Does reflection in 2D equal 180° rotation through 3D?

[madness]

I've noticed the the reflection of a 2-d shape about some line is the same as rotating it by 180 degrees about that line, where it rotates through the third dimension. Does this generalise to higher dimensions? I hope it's clear what I mean here.

 

[Ben Niehoff]

Yes, it generalizes exactly as you would expect. Reflecting in a plane in 3D is the same as rotating about that plane in 4D, etc.

 

[JSuarez]

Not only that, but any rotation can be obtained by two successive reflexions, through different lines. See if you can prove that.