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.