# Classical physics is all reflection in Euclidean a rotation in higher dimension space?

I'm reading book in classical physics where it mentioned that, quote:

" the transformation matrix (of two different cartesian coordinate systems) was orthogonal, so the transformation was reflection or a rotation (Goldstein, Poole, and Safko,2002)".

I was thinking that for a reflection along an axis(say y=x) of a 2D plane(say x-y plane) was equivalent as the rotation along $(y=x,z)$ in a 3D space.

My question was that:

1. In Euclidean space, could all the reflection in a low dimension space being written as a rotation in higher dimension space?

2. Further, if it was true, could it be applied to any other metric space? That was, could all the reflection tensor being written as a rotation tensor.

## 2 Answers

No.

Even for the 2D/3D example you give, a reflection is not equivalent to any rotation.

See "https://en.wikipedia.org/wiki/Orthogonal_transformation" for details on how the transformation matrices differ.

To convince yourself without using any mathematics, hold both hands out in front of you with palms aimed toward you and both thumbs pointing up. One hand will be a reflection of the other. Then keep one hand fixed and see if you can rotate the other one any possible way to make them look the same.

• Of course rotation was different from reflection in a fixed group. The operation involved the rise of dimension and thus an expansion of the structure, asked to be put in a metric form. (Just an example in your case, when you flipped you hand, that was a reflection in a 2D plane, but a simple rotation by Euler angle in 3D space. ) – J C Mar 2 '18 at 1:44
• Well, he is asking about a rotation in more than 3 dimensions, and the hand-test cannot rule that out--at least I can't rotate my hands though the 4th and 5th dimensions. – JEB Apr 5 '18 at 1:50

Just in Euclidean space, it could happen. We only need to expand the space by an extra orthogonal direction, and then we could write reflection with respect to any axis into a rotation, by rotate with respect to the newly created axis and keep the rest axis fixed.

However, in general, it was not true that we could always obtain a solution. Because between any two arbitrary function bases, we were not guaranteed to obtain a transformation.