by TDGperson » Sat Nov 21, 2020 5:30 pm
Earnest
To be sure --> do we need to assume that we have straight edge and compass? yes Or the number of steps can be done with arbitrary non-technological tools?
Does the proof involve other theorems?yes Is a system of equations involved?yes If so, is the system with l^2 = r*pi^2 and another equation? yes The equation of thee intersection between the square and thee circle (I mean, the square can be drawn everywhere in the plane but it must intersect the circle if overlapped to it)no
Proof by contradiction?yes --> so do we need to start by assuming that it is possible to draw a square with side sort of pi or area = pi (which I will call assumption P) and then show that this implies that something is 0 when it can never be 0? yes Or we make assumption P and then proof that something can never be 0 when in reality It is?
Is the irrationality of pi involved in the proof? yes and something else If so, in that it can not be represented by fractions?this In that irrational numbers have relevant other properties? In that there is a theorem about them? In that they can never be represented in a finite number of steps?
Ok I am really think about it but cannot figure it out clearly. Anyway, if you agree I keep trying attacking thee problem. A way is the following. I noticed that polygons can be constructed as a combination of many squares (e.g. octagon is a combination of two squares). The idea is that one can easily move from a polygon to the a square with same area. So, if we managed to see the circle as a combination of polygons we could easily construct a square starting with the n-sided polygon. But this is impossible (i.e. the polygon should have infinitely many sides to approximate a circle)no, OTWT
Hint: There is a property of pi that is very relevant, other than irrationality