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

In summary, the most general motion of a rigid body in 3D consists of a displacement and a rotation. However, in higher dimensions, other types of motions such as multiple rotations can also occur. The concept of "continuous" transformations excludes reflections, and in 6 dimensions, triple rotations are possible as there are 3 mutually orthogonal planes. Isometries preserve distance and angles, and the composition of a rotation and translation is also an isometry. The composition of two simple rotations is another simple rotation if and only if the planes of rotation share a line. Resources or recommendations for learning more about higher dimensional motions are not provided.
  • #1
Hiero
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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?
 
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  • #2
Isn't every continuous change that keeps all distances and angles the same a displacement or rotation by definition?
You get more complex rotations.
 
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  • #3
You can get double rotations, for example.
 
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  • #4
mfb said:
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

Math_QED said:
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.
 
  • #5
Are you considering isometries, or what types of transformations?
 
  • #6
WWGD said:
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.
 
  • #7
Hiero said:
Is the reason for the word “continuous” just to exclude reflections?
Yes.
Hiero said:
So once we hit 6 dimension, since we can choose 3 mutually orthogonal planes, would we have ‘triple rotations’ possible?
Sure.
 
  • #8
Hiero said:
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.
 
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  • #9
mfb said:
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.
 
  • #10
Math_QED said:
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!
 
  • #11
mfb said:
Hiero said:
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.
 
  • #12
Math_QED said:
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?
 
  • #13
Hiero said:
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.
 
  • #14
Hiero said:
Do you have any resources or subject/book recommendations?

Not really, I learned all this at my linear algebra course.
 

FAQ: In higher dimensions, are there more than just rotations and displacements?

1. What are "higher dimensions"?

In mathematics and physics, "higher dimensions" refer to dimensions beyond the three spatial dimensions that we experience in our everyday lives. These dimensions are often represented mathematically as additional axes, such as the fourth dimension being represented as a line perpendicular to the three spatial dimensions.

2. Are there more than just rotations and displacements in higher dimensions?

Yes, there are other types of transformations in higher dimensions, such as reflections, shears, and stretches. These transformations can be represented mathematically as matrices and can be used to manipulate objects in higher dimensions.

3. How do rotations and displacements work in higher dimensions?

Rotations and displacements in higher dimensions work similarly to how they work in three dimensions. In rotations, an object is rotated around a point or axis, while in displacements, an object is moved from one location to another. However, in higher dimensions, these transformations can occur along multiple axes simultaneously.

4. Can we visualize higher dimensions?

It is difficult for humans to visualize dimensions beyond the three spatial dimensions we experience. However, we can use mathematical models and visual aids, such as projections and animations, to help us understand and study higher dimensions.

5. What are some real-world applications of higher dimensions?

Higher dimensions have many practical applications in fields such as physics, engineering, and computer science. They are used to model and solve complex problems, such as in quantum mechanics and computer graphics. They also play a crucial role in understanding and studying the structure of the universe.

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