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Is there plane curves with limit number of operations in which is non-constructible and how do we prove it is non-constructible, i call it non-constructible if we have to plot infinity number of point in order to obtain for every part of curve, for example, the parabola is constructible since we could construct any part of the curve we want if we have long enough string. This link give such method:

Any tools could be using except elctronic device or a object with the curve in it or ruler with marks in it

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Your title does not seem to be complete. Please fix it! – Mariano Suárez-Alvarez Mar 31 '12 at 17:43
It depends what methods you allow - would a sine curve be constructible? – Mark Bennet Mar 31 '12 at 17:46
I find this very hard to follow. – Dylan Moreland Mar 31 '12 at 17:47
@MarkBennet- What do you mean, is there a link for me to understand what you a saying? – Victor Mar 31 '12 at 17:48
@Mark: one can draw a sine curve with a rolling circle with some simple mechanisms attached to it. – Mariano Suárez-Alvarez Mar 31 '12 at 17:57
up vote 2 down vote accepted

I did not manage to include the image into a comment... so here it goes.

This is one way to get a sine curve using a mechanical apparatus:

enter image description here

One needs a non-sliding circle and a few pieces. Use a bit of imagination to picture the actual implementation.

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+1 for trying however i want a curve that is non-construtible. – Victor Mar 31 '12 at 18:12
You are not going to get any example, because your definition of «non contructible» is not precise enough to argue with —there is simply no way one could derive a contradiction from it. Among other problems, it depends on the physical world and the state of our knowledge of the universe. That we cannot think of any way to draw a certain curve today does not preclude that in a few centuries physicists will find an elementary particle which happens to move in exactly that way... – Mariano Suárez-Alvarez Mar 31 '12 at 18:15
To turn your question into one which can be answered you'd need to work a bit to precisely define what you mean by «constructible», and that is not an easy task. Two successful examples of doing this (in other contexts) are the definition of «constructible numbers» which goes back to the greeks, and «origami-constructibility», much more recent. – Mariano Suárez-Alvarez Mar 31 '12 at 18:18
As it stands, your question is not different from «what buildings we cannot build?» and the answer of this question has of course changed in history! – Mariano Suárez-Alvarez Mar 31 '12 at 18:21

I think if we tried to construct $sin(\frac 1 x)$ we would get into trouble. And $f(x) = $ $x$ $sin(\frac 1 x)$ is better behaved, but just as hard, given that we can construct a straight line and do multiplication.

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I think that such features as an arc length which goes to infinity, or an infinite number of turning points, or an infinite number of zeros having an accumulation point on the real line would defy any useful notion of constructible function. But I'm willing to have it argued otherwise. – Mark Bennet Mar 31 '12 at 19:56
Every finite arc of those curves you mention can be constructed mechanically by complicating the device I described. Asking for the complete curve is unreasonable, because cannot even draw a complete straight line! – Mariano Suárez-Alvarez Mar 31 '12 at 20:35

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