How Can Drawing on a Cylinder Lead to Solving the Quadrature of the Circle?

[kleinwolf]

Take a sheet of paper, a straightliner and a compass.

Make a cylinder out of the sheet of paper and use the compass to draw on the cylinder a "circle" with the same radius as the cylinder (let's put that radius R=1(whichever unit you want)).

Unfold the sheet and take the big axis of the closed curved obtained. One can show, this length is : $$L=\frac{\pi}{3}$$.

Triple this distance with compass, and you can now construct a square which perimeter is $$4\pi$$

Now construct a circle of radius 2, and you built a circle of same perimeter as the square above...

 

[T@P]

wait how do you draw a circle on a cylinder? do you mean to go all the way around? because that would give you only $$2 \pi r$$ i don't see where the $$\pi/3$$ comes from...

*edit* maybe you mean to draw it on sideways, and even then, why would you want a square with the same perimeter as a circle? forgive the naiveness of my questions (also, exactly how do you draw a circle on sideways like that if you don't want to crush your paper cylinder?)

 

[kleinwolf]

Well the point is exactly that taking the compass (carefully) on the cylinder doesn't draw a circle (neither an ellipse by the way).

If you want : look at the cylinder from the basis circle : take the radius of the cylinder on your compass : put (carefully), the compass on the surface of the cylinder. You immediatly see that this does not make the complete turn around of the cylinder, but only 60°=Pi/3...Is that clearer ?

Since you now unfold the paper, u get a length that is multiple of Pi u did the quadrature of the circle.