I'm a bit confused about parity transformations (reflections).
A parity operator $\pi$ takes a vector $(x, y, z)$ to $(-x, -y, -z)$. So in a $3$ dimensional space this takes a vector and points it along its "old tail".. I know that $\pi$ has determinant $-1$ so this can't be done with an $SO(3)$ matrix (rotation), but in my head it doesn't quite work.. I mean I can totally see a vector being taken and continuously moved to the antipodal point! For instance if you have $(0, 0, 1)$ I can take it to $(0, 0, -1)$ by rotating it 180° around the $x$ axis for instance, this being done with an orthogonal matrix! And isn't that a reflection?!?
Thanks!!
You can rotate any single vector to its antipodal vector. You cannot rotate all vectors at once to their antipodes by a single rotation, or even by a composition of rotations (for instance, for a single rotation the vectors on the axis won't move at all). The orientation is the simplest explanation why this is not possible. Try for some familiar shape (for instance the contour of your country) drawing all the antipodal points on a sphere, and you will see that you get an isometric image, but that it has the opposite orientation.
For instance, here it is for the U.S.A.