Any rotation in the plane is really two mirror flips in disguise
Stand between two mirrors set at an angle, and reflecting twice turns out to be the same as rotating by double that angle. That fact sits at the heart of the orthogonal group, the collection of every motion that keeps distances intact while holding one point fixed: all the rotations and reflections of space.
In n dimensions this group is written O(n). Its members can be pictured as rigid moves around the origin, or as square matrices whose inverse is simply their transpose, combined by multiplication. The name comes from a defining habit: up to a uniform change of scale, these are exactly the linear maps that send perpendicular directions to perpendicular directions. The group is compact and is both a Lie group and an algebraic group.
Every such matrix has determinant 1 or minus 1, and that splits the group neatly in two. The determinant 1 half, the special orthogonal group SO(n), holds the rotations, which preserve orientation. The minus 1 half holds moves that turn space into its mirror image, and it cannot stand alone as a group, since composing any two of them gives determinant 1 and lands back among the rotations. In the plane, SO(2) is commutative, so the order of two rotations never matters, but from three dimensions upward it no longer is.
Reflections are the basic building blocks. A reflection flips space across a hyperplane, a flat slice one dimension lower. In the plane, a rotation by some angle equals two reflections whose mirror lines meet at half that angle, and in general at most n reflections are enough to produce any element of O(n). The Cartan-Dieudonne theorem extends this to a much broader algebraic setting.
Three dimensions give a result with a famous name. Euler's rotation theorem says that any rotation of ordinary space, other than doing nothing, turns about one fixed axis by one fixed angle. More generally, every orthogonal transformation can be broken into independent turns in separate planes plus some directions that are either kept or flipped, and it counts as a rotation exactly when an even number are flipped.
Source: Orthogonal group
