S*^{2},
(*S*^{3})^{R} = *S*,

For a figure *f* with the symmetry group *G*_{f}, which
consists only of direct symmetries, it is possible to have the
* enantiomorphism* -

"Left" and "right" rosette with the symmetry group |

The results of composition of plane isometries are
different * categories of groups* of isometries of the space

Five plane Bravais lattices. |

Because the symmetry groups of friezes *G*_{21} are
groups of isometries of the plane *E*^{2} with an invariant line,
they cannot have rotations of an order greater than 2.

For the symmetry groups of ornaments *G*_{2} so-called
* crystallographic restriction* holds, according to which
symmetry groups of ornaments can have only rotations of the order

In isometry groups all distances between points under the
effect of symmetries remain unchanged and the * congruence*
of homologous figures is preserved. Consequently, the same holds
for all other geometric properties of such figures, so that the

The next class of symmetry groups we shall consider are
the * similarity symmetry groups*. A

(i) **central dilatation***K* (or simply dilatation), a
transformation which to each vector (*A*,*B*)
assigns the vector (*A*',*B*'), such that
*A*' = *K*(*A*), *B*' = *K*(*B*)
and (*A*',*B*') = *k*(*A*,*B*),
where the coefficient of the dilatation is
*k* Î Â\{-1,0,1} ;

(ii) **dilative rotation***L*, the commutative composition of
a central dilatation *K* and a rotation, with a common invariant
point;

(iii) **dilative reflection***M*, the commutative
composition of a dilatation *K* and a reflection in the
reflection line containing the invariant point (center) of the
dilatation *K* (Figure 1.11).

(a) Dilatation; (b) dilative rotation; (c) dilative reflection. |

Those transformations are, in the given order, isomorphic
with the isometries of the space *E*^{3}: translation, twist and
glide reflection. They make possible the extension of the
symmetry groups of rosettes *G*_{20} by the external
automorphism, having as the result **similarity symmetry
groups***S*_{20} that we will, thanks to the existence of the
invariant point, call the * similarity symmetry groups of
rosettes*.

Dilatations *K* and dilative rotations *L* are direct,
while dilative reflections *M* are indirect transformations. They
all possess the properties of equiangularity and equiformity. All
other aspects of similarity symmetry groups (the problems of
enantiomorphism, fundamental regions, tessellations,¼)
will be discussed analogously to the case of isometry groups.

Further generalization leads to * conformal
transformations* or circle preserving transformations of the
plane

Circle inversion. |

Besides the circle inversion *R*_{I}, by composing it with
isometries maintaining invariant the circle line *c* of the
inversion circle *c*(*O*,*r*) - with a reflection with reflection
line containing the circle center *O* or with a rotation with the
rotation center *O*, we have two more conformal transformations:

(i) **inversional reflection***Z*_{I} = *R*_{I}*R*
= *RR*_{I}, the
involutional transformation, the commutative composition of a
reflection and a circle inversion;

(ii) **inversional
rotation***S*_{I} = *SR*_{I} = *R*_{I}*S*,
the commutative composition of a
rotation and a circle inversion (Figure 1.13).

(a) Inversional reflection; (b) inversional rotation. |

Those three conformal symmetry transformations, besides
isometries and similarity symmetry transformations, constitute
the finite and infinite **conformal symmetry groups***C*_{21},
*C*_{2} - * conformal symmetry groups of rosettes* in

As an extension of the symmetry groups of rosettes
*G*_{20} we have the **finite conformal symmetry groups***C*_{21} isomorphic with the symmetry groups of **tablets***G*_{320}. As a further extension of finite conformal symmetry
groups *C*_{21} by the similarity symmetry transformations *K*,
*L*, *M*, we get the **infinite conformal symmetry groups***C*_{2}. The similarity symmetry groups *S*_{20} and the infinite
conformal symmetry groups *C*_{2} are isomorphic with the line
symmetry groups of the space *E*^{3} - the symmetry groups of
**rods***G*_{31}. In line with the isomorphism mentioned, all
similarity symmetry and conformal symmetry transformations offer
a reflectional (canonic) representation by, at most, four
reflections (reflections and circle inversions). By applying this
isomorphism, ornamental motifs which correspond to the similarity
symmetry and conformal symmetry groups, satisfy one more scope of
painting: adequate interpretation of space objects in the plane.
The plane structures obtained are called * generalized
projections* of the symmetry groups of tablets