3.1 The Symmetry Groups of RosettesDefinition 3.1.1. A subgroup D of D_{E}_{2} is the symmetry group of rosettes (abbreviated: the rosette group) if it does not contain translations. This name originated from the ornamental art [6, 8]. The elements of a rosette group are the symmetries of a rosette. The symmetry groups of rosettes will be denoted by R. So, the translation subgroup of each rosette group is trivial, while the point group is the realization of some subgroup of O_{2}. The lattice of every rosette group is the point (0,0). Let`s suppose that the rotation M_{f} from O_{R} is realized as (0,M_{f}) in R. The group where M_{f} is realized as a rotation about (0,0) is isomorphic to the rotation group about any other point x, x ¹ (0,0) (the isomorphism f : G® G(x,0)). Let`s suppose that the rotation from O_{R} is realized in R as (x,M_{y}), x ¹ (0,0). Then
The transformation obtained is a translation, so it follows that the rosette group contains a translation, and we have a contradiction. Without loosing generality, let the point (0,0) be the invariant point of rotations from R. All indirect transformations from O_{R} must be realized as reflections in lines which contain the point (0,0). In the contrary, by composing a reflection with the product of reflection and rotation, we will obtain in R a new rotation about some point different from (0,0). Theorem 3.1.1. The point (0,0) is invariant under all the transformations from R, this means
From Theorem 3.1.1. we see that the groups O_{R} and R are isomorphic. The group R is discrete, so O_{R} contains M_{q} such that "M_{Q} Î O_{R}, q £ Q (otherwise, Orb_{OR} would not be discrete). Let M_{Q} Î O_{R} and let n = min{N | Nq ³ Q}. From the way that n was chosen, we see that 0 £ nq-Q £ q. From M_{q}^{n} = M_{n}_{q} and M_{n}_{q}M_{-Q} = M_{n}_{q-Q} (The Angle Addition Theorem) it follows that M_{n}_{q-Q} Î O_{R} . Having (by a choice of q) nq-Q ³ q, we have: nq-Q = q. Rotations from R form a group, so: Lemma 3.1.1. The rotational subgroup of a rosette group R is generated by a rotation R = áM_{q} ñ, q Î [ 0,2p] . (1) In the case that O_{R} contains only direct transformations,
the unique possibility is R = áM_{[(2p)/
m]}ñ,
m Î N. Such a rosette group is isomorphic
to a cyclic group C_{m}.
(2) Let O_{R} contains n indirect transformations and a rotation M of the order m, so the number of rotations in O_{R} is m. Hence, for an indirect transformation S, the compositions SM, SM^{2}, ¼, SM^{m} are mutually different indirect transformations from O_{R}, so m £ n. On the other hand SMS, SM^{2}S, ¼, SM^{m}S are mutually different direct transformations from O_{R}, and we have n £ m. Therefore, m = n. We see that all indirect transformations from R have the form (0,M_{[l(2p) / m ]})(0,S), where l Î N and S is an indirect transformation from O_{R}. Hence, R = á(0,M_{[(2p)/
m]}),(0,S) ñ. Such
a group R is isomorphic to a dihedral group D_{m}.
Theorem 3.1.2. Every rosette group is isomorphic to a cyclic or dihedral group. According to H. Weyl [9], this theorem is attributed to Leonardo. |