0
$\begingroup$

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: http://mathdemos.org/mathdemos/conic_via_locus/

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

  • 0
    There are constructions with straight edge and compasses, others are possible if you are allowed to mark the straight edge. Mechanical linkages can achieve things - a straight line without having to have a straight edge. Mechanical devises can put constraints on movement or control it. It does rather depend on what is allowed.2012-03-31

2 Answers 2

2

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.

  • 0
    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!2012-03-31
0

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.

  • 0
    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!2012-03-31