Squaring the Circle: A Practical Squaring Construction

[Lasand]

A Practical Squaring of the Circle.

Does anyone know if this method of construction been done before?

https://imageshack.com/i/exFv7X1Wj

 

[Doug Huffman]

Squaring the Circle is practically an oxymoron for the transcendental nature of Pi as Lindermann-Weierstrass theorem. Do you mean approximating the area of a circle with a square figure?

 

[Lasand]

I'll post some more images. I went for the diagonal first.

https://imageshack.com/i/p9iQ6q4njhttps://imageshack.com/i/ipMPcPZ0j

https://imageshack.com/i/exGgaxiUj

https://imageshack.com/i/iqvEgAqSj

 

[Mark44]

A Practical Squaring of the Circle.

Does anyone know if this method of construction been done before?

https://imageshack.com/i/exFv7X1Wj

For starters, $\pi \neq 3.14$.
"Squaring the circle" was a problem that the ancient Greeks came up with; namely, to construct a square having the same area as that of a given circle, using only a compass and straightedge. See http://en.wikipedia.org/wiki/Squaring_the_circle.

 

[HallsofIvy]

As Mark44 said, "squaring the circle" refers to using only straight edge and compasses to construct a square whose area is exactly the same as the area of a given circle. And as Doug Huffman said, it has long been know that, because "$\pi$" is "transcendental", that is impossible.

The basic idea is that it can be proven that, given a "unit length", that is a line segment taken to represent the length one, using straight edge and square we can construct only lengths that are "algebraic of order a power of two". If we were able, starting with a circle of radius 1, so area $\pi$, to construct a square of area $\pi$, then we would have constructed a line segment of length $\sqrt{\pi}$ which, like $\pi$ is "transcendental", not "algebraic" of any order. What you have done here is construct a square whose area is almost the same as the give circle but not exactly the same so you have not "squared the circle".

The two other famous "impossible constructions", "trisecting the angle" (given an angle, use only a straight edge and compasses to construct an angle 1/3 as large) and "duplicating the cube" (given a cube, use three dimensional analogues of straight edge an compasses to construct a cube with twice the volume) can be proven impossible in a similar way- these would both be equivalent to constructing the root of a cubic equation, thus constructing a number that is "algebraic of order 3", algebraic alright but its order is not a "power of two".

There are, of course, many ways of constructing thing that are approximately correct and there are even way of doing them exactly using tool other than just straight edge and compasses.

 

[zoki85]

Impossible with ruler and compass, but possible with quadratix