3. Discrete Subgroups of Euclidean GroupHaving represented the isometries of Euclidean plane as an algebraic structure, we will restrict our interest to the discrete subgroups of Euclidean group (see Appendix, Definition A.3.1). The considered groups are discrete, so in every such group we can choose a translation whose vector is minimal. The set of translations is independent if their vectors are independent. According to the number of independent translations contained in a particular group, there are three classes of discrete subgroups of Euclidean group E_{2}. The first is the class of discrete subgroups of E_{2} without translations - the symmetry groups of rosettes. This class is infinite. The second class contains the groups with a translation subgroup generated by one single translation - the symmetry group of friezes. That class contains 7 non-isomorphic symmetry groups. The third class are the wallpaper groups. Their translation subgroup is generated by two independent translations, and this class contains 17 non-isomorphic groups. The classification of the discrete subgroups mentioned will be considered in the next three chapters. In order to simplify, we will introduce few new concepts: (1) E_{2} is isomorphic to the semidirect product of the orthogonal group O_{2} and translational group T_{2} (see Chapter 2). The realization of the subgroup O_{2} in some discrete subgroup D of the Euclidean group E_{2} we will call the point group of D, and denote it as O_{D}. The realization of T_{2} we will call the translational group of D, and denote it as T_{D} . (2) In the further text, a subgroup of the discrete transformations of Euclidean group will be denoted as D_{E}_{2}; (3) We introduce the notion of the lattice of a discrete subgroup D. The lattice is a set of images of the origin, when T_{D} acts on it: Definition 3.1. The lattice of a discrete Euclidean group, denoted as R_{D}, is Orb_{TD}(0,0) (for the definition of orbit see Appendix). (4) Every point of a lattice R_{D} can be represented as t_{x} = (x,I) Î T_{D}. As T_{D} is the kernel of homomorphism p: D ® O_{K} , T_{D} is the normal subgroup of D (for the proof see [3]), and we have st_{x} s^{-1}Î T_{D}, sÎ O_{D}. As s = (v ,M) (see Chapter 2):
So, when O_{D} acts on the lattice, the lattice remains invariant.
Because y is an isomorphism:
This contradicts the fact that t_{2} and g_{2} don't commute. A rotation of order 2 can have as it's y-image a rotation of order 2 or a reflection. The same holds for reflections. Let r_{1} be a rotation from D_{1} such that y(r_{1}) = s_{2}, where s_{2} is a reflection from D_{2}. Let t_{2} be a translation from D_{2}, such that s_{2}t_{2} is a glide reflection. Now we have
where t_{1} is the translation from D_{1}. Because the product of a rotation of order 2 and the translation is a rotation of order 2, we will have here the isomorphism which sends a glide reflection into a rotation, and that is the contradiction. Theorem 3.2. The isomorphism of two subgroups of D_{E}_{2} preserves the type of isometries. |