Note of Michael Artin "Algebra" Chapter 6 "Symmetry" (to complete)

6.1 SYMMETRY OF PLANE FIGURES

Bilateral, rotational, translational, glide symmetry, and their combinations.

6.2 ISOMETRIES

6.2.1 Def. (Distance, Isometry)

The distance between points of \(\mathbb{R}^n\) is the length \(|u-v|\) of the vector \(u-v\). An isometry of \(n\)-dimensional space \(\mathbb{R}^n\) is a distance-preserving map \(f\) from \(\mathbb{R}^n\) to itself, a map such that, \(\forall u \in \mathbb{R}^n\),

\[|f(u) - f(v)|=|u-v|. \]

An isometry will map a figure to a congruent figure.

6.2.2. Examples.

• (a) Orthogonal liear operators are isometries.
  • Orthogonal operator preserve dot product \(\rightarrow\) distance.
• (b) Translation \(t_a(x) = x + a\) by vector \(a\) is isometry.
• (c) The composition of isometries is an isometry.

6.2.3 Theo.

The following conditions on a map \(\mathbb{R}^n \to \mathbb{R}^n\) are equivalent:

• (a) \(\varphi\) is an isometry that fixes the origin: \(\varphi(0)=0\),
• (b) \(\varphi\) preserves dot products: \((\varphi(v)\cdot \varphi(w)) = (v \cdot w), \forall v,w\),
• (c) \(\varphi\) is an orthogonal linear operator.

6.2.4 Lemma.

\[x,y \in \mathbb{R}^n, (x \cdot x) = (y\cdot y) = (x\cdot y) \rightarrow x=y \]

6.2.7 Cor.

Every isometry \(f\) of \(\mathbb{R}^n\) is the composition of an orthogonal linear operator and a translation.
If \(f\) is an isometry and if \(f(0)=a\), then \(f = t_a\varphi\), where \(t_a\) is a translation and \(\phi\) is an orthogoal linear operator, and this expression for \(f\) is unique.

6.2.9 Cor.

The set of all isometries of \(\mathbb{R}^n\) forms a group that we denote by \(M_n\), with composition of functions as its law of composition.

The Homomorphism \(\pi: M_n \to O_n\), defined by dropping the translation part of an isometry \(f\).

6.2.10 Prop.

The map \(pi\) is a surjective homomorphism. Its kernel is the set \(T=\{t_v\}\) of translations, which is a normal subgroup of \(M_n\).

6.2.11 Change of Coordinates

Our change in coordinates will be given by some isometry, let's denote it by \(\eta\). Let the new cooirdinate vectors of \(p\) and \(q\) be \(x'\) and \(y'\). The new formula \(m'\) for \(f\) is the one such that \(m'(x')=y'\). We also have the formula \(\eta(x') = x\) analogous to the change of basis formula \(PX'=X\).

\[m' = \eta^{-1} m\eta \]

6.2.12 Cor.

The homomorphism \(\pi\) does not change when the origin is shifted by a translation.

6.2.13 Orientation

The determinant of an orthogonal operator \(\varphi\) on \(\mathbb{R}^n\) is \(\pm 1\). There are 2 orientations, orientation-preserving and orientation-reversing. The map

\[\sigma: M_n \to \{ \pm 1\} \]

is a group homomorphism.

6.3 ISOMETRIES OF THE PLANE