SYMMETRY IN

A similarity symmetry group is any group of similarity
transformations (H.S.M. Coxeter, 1969, pp. 72), at least one of
which is not an isometry. According to the theorem on the
existence of an invariant point of every discrete similarity
transformation that is not an isometry (E.I. Galyarski,
A.M. Zamorzaev, 1963; H.S.M. Coxeter, 1969),
all the similarity
symmetry groups of the plane *E*^{2} belong to the similarity
symmetry groups with an invariant point. From the relationships:
*S*_{2} = *S*_{20}, *S*_{21} =
*S*_{210}, *S*_{210} Ì *S*_{2}, we can
conclude that for a full understanding of the similarity symmetry
groups of the plane *E*^{2}, it is enough to analyze the similarity
symmetry groups of the category *S*_{20}. Owing to the existence
of an invariant point, the similarity symmetry groups of the
category *S*_{20} are also called the similarity symmetry groups
of rosettes *S*_{20}, and the corresponding figures possessing
such a symmetry group are called similarity symmetry rosettes.