The discussion revolves around the dimensionality of a curve, specifically whether it is one-dimensional or two-dimensional. Participants explore this concept through intuitive reasoning and examples, considering both mathematical definitions and physical interpretations.
Participants express differing views on the dimensionality of curves, with some supporting the one-dimensional perspective and others challenging it by introducing conditions or alternative interpretations. The discussion remains unresolved with multiple competing views present.
Some arguments depend on the definitions of dimensionality and the context in which curves are considered, such as their embedding in higher-dimensional spaces. The discussion also highlights the ambiguity in interpreting movement along curves.
kini.Amith said: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?
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 said: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".