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3.1 The Symmetry Groups of Rosettes




Definition 3.1.1. A subgroup D of DE2 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 Mf from OR is realized as (0,Mf) in R.

The group where Mf 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 OR is realized in R as (x,My), x ¹ (0,0). Then

(x,My)(0,Mf)(x,My)-1(x,My)-1 = (ae1+be2,MyMfMy-1Mf-1) = (ae1+be2 ,0) Î R,     a,b Î R.

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 OR 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 OR and R are isomorphic.

The group R is discrete, so OR contains Mq such that "MQ Î OR,     q £ Q (otherwise, OrbOR would not be discrete). Let MQ Î OR and let n = min{N | Nq ³ Q}. From the way that n was chosen, we see that 0 £ nq-Q £ q. From Mqn = Mnq and MnqM-Q = Mnq-Q (The Angle Addition Theorem) it follows that Mnq-Q Î OR . 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 = áMq ñ,    q Î [ 0,2p] .

(1) In the case that OR 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 Cm.

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(2) Let OR contains n indirect transformations and a rotation M of the order m, so the number of rotations in OR is m. Hence, for an indirect transformation S, the compositions SM, SM2, ¼, SMm are mutually different indirect transformations from OR, so m £ n. On the other hand SMS, SM2S, ¼, SMmS are mutually different direct transformations from OR, 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 OR.

Hence, R = á(0,M[(2p)/ m]),(0,S) ñ. Such a group R is isomorphic to a dihedral group Dm.

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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.

 

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