Squaring The Circle (thought experiment)

[moe darklight]

OK, so I understand mathematically why one can't square a circle, but when I do the following thought experiment I can't see how one could not square the circle:

- I tie a string at both ends and lay it on a table so that it forms a perfect circle.
- Now I place four pins, at equal distances from each other, so that the center of the square they form is also at the center of the circle formed by the string.
- Now I start moving the pins apart in small increments, but in equal increments, so that they remain at equal distances.
- Eventually all four pins will be touching the string (circle), and as I keep moving them apart, they will begin to distort the shape of the string.
- Finally they will reach a limit, at this point, they will have distorted the string into a square, which should have the exact same area as that of the initial circle.

I don't know if it makes sense written down so here I drew it: http://38.114.207.18/980826fdf52d0d9c4e1f36c9428704cc4g.jpg"

I assume then that what they mean is that, though it is impossible to mathematically determine the exact distance by which those pins are separated, it is not physically impossible to square a circle, right? or am I missing something.

 

[D H]

Moe, you have come up with a way to make a square that has the same perimeter as a circle. That does not mean they have the same area (which they don't, since \pi \ne 4).

 

[moe darklight]

Moe, you have come up with a way to make a square that has the same perimeter as a circle. That does not mean they have the same area (which they don't, since \pi \ne 4).

uh oh! I guess I did. thanks.

my uncanny ability to overlook the obvious strikes yet again!

 

[robert Ihnot]

Right, like if we took a square with sides of 1, A=1x1=1. But if that same perimeter was divided up as a rectangle with sides of 1.5 and .5, then the area would be 3/4. And going on that way, P=4, sides of 1/n and 2-1/n, we can reduce the area to as little as we want.

 

[moe darklight]

haha yea, I did it without realizing that it wrongfully assumes that area and perimeter are proportional.

 

[robert Ihnot]

However, the nature of pi is that AS LONG AS IT IS A CIRCLE, the ratio of the perimeter to the diameter is the constant pi, as Euclid has shown.

 

[tiny-tim]

I assume then that what they mean is that, though it is impossible to mathematically determine the exact distance by which those pins are separated, it is not physically impossible to square a circle, right? or am I missing something.

Hi moe!

(why has everyone got side-tracked? )

They just mean that Euclid couldn't do it … in other words, you can do it … just not with a straight-edge and compass (like trisecting an angle, or finding a cube root)!

For loads of detail, see: http://en.wikipedia.org/wiki/Squaring_the_circle.

( best signature ever …)