Rotation is kind of a 2D thing. Even in 3D or higher, it's equivalent to slicing things into planes and rotating them in 2D. And it doesn't exist in 1D.
Is there something that doesn't exist in 2D (or 3D or etc.), but only in higher dimensions that preserved euclidean congruence, but isn't the usual translation/rotation? :>
AH reflection would satisfy that, wouldn't it!
And..that is quite similar! o,o
If we wanted to exclude it, perhaps we should say it must preserve chirality as well? But then how do we define chirality outside of rotation? XD
Also, reflection exists in 2D as well so it can't be the answer xD
(Heck, it exists in 1D! o,o )
@codepuppy No, an isometry is always an orthogonal transformation. Generally the latter is taken as the definition of a rotation/reflection.
@codepuppy (Well, a linear isometry. But all isometries are affine, so they are all combinations of a translation and an orthogonal transformation).
@axiom @QContinuumHypothesis Yus I know about affine transforms and matrices :>
But not all of them are congruence-preserving (thanks @axiom for showing me the word "isometry"! :D ), like skewing and scaling (non-uniform scaling doesn't even preserve similarity!)
However, just looking at linear transform matrices, you probably wouldn't come up with rotation as its own thing--at least I wouldn't! (I mean who would think, well if we make these coefficients be these bizarre complex super-nonlinear *periodic* special functions of some other new variables, then we can make a subclass of linear transforms that preserves this property apart from simple translation/addition! (which I think would be obvious))
Being like 'hey what if we just multiplied everything by a constant' would probably be a thing we'd think of, and that's scaling :3
So I wonder..how do we know there isn't some other funky subclass that does a similar thing in 4 or 29 or 8,243,721 dimensions? XD
@codepuppy @QContinuumHypothesis I've already told you, the isometries are exactly the orthogonal transformations. There is no other transformation in any dimension.
@codepuppy Rotation about more than one axis at a time.
@jhertzli Oh yeah, I forgot about that! :D
But now that I think about it, I thought Euler proved that that's equivalent to a single rotation around a very precisely chosen axis, or am I totally wrong about that?
(I thought that--it's just that in physics, it's easier to describe an object's motion as multiple rotations than by a time-varying axis. But I might be wrong!)
@codepuppy @jhertzli The composition of rotations is a rotation.
@codepuppy The more general case of rotation/translation/reflection is linear transformations, which work in any number of dimensions...
Oh but then isn't euclidean congruence defined in terms of translation/rotation? XD
Could we define it as 'transforms that preserve the inner product between any two vectors, and the length of any vector' or would that be not equivalent to translation+rotation in 2D/3D? :>