Why does physics consider only 3 of 6 dimensions

[Thuring]

This may seem a very elementary question, but I don't believe it is; so I put it in the advanced section. I'm mathematically experienced, and this question has stumped Ph Ds.

I haven't figured out why space is usually described in terms of only 3 spatial dimensions rather than six: x,y,z, Tx, Ty, Tz. For instance spin needs angular motion, as does torque. Torque is not along the z-axis, for instance, like the cross-product would lead one to believe, but "around" the z-axis. In spacecraft structural dynamics, we always use 6 dimensions in order to describe torque and bending. Without rotation, a pinned interface would be the same as a fixed interface. Once you get the problem in terms of vectors and matrices, the math is the same with 3 or 6 dimensions.

Perhaps physics assumes that particles are points, so a rotation is meaningless (except spin is used as well as particle angular momentum). If only 3 dimensions is required for a particular application, fine, but it should be ADMITTED that the other 3 are simply being ignored. (Yes, I know 10 - 11 are used for string theory.) Perhaps, the rotations may be considered compacted dimensions. :-)

I must be missing something ...

 

[Orodruin]

Rotations are perfectly well defined within a three-dimensional description. You may be mixing up the number of spatial dimensions with the dimensions of phase space or configuration space. A system of N particles a priori has a 3N-dimensional configuration space and physicists are perfectly happy in working with that.

Also, the thread level tag is to set your understanding of the subject. Putting it to A indicates that you understand the subject at the level of a graduate student or better and expect answers to be aimed at that level.

 

[ZapperZ]

Just to add what Orodruin said, physics also deal with infinite dimension in Hilbert space. So I don't know what all this fuss about us dealing only in 3 or 6-dimensional space.

Lost in all of this is that we don't demand that there has to be such-and-such number of spatial coordinates. That is wagging the tail of the dog! You solve the dynamics of the system, and if the minimum number of dimensional space can satisfactorily describes the dynamics, you are done! You don't simply go in with a preconceive idea that there has to be a set number of dimensions. You use what is required and demanded by the situation.

Zz.

 

[Thuring]

Thanks for your response.

I fully understand the difference between degrees of freedom and dimensions, and have worked with them for years. As I said, the rotational dimensions are orthogonal to the translation dimensions meaning that you cannot describe rotations with translations just as you can't describe x with any combination of y and z. In structural dynamics, we use 6 dimensions regularly in order to calculate torque and bending.

P.S. I do have a graduate understanding of this, but something is missing ... I put this in "A" because people think this is a simple question. I've had very smart people not be able answer this.

 

[ZapperZ]

Thanks for your response.

I fully understand the difference between degrees of freedom and dimensions, and have worked with them for years. As I said, the rotational dimensions are orthogonal to the translation dimensions meaning that you cannot describe rotations with translations just as you can't describe x with any combination of y and z. In structural dynamics, we use 6 dimensions regularly in order to calculate torque and bending.

P.S. I do have a graduate understanding of this, but something is missing ... I put this in "A" because people think this is a simple question. I've had very smart people not be able answer this.

And in solid state physics, we work in 6D phase space regularly, because we deal with x, y, z, p_x, p_y, and p_z.

But you are not asking about "rotational dimensions", but rather spatial. Unless you think that your engineering problems need more than 3 dimensional space, I do not see what the issue is here.

Zz.

 

[Orodruin]

But your premise is wrong then. Of course you deal with more than 3 dimensions in structural dynamics, but they are not the three spatial dimensions. They are dimensions of a configuration space, not dimensions of the underlying space. I would say you are definitely confusing spatial dimensions with dimensions of configuration space.

 

[Thuring]

ZapperZ, I'm afraid that your answer is probably the best I'll get: use whatever works. It's a common theme. However, physicists keep saying there are 3 dimensions, not that "let's assume 3 dimensions because it's all we need". I should probably stop "fussing".

 

[Orodruin]

I think you are overinterpreting what physicists mean when they say "we need three dimensions". What is implied is "we need 3 spatial dimensions", not "we only need to worry about three-dimensional configuration spaces".

 

[Thuring]

Ahhh, key word "Configurational Space"

Wikipedia:
The set of coordinates that define the position of a reference point and the orientation of a coordinate frame attached to a rigid body in three-dimensional space form its configuration space, often denoted where represents the coordinates of the origin of the frame attached to the body, and represents the rotation matrices that define the orientation of this frame relative to a ground frame. A configuration of the rigid body is defined by six parameters, three from and three from , and is said to have six https://www.physicsforums.com/x-dictionary:r:'Degrees_of_freedom_(mechanics)?lang=en&signature=com.apple.DictionaryApp.Wikipedia'.

I'll have to read the article thoroughly. Thanks

 

[Dale]

I haven't figured out why space is usually described in terms of only 3 spatial dimensions rather than six: x,y,z, Tx, Ty, Tz. For instance spin needs angular motion, as does torque.

Neither spin nor torque are locations in space. You are confusing the number of degrees of freedom of a rigid body with the number of spatial dimensions.

The other thing you might be thinking of is the isometries of space. A three dimensional Euclidean space has 6 isometries already. So there is no need to add extra dimensions to generate the 6 isometries.

 

[Thuring]

I'm definitely missing something. I am still studying the full meaning of R3 X SO(3) .

But it seems to simply describe translation and rotation. Not quite the point of my question. And, (I need to look again) seems the translations get the status of "dimension" and the rotation gets the status of "Configuration Space". Or, I've missed something... I'm wondering if it's just semantics.

Anything, including point particles, (my contention) need 6 of something, unless a statement is made that the rotational dimensions can be ignored. The rotations "should" have no less status than the translational, at least mechanically.

Yes, Dale, you are right. So, what is the difference btw DoF and dimensions? (We can consider just one "particle"; I know about multiple nodes, particles, whatever.)

I'm going to have to look up isometries.

Thanks for all of yous' help, and your patience.

 

[Dale]

Anything, including point particles, (my contention) need 6 of something,

I think this is clearly wrong. A point has no orientation in space, only position. So what does it need 6 of?

We can consider just one "particle"; I know about multiple nodes, particles, whatever

I think you need to go further than that. If you are talking about space then you need to consider just space, no particles.

If you are including particles then you are no longer just talking about space but about space and objects. You can no longer consider the joint configuration to represent space. (Nothing particularly wrong with that, but it seems to be contrary to what you are getting at)

In any case. If your model for space is a 3D Euclidean space then you automatically get 6 isometries. This also generalized correctly to 2D or ND.

 

[Thuring]

"I think this is clearly wrong. A point has no orientation in space, only position. So what does it need 6 of?"

My "contention" is that a point does have orientation, and orientation is spatial and a "dimension". Now, you may set the orientation to Tx = Ty = Tz = 0, and leave it there. If it doesn't need 6 dimensions, just admit that rotation is being ignored. You may leave them out of the DoFs used for an analysis as you may leave out any other DoF at particular nodes.

Perhaps torque, electron spin, or angular momentum, would be an example. (I don't wish to blow this up into electron dynamics) Angular momentum must be rotational, right? .. ANGULAR otherwise it's along the same basis as, say, Z translation. Translation and rotation along the same basis ?

Don't forget x, y, z, Tx, Ty, Tz are orthogonal. It's not like they're not related.

"If you are including particles then you are no longer just talking about space but about space and objects."

I don't need particles but I'm sure my bias is in that direction, being a dynamist. What is the difference ?

But then, what if I used cylinder coords, where would I get my 6 orthog dims ?

W.R.T. isometrics, I don't think symmetries are what I'm looking for.

I appreciate you all for your efforts. This has been my number 1 question for about 5 years; doesn't look like I'm going to break easily .. lol

 

[Vanadium 50]

and this question has stumped Ph Ds.

This is a bad, bad way to start. Just because someone doesn't agree with you doesn't mean they are "stumped". Worse, this is a classic start to a crackpot thread - do you really want to be lumped in with them? I think not.

To specify the position of an object in space requires 3 numbers. To specify the position and orientation requires 6 (technically only 5) numbers. We all agree on that, right? When someone says "we live in 3 spatial dimensions", they are referring to the former statement.

 

[Dale]

My "contention" is that a point does have orientation,

Please provide a professional reference for this. This is, to my knowledge, completely opposed to Euclidean geometry, Riemannian geometry, Lorentzian geometry, and pseudo-Riemannian geometry. In none of these do points have orientations. This forum does not permit personal speculation.

Perhaps torque, electron spin, or angular momentum, would be an example.

If that is an example then you are not talking about dimensions of space but configurations of matter. Again, there is nothing wrong with that, but I didn't think that was what you wanted to do.

Clearly, once you introduce matter your configuration space has a larger dimensionality than 3D Euclidean space. But that is no longer a statement about the dimensionality of space itself. So I think that you need to decide: do you want to discuss space alone or do you want to discuss matter and space together. If you want to discuss space alone then 3 dimensions are enough, but if you want to discuss space and matter together then 6 dimensions is insufficient.

 

[Thuring]

Ah, then the key is (as others have also suggested) is the difference between the nature of space and describing it, and the nature of matter within the space (thus the "Configuration Space" discussion); the difference between dimensions and configurations. And I've been stubbornly combining the two. So, a dof doesn't need a corresponding dimension.

I believe I understand now !

You were all correct. Thanks everyone for your tremendous patience!