Are all lines one-dimensional ?

[phinds]

This has GOT to be a stupid question, but I have to ask because something I was reading got me a bit messed up about dimensions.

The argument is that all lines are one-dimensional because to a one-dimensional creature (yeah, this is a thought experiment), only one number is needed to specify where it is on the line (relative to some arbitrary "center" point).

This would seem to apply even for something like a helix, but I don't see how you can call a helix one-dimensional.

What am I missing?

Thanks

 

[quasar987]

Hi phinds. What you seem to be "missing" is that this property that to a one-dimensional creature, only one number is needed to specify where it is on the line is by definition what it means for an object to be one dimensional! :)

So that is why one can call a helix one-dimensional: because if you're a point particle living in a universe made out of a helix, you can only move along on direction: fowards and backward. This, by definition, makes the helix a one-dimensional object.

 

[phinds]

Hi phinds. What you seem to be "missing" is that this property that to a one-dimensional creature, only one number is needed to specify where it is on the line is by definition what it means for an object to be one dimensional! :)

So that is why one can call a helix one-dimensional: because if you're a point particle living in a universe made out of a helix, you can only move along on direction: fowards and backward. This, by definition, makes the helix a one-dimensional object.

Makes sense. I guess I'm guilty of the same problem exhibited by folks who insist that the baloon analogy is flawed because the baloon DOES have a center without understanding that in the analogy it is ONLY the surface that is to be considered, like a spherical 2D flatland.

So ANY line, regardless of shape in the 3D world, is truly 1D. I'm still having trouble w/ it, but as I said, it makes sense.

Thanks.

 

[micromass]

Makes sense. I guess I'm guilty of the same problem exhibited by folks who insist that the baloon analogy is flawed because the baloon DOES have a center without understanding that in the analogy it is ONLY the surface that is to be considered, like a spherical 2D flatland.

So ANY line, regardless of shape in the 3D world, is truly 1D. I'm still having trouble w/ it, but as I said, it makes sense.

Thanks.

You're trying to say that a Helix would be 3-dimensional since it is a part of $\mathbb{R}^3$. OK, this is fine. But it is also 4-dimensional then, since it's also part of $\mathbb{R}^4$. And it's also 5-dimensional and 6-dimensional.

The thing is that the embedding of the Helix in a larger space shouldn't be considered. The dimensional should be dependent only of the helix itself, not of a larger ambient space. With this notion, the only thing that make sense is to call the dimension 1.

You should consider the point of inhabitants living on the helix. These little creatures have no notion of a bigger space (just as we have no notion of a bigger space in which our universe embeds). These creatures should be able to tell the dimension of their space. And since they can only move forward and backwards, the dimension is 1.
Moreover: the inhabitants of the space can impossibly distinguish their helix and a straight line.

 

[phinds]

You're trying to say that a Helix would be 3-dimensional since it is a part of $\mathbb{R}^3$. OK, this is fine. But it is also 4-dimensional then, since it's also part of $\mathbb{R}^4$. And it's also 5-dimensional and 6-dimensional.

The thing is that the embedding of the Helix in a larger space shouldn't be considered. The dimensional should be dependent only of the helix itself, not of a larger ambient space. With this notion, the only thing that make sense is to call the dimension 1.

You should consider the point of inhabitants living on the helix. These little creatures have no notion of a bigger space (just as we have no notion of a bigger space in which our universe embeds). These creatures should be able to tell the dimension of their space. And since they can only move forward and backwards, the dimension is 1.
Moreover: the inhabitants of the space can impossibly distinguish their helix and a straight line.

Yes, as I believe my post #3 makes clear, I got it.

 

[quasar987]

So ANY line, regardless of shape in the 3D world, is truly 1D. I'm still having trouble w/ it, but as I said, it makes sense.

It depends what you mean by "line" exactly.

 

[phinds]

It depends what you mean by "line" exactly.

Hm ... I'm not sure how to get around that without circular reasoning. I can only say that by "line" I mean a one-dimensional construct, which of course is self-answering to my original question. What other kind of line is there?

 

[quasar987]

There is the notion of line, usually called a curve , as a continuous map (or its image) from R (or some subinterval of R) into some R^n.

Strangely enough, it is possible to map continuously R onto R^n for any n! With this definition of "line", one would be hard pressed to call these one-dimensional :)

But it seems that by "line", the text you are reading actually means what is called a 1 dimensional manifold. Because the motivating idea behind the notion of an n-dimensional manifold is precisely that it is a space such that a point particle living in that space can only move along n independant directions! For the precise definition, I guess you can consult wikipedia if you are interested.

 

[HallsofIvy]

Yes, as I believe my post #3 makes clear, I got it.

What phinds meant in his post #5 was "Alright, already, I got it! Stop yelling at me!" :tongue:

 

[phinds]

What phinds meant in his post #5 was "Alright, already, I got it! Stop yelling at me!" :tongue:

Yes, thanks for that.

 

[phinds]

There is the notion of line, usually called a curve , as a continuous map (or its image) from R (or some subinterval of R) into some R^n.

Strangely enough, it is possible to map continuously R onto R^n for any n! With this definition of "line", one would be hard pressed to call these one-dimensional :)

But it seems that by "line", the text you are reading actually means what is called a 1 dimensional manifold. Because the motivating idea behind the notion of an n-dimensional manifold is precisely that it is a space such that a point particle living in that space can only move along n independant directions! For the precise definition, I guess you can consult wikipedia if you are interested.

I appreciate the additional information, but what you are saying is beyond me.

 

[quasar987]

Which part?

 

[phinds]

Which part?

I understood the word "curve"

 

[quasar987]

haha, I see. Let me rephrase.

You mentioned the helix as a line in 3d space. Notice that this line does not intersect itself. But if you allow your line to intersect itself, then it is a theorem of mathematics that you can "weave" any solid 3d object out of a line! (This is not obvious: since a line has no width or dept, one imagines that even with an infinitely long piece of thread, I could never completely fill up even the smallest of 3-d cube!)

So with this definition of line, a line can hardly be called one-dimensional! This notion of line is called in the math literature a curve.

Then there is the more intrinsic point of view, which is the one adopted by the text you're reading. And this is to say: a line, by definition, is a space in which there is only one degree of freedom. In other words, in which a point particle can only move along one direction: foward of backward with respect to some arbitrarily chosen "mark point" on said line. This notion of line is called in the math literature a one-dimensional manifold.

 

[phinds]

haha, I see. Let me rephrase.

You mentioned the helix as a line in 3d space. Notice that this line does not intersect itself. But if you allow your line to intersect itself, then it is a theorem of mathematics that you can "weave" any solid 3d object out of a line! (This is not obvious: since a line has no width or dept, one imagines that even with an infinitely long piece of thread, I could never completely fill up even the smallest of 3-d cube!)

So with this definition of line, a line can hardly be called one-dimensional! This notion of line is called in the math literature a curve.

Then there is the more intrinsic point of view, which is the one adopted by the text you're reading. And this is to say: a line, by definition, is a space in which there is only one degree of freedom. In other words, in which a point particle can only move along one direction: foward of backward with respect to some arbitrarily chosen "mark point" on said line. This notion of line is called in the math literature a one-dimensional manifold.

Much more clear. Thanks. Yes, I did assume that if a line intersected itsef, things would get more complicated and the 1D creature did not have just ONE number that could tell him where he was.

 

[phinds]

Phinds:

You can unambiguously describe a point in the helix if you parametrize by arclength, i.e., using a single parameter. In this sense, it is 1-dimensional.

Basically, designate any point as the 0-point, and then, by knowing how far along you are in the spiral/helix, you can determine where you are.

I don't get how what you have just said is any different than post #2

 

[Sina]

Dimensions are usually defined locally. Take a local piece from the curve (that is a segment), by streching it you can make it straight and map it to some set of R. This makes it of dimension 1.

If you know what a tangent space is, you may also try to make this correspondance. Given some curve, surface or whatever space put in R^n, at each point p of the curve, how many independent vectors do I need to construct the tangent space of p? In here, tangent vectors are assumed to be attached to the point which they are tangent to.

So for a curve, tangent space to a point p on that curve is the straight line that passes through p and is tangent to the curve. A tangent vector is defined to be a vector that resides in this line and has base point p.

Given a point p on your space and a vector tangent to it, imagine the "projection" of that tangent vector to the space (considering the line again, tangent vectors are not necessarily in your space). With a bit of imagination you can I guess believe that enough tangent vectors can be used to define coordinates around a point by "projecting" them to the space. So number of independent tangent vectors is also equal to the dimension of your manifold

(although this argument is usually built in the reverse order to intiutively argue that given a dimension n space, its tangent space at each point is also n dimensional)

 

[Sina]

For instance, if the creatures living in that 1-dimensional curve, somehow using laws of physics let's say can find a way to measure the "curvature" of that curve (for instance if the curvature of the curve somehow changes the force they feel at that point), then they might get an idea of what their space looks like in the bigger n dimensional space although they don't feel it (given that they have enough mathematics :p)

 

[Alesak]

Yeah, it kinda sucks to be a physicist living in an embedded submanifold :P