3.II.14E

Show that every isometry of Euclidean space $\mathbb{R}^{3}$ is a composition of reflections in planes

What is the smallest integer $N$ such that every isometry $f$ of $\mathbb{R}^{3}$ with $f(0)=0$ can be expressed as the composition of at most $N$ reflections? Give an example of an isometry that needs this number of reflections and justify your answer.

Describe (geometrically) all twelve orientation-reversing isometries of a regular tetrahedron.

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