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3. Discrete Subgroups of Euclidean Group

Having 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 E2. The first is the class of discrete subgroups of E2 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) E2 is isomorphic to the semidirect product of the orthogonal group O2 and translational group T2 (see Chapter 2). The realization of the subgroup O2 in some discrete subgroup D of the Euclidean group E2 we will call the point group of D, and denote it as OD. The realization of T2 we will call the translational group of D, and denote it as TD .

(2) In the further text, a subgroup of the discrete transformations of Euclidean group will be denoted as DE2;

(3) We introduce the notion of the lattice of a discrete subgroup D. The lattice is a set of images of the origin, when TD acts on it:

Definition 3.1. The lattice of a discrete Euclidean group, denoted as RD, is OrbTD(0,0) (for the definition of orbit see Appendix).

(4) Every point of a lattice RD can be represented as tx = (x,I) Î TD. As TD is the kernel of homomorphism p: D ® OK , TD is the normal subgroup of  D (for the proof see [3]), and we have stx s-1Î TD, OD. As s = (v ,M) (see Chapter 2):

stx s-1
= (v,M)(x,I)(M-1(v),M-1
= (v,M)(x-M-1(v),M-1)
= (v+M(x-M-1(v)),MM-1)
= (v+M(x)-v,I)
= (M(x),I)
Þ (M(x),I ) Î TD Þ M(x) Î RD.

So, when OD acts on the lattice, the lattice remains invariant.

Theorem 3.1. The orthogonal subgroup OD of a discrete subgroup D of Euclidean group preserves the lattice RD.

(5) If two discrete subgroups of Euclidean group are isomorphic, then their isomorphism preserves the type of isometries.

The function sending one group of isometries into another preserves the type of transformations if it sends translations in translations, rotations in rotations, reflections in reflections and glide reflections in glide reflections. The proof follows: let D1 @y D2, where D1, D2 Î DE2. As isomorphism preserves the order, every rotation from D1 of an order higher than 2 goes into a rotation of the same order from D2.

Translations have no finite order, so that the image of a translation from D1 must be either a translation or a glide reflection. Let`s suppose that the translation t1Î D1 goes to a glide reflection g2 Î D2. Let t2 be the translation from D2 which does not commute with g2 (every translation whose vector is not parallel with the glide reflection axis will satisfy that condition). There exists x1 Î D1, such that y(x1) = t2. Hence, x1 must be either a translation or a glide reflection. So x12 is a translation. As every two translations commute, we have

x12t1 = t1x12

Because y is an isomorphism:

t22g2 = y(x12)y(t1) = y(x12t1) = y(t1x12) = y(t1)y(x12) = g2t22

This contradicts the fact that t2 and g2 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 r1 be a rotation from D1 such that y(r1) = s2, where s2 is a reflection from D2. Let t2 be a translation from D2, such that s2t2 is a glide reflection. Now we have

y-1 (s2t2) = y-1(s2)y-1(t2) = r1t1,

where t1 is the translation from D1. 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.

So we proved:

Theorem 3.2. The isomorphism of two subgroups of DE2 preserves the type of isometries.

From Theorem 3.2. follows that:

Corollary 3.1. If two subgroups of DE2 are isomorphic, their orthogonal groups are isomorphic as well.

Let D1, D2 Î DE2, D1 @ fD2, and let d1 Î D1 be decomposed in D1 in the product of two isometries. Then, from Theorem 3.2. follows that f(d1) is D2-decomposable in two isometries.

Corollary 3.2. If two subgroups of DE2 are isomorphic, then their elements can be decomposed in the same types of isometries.

Corollary 3.3. If two subgroups of DE2 are isomorphic, then the product of two elements from one group and the product of their isomorphic images from the other group belong to the same type of isometries.