Is Curve S One-Dimensional or Two-Dimensional?

[kini.Amith]

Is a curve (Say 'S') 1d or 2d? I ask this question because for so long i was under the impression that it was 2d, since we need a 2d cartesian plane to draw and describe a curve. But then i read in a popular book that it was 1D, which is hard to believe. so which is it?

 

[DrRocket]

Is a curve (Say 'S') 1d or 2d? I ask this question because for so long i was under the impression that it was 2d, since we need a 2d cartesian plane to draw and describe a curve. But then i read in a popular book that it was 1D, which is hard to believe. so which is it?

A curve is 1-dimensinal --- it "locally" looks like a line. A small piece of a curve, looked at under an idealized microscope would look just like a small piece of an ordinary straight line.

A curve may be embedded in a 2-dimensional space or an n-dimensional space where n can be large, but the curve itself is 1-dimensional.

 

[Landau]

Intuitively: imagine living on a curve (forgetting about the "embedding", thinking of the curve as everything there is), then you can only go forward or backwards, i.e. moving along the curve. Phycisists would say something like "there is only one degree of freedom".

 

[kini.Amith]

k. thnks

 

[rathat]

Never thought of it like that.

 

[Sakha]

Intuitively: imagine living on a curve (forgetting about the "embedding", thinking of the curve as everything there is), then you can only go forward or backwards, i.e. moving along the curve. Phycisists would say something like "there is only one degree of freedom".

Lets say the curve is positively sloped. If you move forward (to the right), you would also be moving 'up'. Doesnt that means its 2D?

 

[Landau]

No, as I said you should forget about the curve being embedded in the plane, the curve 'is' the whole world. For example, an ant walking along a curve, or a tightrope walker walking along a thin rope. Then the only possible directions to go are forward and backwards; there is no 'up' or 'down'.

 

[analmux]

Just to stir up the haziness a bit...if a set can be described as a HOLOMORPHIC curve, then it IS 2-dimensional.

 

[asdfghjklqqww]

nice work analmux... epic win!