Is Space Truly Three Dimensional?

[StarThrower]

Has anyone here ever actually tried to prove (analytically) that space is three dimensional? I've never seen a good proof. Here is my basic idea behind a proof:

Let us proceed as if we know what an infinite straight line is.

Consider two infinite straight lines that have one point in common, point X. There will be four vertical angles formed. If they are all equal, each of them is a right angle, and the two lines are perpendicular.

Now, consider the following question.

How many more infinite straight lines can pass through point X, such that all vertical angles formed are right angles.

Let us define the dimensionality of space to be the maximum number of infinite straight lines that can meet at a point, such that all the lines are mutually perpendicular.

I put this in the SR thread, because I didn't really know where else to put it, so change the location if you want.

Regards,

StarThrower

 

[StarThrower]

I did a google search on "dimensionality space proof three" here are some sites that actually make the question seem interesting:

http://scholar.uwinnipeg.ca/courses/38/4500.6-001/Cosmology/dimensionality.htm

http://members.aol.com/rick3in1/nature/geometry.htm

My idea was the same as that of Ptolemy's, simply say that three mutually orthogonal straight lines can meet at any point in space, but not more than three.

Since space is three dimensional, this is completely right.

However, at this site you see the rather "boneheaded" statement that Ptolemy's proof doesn't satisfy them because there could be more dimensions that they cannot visualize.

My first comment is that the word 'visualize' as used in their statement is completely meaningless.

But then again, they would argue that I am trying to 'visualize' three mutually orthogonal straight lines. But then I would argue that we can at least construct something like what I am thinking of in reality.

Anyways, I am interested in any thought's on Ptolemy's and my approach to proving that space is three dimensional.

When you stop to think about it, this post isn't even stupid because of all the string theorists out there.

Regards,

Star

 

[TeV]

When you stop to think about it, this post isn't even stupid because of all the string theorists out there.

 

[turin]

I think that rotations add three more closed dimensions (dimensions of orientation). Then, you can imagine a 6-D space with six intersecting lines, all mutually orthogonal. The three obvious lines that you have mentioned, and three lines that "you cannot see." Each of the unseen lines represents the set of all possible orientations about about some axis. Note: The orientation of an object cannot be changed by any combination of the three translational degrees of freedom represented by the first three lines mentioned when applied to the object as a whole. Requiring the three axes to be orthogonal to each other takes care of the orthogonality of the three lines of rotations. A rotation and a translation are completely independent (the one does not affect the other), and so they are orthogonal to each other in some since. I shall suggest that they are orthogonal in the same sense that the three lines of translation are orthogonal to each other. Points separated by 2 pi along the lines of orientation are identified, which is the sense in which these other three dimensions are closed.


Another way to look at it is to consider configuration space of N identifiable point particles. Assume that orientation is meaningless for a point particle, but that each particle has three traslational degrees of freedom. Then, the configuration has 3 N degrees of freedom. All of these degrees can be made mutually orthogonal.


Edit:
Let me try one more appeal, similar to the first. In the absense of any object, and just considering the space itself, three lines can be made to intersect at a point and all make 90^o angles with each other. Then, these lines can be rotated about this same point of intersection. Thus, the orientation of the lines allows for three more degrees of freedom very similar to the ones that I first tried to describe, but now with no need for a reference object. You may say, "but there is need for a reference orientation." My response to that would be, "but, since I am suggesting 6 dimensions, then the 3-D point through which the lines intersect is now no longer sufficiently specified. Therefore, the null/initial/reference orientation is merely the other three coordinate values of the 6-D point of intersection."

Some features:

Any two lines in the 3-D suggestion form a 2-D subspace. In this space, say x-y space, the intersection point need only be specified by (x,y)_intersection. In order to introduce the third dimension, the thrid coordinate value must also be specified, just as the referece orientation must also be specified.

Rotations about any two perpendicular axes should also form a subspace of orientation, I think, for these orientations to be considered dimensions. Hmm, I'll have to think about that one. I know that the rotations are not commutable as are translations. I wonder if that invalidates the orientations as dimensions.

Also, the set of all orientations about some axis together with the set of all translations in one direction should, I think, form a 2-D subspace. This seems even more problematic.

 

[StarThrower]

I think that rotations add three more closed dimensions (dimensions of orientation). Then, you can imagine a 6-D space with six intersecting lines, all mutually orthogonal. The three obvious lines that you have mentioned, and three lines that "you cannot see." Each of the unseen lines represents the set of all possible orientations about about some axis. Note: The orientation of an object cannot be changed by any combination of the three translational degrees of freedom represented by the first three lines mentioned when applied to the object as a whole. Requiring the three axes to be orthogonal to each other takes care of the orthogonality of the three lines of rotations. A rotation and a translation are completely independent (the one does not affect the other), and so they are orthogonal to each other in some since. I shall suggest that they are orthogonal in the same sense that the three lines of translation are orthogonal to each other. Points separated by 2 pi along the lines of orientation are identified, which is the sense in which these other three dimensions are closed.


Another way to look at it is to consider configuration space of N identifiable point particles. Assume that orientation is meaningless for a point particle, but that each particle has three traslational degrees of freedom. Then, the configuration has 3 N degrees of freedom. All of these degrees can be made mutually orthogonal.


Edit:
Let me try one more appeal, similar to the first. In the absense of any object, and just considering the space itself, three lines can be made to intersect at a point and all make 90^o angles with each other. Then, these lines can be rotated about this same point of intersection. Thus, the orientation of the lines allows for three more degrees of freedom very similar to the ones that I first tried to describe, but now with no need for a reference object. You may say, "but there is need for a reference orientation." My response to that would be, "but, since I am suggesting 6 dimensions, then the 3-D point through which the lines intersect is now no longer sufficiently specified. Therefore, the null/initial/reference orientation is merely the other three coordinate values of the 6-D point of intersection."

Some features:

Any two lines in the 3-D suggestion form a 2-D subspace. In this space, say x-y space, the intersection point need only be specified by (x,y)_intersection. In order to introduce the third dimension, the thrid coordinate value must also be specified, just as the referece orientation must also be specified.

Rotations about any two perpendicular axes should also form a subspace of orientation, I think, for these orientations to be considered dimensions. Hmm, I'll have to think about that one. I know that the rotations are not commutable as are translations. I wonder if that invalidates the orientations as dimensions.

Also, the set of all orientations about some axis together with the set of all translations in one direction should, I think, form a 2-D subspace. This seems even more problematic.

Thank you for this interesting response.

1. Can I view some mathematics about what you are saying, something involving sine and cosine perhaps.

2. I think objects in space are a necessary part of understanding rotations, so we need to keep this fact around, otherwise the mathematical analysis will be flawed.

3. I started out by saying, let us presume that we understand what an infinite straight line is. What if I were to assert that an infinite straight line cannot rotate? How would that affect your mathematical analysis of rotations in space?

 

[DrMatrix]

You only need 2 coorinates to specify rotation of a 3d object. Consider an arrow at the center of a shpere pointing north. You only need latitude and longitude to specify any rotated position. But, while rotation could be considered dimensions in a phase space, they are not physical dimensions. The point $(x_1,x_2,x_3)$ is near $(x_1+dx,x_2,x_3)$ but not touching it. If you use $(x_1,x_2,x_3,r_1,r_2)$, then a small rotation should move the point to $(x_1,x_2,x_3,r_1+dr,r_2)$. This would specify the same "place".

 

[DrMatrix]

Oops. Two coordinates are needed to specify an axis. You'd need a third to specify rotation about that axis. Sorry. But I maintain rotational coordinates are not physical dimensions.

 

[turin]

... I maintain rotational coordinates are not physical dimensions.

Fair enough. The more I've been thinking about it, the more I am agreeing with you. At least, the coordinates of the orientation are too different to put them in the same category as the coordinates of the position. I'm still thinking about it.

Here's my current investigation:
I am mapping the 3-D orientation supspace into a cube of edge 2 π. Actually, it is more like a lattice with simple cubic primitive cells of edge 2 π. But, this can also be accomplished, as I understand, by considering only one cell and identifying the opposing faces (the 3-D analog to the 2-D environment of the atari game "Asteroids"). What I'm investigating at this point is whether or not this topology can represent not only particular orientations, but transitions of orientations (rotations), as translations in the cube (under the identifications in the case of rotations > 2 π or any rotation that takes the configuration out of the cube). What concerns me is that I don't think rotations are commutative, so I don't know what that would mean for translations in the cube representing rotations: would I have to forbid certain types of translations, etc.

 

[Severian596]

I like this question.

I like the analogy of proposing that two stick figures are completely happy living on the printed page. Using some abstract thinking one of them can CONCEPTUALIZE a 3rd dimension by viewing a number of "slides" in 2-dimensional cross sections and then hypothesizing that if there were a 3rd dimension, the sum of these cross sections would actually construct an object in 3 dimensions.

The problem is these figures can never actually visualize 3 dimensions. To them visual cues are always 2-D; length and width. So how would they prove (in quantitative data expressed ONLY in 2 dimensions) the existence or non-existence of a 3rd dimension?

Any ideas or opinions?

 

[turin]

I have come to two conclusions:

1) The 90^o requirement is not necessary. Any angle is sufficient to indicate a dimension. If three lines can intersect at a point and all make nontrivial angles with each other, and if the sum of any two of these is never equal to the third, then there are at least three dimensions.

2) Something is definitely wrong with the orientation dimensions. This occurred to me as I was trying to learn java. I got this demo folder with the development tools, and one of the demo categories was molecules, and the other was wire frames. In both of these demo categories, a picture of some 3-D object is presented. Dragging the mouse changes the orientation of the 3-D object (these demos are very cool, and I am now extremely excited about learning java). Sometimes I got frustrated by trying to orient the object the way I wanted it, until I realized that a right-left drag rotated the image about the y-axis, and an up-down drag rotated the object about the x-axis. Then, I was a little suprised when I realized that only two axes of rotation were needed to orient the object in any way I wanted. I still hold that the orientation itself requires three specifications for a rigid body, but only two degrees of freedom are needed to adjust the orientation to any arbitrary orientation. This has me a little confused.