# B In higher dimensions, are there more than just rotations and displacements?

#### Hiero

In 3D the most general motion of a rigid body consists of a displacement and a rotation.

In higher dimensions is this still the most general motion? Or are there unexpected ways of moving with more freedom?

One subtlety, for example, is that we would have to allow for multiple rotations since two rotations do not necessarily compose into a single rotation in dimensions higher than three.

Are there any other surprises, or is it just displacements and rotations all the way up?

#### mfb

Mentor
Isn't every continuous change that keeps all distances and angles the same a displacement or rotation by definition?
You get more complex rotations.

#### Math_QED

Homework Helper
You can get double rotations, for example.

#### Hiero

Isn't every continuous change that keeps all distances and angles the same a displacement or rotation by definition?
You get more complex rotations.
I never thought of it in that definition, but I suppose that’s fitting! Is the reason for the word “continuous” just to exclude reflections?

What I had in mind by ‘rotation’ was a motion occurring in a plane with all lines orthogonal being unaffected. I now think this is called a “simple rotation,” at least in this Wikipedia page: https://en.m.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space

You can get double rotations, for example.
I suppose double rotations are like a composition of simple rotations in independent planes (which can’t be reduced to a simple rotation) but I’m not sure if that’s all there is to it because I don’t find Wikipedia particularly readable.

So once we hit 6 dimension, since we can choose 3 mutually orthogonal planes, would we have ‘triple rotations’ possible?

Any other resources will be much appreciated!
(I kept getting computer science related search results...)

Thanks.

#### WWGD

Gold Member
Are you considering isometries, or what types of transformations?

#### Hiero

Are you considering isometries, or what types of transformations?
I’ve never heard of these. They preserve distance but not angles? Can I have an example maybe?

The image I had in mind when asking this question was a continuous transformation between two Cartesian bases. Specifically I was doing a classical rigid body dynamics problem, where we prefer to work with a (or the) principle basis, but we relate it to an inertial basis. I was just wondering what might be different in higher dimensions.

I just want to start a discussion or get some resources on higher dimensional ‘rigid’ motions.

Thanks.

#### mfb

Mentor
Is the reason for the word “continuous” just to exclude reflections?
Yes.
So once we hit 6 dimension, since we can choose 3 mutually orthogonal planes, would we have ‘triple rotations’ possible?
Sure.

#### Math_QED

Homework Helper
I’ve never heard of these. They preserve distance but not angles? Can I have an example maybe?
An isometry of the Euclidean space $\mathbb{R}^n$ is a map that preserves the Euclidean norm. Since the inproduct can be written in terms of the norm, such an isometry also preserves angles.

#### Math_QED

Homework Helper
Isn't every continuous change that keeps all distances and angles the same a displacement or rotation by definition?
You get more complex rotations.
Compose a non-trivial rotation and a non-trivial translation. This is an isometry (and therefore continuous). Thus this map preserves distance and angles. Yet this is not a rotation or displacement.

#### Hiero

An isometry of the Euclidean space $\mathbb{R}^n$ is a map that preserves the Euclidean norm. Since the inproduct can be written in terms of the norm, such an isometry also preserves angles.
I see... I didn’t before realize that $a\cdot b = (|a+b|^2-|a-b|^2)/4$

That’s good to know, thanks!

#### Hiero

So once we hit 6 dimension, since we can choose 3 mutually orthogonal planes, would we have ‘triple rotations’ possible?
Yes.
This confused me because I kept thinking, in 6D we could decompose the space into two orthogonal 3D subspaces, and in each 3D subspace the rotations will compose into simple-rotations and so I thought that 6D would only require two simple rotations (a double rotation).

Of course this train of thought is silly because each of the 3D subspaces only contain 3 (independent) planes of rotation, giving 6 planes but in 6 dimensions we would expect 6choose2 = 15 planes of rotation. The 9 missing planes have one vector in each of the 3D subspaces.

A 6D basis has 15 ways to choose two basis elements which describe planes, but only 3 mutually orthogonal planes span the whole space.

Is the following true?
The composition of two simple rotations is another simple rotation if and only if the planes of rotation share (at least) a line.

#### Hiero

Compose a non-trivial rotation and a non-trivial translation. This is an isometry (and therefore continuous). Thus this map preserves distance and angles. Yet this is not a rotation or displacement.
Do you have any resources or subject/book recommendations?

#### mfb

Mentor
This confused me because I kept thinking, in 6D we could decompose the space into two orthogonal 3D subspaces, and in each 3D subspace the rotations will compose into simple-rotations and so I thought that 6D would only require two simple rotations (a double rotation).
You can have a rotation in the plane made out of the two rotation axes in their 3D subspaces. It will rotate things out of the 3D subspaces, of course.

#### Math_QED

Homework Helper
Do you have any resources or subject/book recommendations?
Not really, I learned all this at my linear algebra course.

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