Can you have a curved line without a plane?

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SUMMARY

The discussion centers on the concept of curves existing without a plane, specifically referencing the helix defined by the equations x = cos(t), y = sin(t), z = t. Participants clarify that a curve does not necessarily need to lie within a single plane, as demonstrated by three-dimensional curves. The conversation also touches on the minimum dimensional requirements for constructing curved lines, with some suggesting that curves with Hausdorff dimensions can exist in lower dimensions. The need for a deeper understanding of intrinsic curvature and differential geometry is highlighted as essential for grasping these concepts.

PREREQUISITES
  • Understanding of three-dimensional curves and their properties
  • Familiarity with the helix equation: x = cos(t), y = sin(t), z = t
  • Basic knowledge of Hausdorff dimensions and fractals
  • Introduction to differential geometry concepts
NEXT STEPS
  • Research the properties of three-dimensional curves and their embeddings
  • Study the concept of Hausdorff dimensions and their implications in geometry
  • Learn about intrinsic curvature and its significance in differential geometry
  • Explore the relationship between curves and planes in higher-dimensional spaces
USEFUL FOR

Mathematicians, students of geometry, and anyone interested in the properties of curves in various dimensions, particularly those exploring concepts in differential geometry.

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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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