Can you have a curved line without a plane?

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Discussion Overview

The discussion revolves around the concept of whether a curved line can exist without being situated in a plane. Participants explore the definitions and implications of curves in different dimensions, including examples from three-dimensional space and the mathematical properties of curves.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants question the meaning of "curve a line" and seek clarification on whether it refers to a line that is not straight.
  • One participant provides the example of a helix, which exists in three-dimensional space and does not fit in any single plane, suggesting that a curve does not necessarily require a plane.
  • Another participant emphasizes that a curve can exist without being confined to a single plane, proposing that the question may be about the minimum dimensions required for a curve.
  • There is mention of curves with Hausdorff dimensions, including fractals, and whether they can be embedded in lower-dimensional spaces.
  • Some participants express confusion about the relationship between curves and planes, with one noting that a curve can exist in a volume rather than being limited to a plane.
  • A later reply suggests that the discussion may require concepts from differential geometry to fully explain the nature of curves and their embeddings.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the original question. There are multiple competing views regarding the relationship between curves and planes, and the discussion remains unresolved with differing interpretations of dimensionality and embedding.

Contextual Notes

Some participants express uncertainty about technical terminology and the limits of their questions, indicating that definitions and assumptions may vary among contributors.

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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?
 
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What do you mean by "curve a line"?
 
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.
 
gel said:
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.
 
HallsofIvy said:
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 .
 
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.
 
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.
 
ok, the answer is yes. HallsofIvy's post gives an example.
 
gel said:
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.
 
  • #10
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.
 
  • #11
gel said:
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.
 
  • #12
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.
 
  • #13
jim mcnamara said:
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).
 
  • #14
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.
 
  • #15
HallsofIvy said:
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 differential geometry to explain?
 

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