I In Which N-Dimensional Spaces Are Rotations Chiral?

[Hornbein]

Rotations in 2D space are chiral, not chiral in 3D space. An acquaintance says rotations are chiral only in 2 and 4D space. I think rotations are chiral in all even-dimensional spaces but don't feel sure about this. According to Wikipedia a figure is chiral if and only if its symmetry group contains no orientation-reversing isometry. This is too technical for me. Any help? If rotations in 6D space are chiral then I'll feel certain I was right.

 

[renormalize]

For the ignorant (e.g., me) can you define chiral rotation? Is it related to parity transformation?

 

[Hornbein]

For the ignorant (e.g., me) can you define chiral rotation? Is it related to parity transformation?

According to Wikipedia a figure is chiral if and only if its symmetry group contains no orientation-reversing isometry. I don't understand this. All I know for sure is that rotation in 2D is chiral. To an observer it will always appear to be rotating one way or the other. There is no way to see it any other way.

Rotation in 3D is not chiral because it you stand on your head the rotation appears to be going the other way.

Rotation in 4D must be chiral because quantum spin is 4D. Positive quantum spin can never become negative spin no matter how you look at it. It's also apparent geometrically.

My expectation is that rotation is chiral in even dimensional spaces and not in odd dimensional spaces. But I don't know how to prove this and the jargon is too thick to learn easily.

 

[jbergman]

According to Wikipedia a figure is chiral if and only if its symmetry group contains no orientation-reversing isometry. I don't understand this. All I know for sure is that rotation in 2D is chiral. To an observer it will always appear to be rotating one way or the other. There is no way to see it any other way.

Rotation in 3D is not chiral because it you stand on your head the rotation appears to be going the other way.

Rotation in 4D must be chiral because quantum spin is 4D. Positive quantum spin can never become negative spin no matter how you look at it. It's also apparent geometrically.

My expectation is that rotation is chiral in even dimensional spaces and not in odd dimensional spaces. But I don't know how to prove this and the jargon is too thick to learn easily.

As I think of it, all rotations are orientation-preserving. A reflection is orientation reversing. An even number of reflections is equivalent to a rotation.

I don't quite get the standing on your head example and not sure how this relates to chirality.

 

[fresh_42]

If chirality is defined by the symmetry group of an object, then it is simply the question whether this symmetry group is a subgroup of all permutation of the vertices $S(n)$ or a subgroup of the alternating group $A(n)$ of even permutations of the vertices. I don't think that this can be connected to the dimension.

 

[jbergman]

If chirality is defined by the symmetry group of an object, then it is simply the question whether this symmetry group is a subgroup of all permutation of the vertices $S(n)$ or a subgroup of the alternating group $A(n)$ of even permutations of the vertices. I don't think that this can be connected to the dimension.

How does that work for objects with

According to Wikipedia a figure is chiral if and only if its symmetry group contains no orientation-reversing isometry. I don't understand this. All I know for sure is that rotation in 2D is chiral. To an observer it will always appear to be rotating one way or the other. There is no way to see it any other way.

Rotation in 3D is not chiral because it you stand on your head the rotation appears to be going the other way.

Rotation in 4D must be chiral because quantum spin is 4D. Positive quantum spin can never become negative spin no matter how you look at it. It's also apparent geometrically.

My expectation is that rotation is chiral in even dimensional spaces and not in odd dimensional spaces. But I don't know how to prove this and the jargon is too thick to learn easily.

After reading the wikipedia article, I am not sure it makes sense to talk about a rotation itself as being chiral. It's more a property of an object. For example a sphere is clearly achiral.

And as you say such an object is chiral if it requires an orientation reversing transformation to map it to it's mirror image.

 

[fresh_42]

The German Wikipedia is a bit less technical:

In geometry, a figure is chiral (and has chirality) if it is not identical to its mirror image, or more precisely, if it cannot be mapped onto its mirror image by rotations and parallel translations alone.

 

[Hornbein]

How does that work for objects with

After reading the wikipedia article, I am not sure it makes sense to talk about a rotation itself as being chiral. It's more a property of an object. For example a sphere is clearly achiral.

And as you say such an object is chiral if it requires an orientation reversing transformation to map it to it's mirror image.

Aha, this is my problem. I'm using a non-standard definition of chirality. I don't know what the standard definition of what I'm looking for is. Informally it's "if and only if in a space there are two or more types of rotation of spheres that are always distinguishable then rotation in that space is chiral." The spheres have to have no recognizable features other than their rotations.

Here in 3D a view from above the North Pole sees the Earth rotating counterclockwise. A view from above the South Pole sees the Earth rotating clockwise. So 3D is non-chiral by my definition. Or looking at a sphere you say it is rotating clockwise. Stand on your head and look at the sphere and it appears to be rotation counter-clockwise relative to you. Or consider the planet Uranus. It has no surface features and its plane of rotation is perpendicular to the ecliptic so no one can say its rotation is clockwise or counter clockwise. All odd-dimensional spaces are like this.

I once convinced myself geometrically that all even-dimensional spaces were chiral but I don't quite trust geometry and others think differently. I want an algebraic proof or a counterexample.

 

[Hornbein]

Aha, this is my problem. I'm using a non-standard definition of chirality. I don't know what the standard definition of what I'm looking for is. Informally it's "if and only if in a space there are two or more types of rotation of spheres that are always distinguishable then rotation in that space is chiral." The spheres have to have no recognizable features other than their rotations.

Here in 3D a view from above the North Pole sees the Earth rotating counterclockwise. A view from above the South Pole sees the Earth rotating clockwise. So 3D is non-chiral by my definition. Or looking at a sphere you say it is rotating clockwise. Stand on your head and look at the sphere and it appears to be rotation counter-clockwise relative to you. Or consider the planet Uranus. It has no surface features and its plane of rotation is perpendicular to the ecliptic so no one can say its rotation is clockwise or counter clockwise. All odd-dimensional spaces are like this.

I once convinced myself geometrically that all even-dimensional spaces were chiral but I don't quite trust geometry and others think differently. I want an algebraic proof or a counterexample.

Here's what ChatGPT has to say about it.

You're essentially looking at the behavior of rotations in different-dimensional spaces and whether they can be consistently distinguished under all transformations (such as reflection or inversion).

From an algebraic standpoint, the key structure governing rotations is the special orthogonal group, SO(n)SO(n)SO(n):

• In odd dimensions, SO(2k+1)SO(2k+1)SO(2k+1) has a determinant of +1+1+1, and its rotations include reflections that flip handedness (parity transformations exist within O(n)O(n)O(n), the full orthogonal group, which contains both proper and improper rotations). This allows different rotations to appear indistinguishable under reflection, making the space non-chiral.
• In even dimensions, SO(2k)SO(2k)SO(2k) is fundamentally different because it has two distinct connected components in its double cover, Spin(2k)Spin(2k)Spin(2k), corresponding to left- and right-handed rotations. This often leads to an inherent chirality.

This supports my conjecture so naturally I like it. What's a double cover?

A double cover is a mathematical concept where one space "covers" another in such a way that each point in the base space corresponds to exactly two points in the covering space.

Example: The Circle and the Line​

A simple example is the map f:R→S1f: \mathbb{R} \to S^1f:R→S1 given by:

f(θ)=eiθf(\theta) = e^{i\theta}f(θ)=eiθ
Here, every point on the unit circle S1S^1S1 corresponds to infinitely many points on R\mathbb{R}R, but if we restrict the domain to [0,2π)∪[π,3π)[0, 2\pi) \cup [\pi, 3\pi)[0,2π)∪[π,3π), each point on S1S^1S1 corresponds to exactly two points. This makes it a double cover when restricted appropriately.

Example: Spin(n)Spin(n)Spin(n) and SO(n)SO(n)SO(n)​

For rotation groups, Spin(n)Spin(n)Spin(n) is a double cover of SO(n)SO(n)SO(n), meaning that every rotation in SO(n)SO(n)SO(n) corresponds to two elements in Spin(n)Spin(n)Spin(n). The map from Spin(n)→SO(n)Spin(n) \to SO(n)Spin(n)→SO(n) identifies these two elements as equivalent, but in the covering space Spin(n)Spin(n)Spin(n), they remain distinct.

This is crucial in physics and mathematics because in even dimensions, these two elements in Spin(2k)Spin(2k)Spin(2k) can correspond to "left-handed" and "right-handed" rotations, which is directly related to the chirality question you're investigating.

It seems to know what it's talking about but I can't say that I understand it. Any objections?

 

[jbergman]

Aha, this is my problem. I'm using a non-standard definition of chirality. I don't know what the standard definition of what I'm looking for is. Informally it's "if and only if in a space there are two or more types of rotation of spheres that are always distinguishable then rotation in that space is chiral." The spheres have to have no recognizable features other than their rotations.

Here in 3D a view from above the North Pole sees the Earth rotating counterclockwise. A view from above the South Pole sees the Earth rotating clockwise. So 3D is non-chiral by my definition. Or looking at a sphere you say it is rotating clockwise. Stand on your head and look at the sphere and it appears to be rotation counter-clockwise relative to you. Or consider the planet Uranus. It has no surface features and its plane of rotation is perpendicular to the ecliptic so no one can say its rotation is clockwise or counter clockwise. All odd-dimensional spaces are like this.

I once convinced myself geometrically that all even-dimensional spaces were chiral but I don't quite trust geometry and others think differently. I want an algebraic proof or a counterexample.

I think we would say that the 3D view is chiral. Since the rotation direction of the sphere changes depending what pole you are standing on.

I believe this might related to what I said before but I need to think about it more. Any 2 reflections is equivalent to a rotation.

So, in a 3D space when looking at a 2D surface embedded in it like a sphere. We can rotate the sphere and do a reflection (standing on your head so to speak) and the combination of those two operations is orientation reversing aka reflection.

I need to think more about the even dimensional case... Anyways interesting question!

 

[Hornbein]

The special orthogonal group is SO(n), not SO(n)SO(n)SO(n):

 

[Hornbein]

I wrote MathStackExchange, who said what ChatGPT wrote was nonsense. They closed the thread without attempting to answer the question.

I'm asking again without the ChatGPT stuff. I hope I don't get in trouble for that. Well I hardly ever use MathStackExchange so if they ban me, so what.