Can you have a curved line without a plane?

[Pjpic]

Sorry to beat a dead horse. But, maybe; if someone could explain this slowly just using words to me, it would help a lot of lay readers (those of us without the calculus).

So, how can you curve a line if you don't have a plane?

 

[gel]

What do you mean by "curve a line"?

 

[HallsofIvy]

The helix given by x= cos(t), y= sin(t), z= t is a three dimensional curve that will not fit in any single plane. It looks like a coiled spring. It is not clear to me why you would think that a plane has anything to do with a curve.

 

[Pjpic]

What do you mean by "curve a line"?

I don't know any technical terms. But I mean to say a line with a curve in it - or a line that is not straight.

 

[Pjpic]

The helix given by x= cos(t), y= sin(t), z= t is a three dimensional curve that will not fit in any single plane. It looks like a coiled spring. It is not clear to me why you would think that a plane has anything to do with a curve.

Sorry, I probably have mispoken some technical term. But, I think of a curved line as drawn on a the plane piece of paper .

 

[gel]

I still don't understand what you're asking. HallsofIvy's post gives an example of a curve in 3 dimensional space. It doesn't lie in any plane, and why should it?
If you're only thinking of curves drawn on a sheet of paper, then it lies in a plane simply because a sheet of paper lies in a plane -- unless you roll it up.

btw, in mathematics a line is usually by definition taken to be straight. A curve can be a line (i.e. straight), but doesn't have to be.

 

[Pjpic]

It doesn't lie in any plane, and why should it?

Sorry, again; I don't know how phrase the limits of the question technically, so that the gist of it is apparent. Maybe something more like: Can you have a curve without, at least, a plane to draw it on.

 

[gel]

ok, the answer is yes. HallsofIvy's post gives an example.

 

[Pjpic]

ok, the answer is yes. HallsofIvy's post gives an example.

Sorry, I'm not understanding . Hallsofivy's answer seems, to me, to say how to have a curve without a single plane (though many planes?). I'm asking how to have a curve without, at least, a plane.

 

[gel]

you're losing me here. What does "have a curve without a single plane" mean?
A curve doesn'y have to lie in a plane, as HallsofIvy showed. What else are you asking.

 

[Pjpic]

you're losing me here. What does "have a curve without a single plane" mean?
A curve doesn'y have to lie in a plane, as HallsofIvy showed. What else are you asking.

Sorry, I'm thinking of a plane as less than a volume. I understand Hallsofivy to be saying a curve does not HAVE to be in a SINGLE plane (in words, use a volumn). I'm asking if you can have a curve without, at least, a plane to draw it on.

 

[jim mcnamara]

Maybe the OP is asking:
'what is the minimum number of dimensions required to construct a curved line?'

The OP probably expects the answer to be "2". Which appears to be what he/she means by a plane. You've already shown that some curves require 3 dimensions.

Curves with Hausdorff dimension == 'fractal' can be embedded in a 2D plane for example. Can a curve with dimension less than 2 be embedded? It seems that would be the answer to the question. I don't know the answer. Or if such a thing can be shown to exist.

 

[Pjpic]

Maybe the OP is asking:
'what is the minimum number of dimensions required to construct a curved line?'

The OP probably expects the answer to be "2". Which appears to be what he/she means by a plane. You've already shown that some curves require 3 dimensions.

Curves with Hausdorff dimension == 'fractal' can be embedded in a 2D plane for example. Can a curve with dimension less than 2 be embedded? It seems that would be the answer to the question. I don't know the answer. Or if such a thing can be shown to exist.

Thanks, that seems like a common sense answer. But it begs the question that many lay people have: {excuse the terminology} but why can't that answer be extrapolated to requiring a curved volume (3 dimesion) to be embedded in a hypervolume (4 dimension).

 

[HallsofIvy]

I would be very surprised if "lay people" asked about curved volumes embedded in a hyperplane but I can see why they would not get an answer- the basic assumption of that question is incorrect: you certainly can embed a curved volume in a hyperspace, exactly analogous to the situation for a curved path in 3 space.

 

[Pjpic]

I would be very surprised if "lay people" asked about curved volumes embedded in a hyperplane but I can see why they would not get an answer- the basic assumption of that question is incorrect: you certainly can embed a curved volume in a hyperspace, exactly analogous to the situation for a curved path in 3 space.

But, if I'm reading posts about "intrinsic curvature" correctly, you don't HAVE to imbed it hyperspace in the way a curved line has to be imbedded in 2 dimensional space. It seems like this is where the lay people, me included, on these sites are getting lost. Does this require differntial geometry to explain?