3.1 The Symmetry Groups of Rosettes

Definition 3.1.1. A subgroup D of D_E2 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 O₂. 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

OrbR(0,0) = (0,0).

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_qⁿ = M_nq and M_nqM_-Q = M_nq-Q (The Angle Addition Theorem) it follows that M_nq-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, SM², ¼, SM^m are mutually different indirect transformations from O_R, so m £ n. On the other hand SMS, SM²S, ¼, SM^mS 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.