Squaring The Circle (thought experiment)

In summary, the conversation discusses the mathematical concept of squaring a circle and how it is impossible to do so using only a straight-edge and compass. However, the conversation also explores a thought experiment where the perimeter of a square can be made to equal the circumference of a circle, but the areas will still be different. Ultimately, the conversation concludes that it is not physically impossible to square a circle, but it cannot be done using traditional Euclidean geometry methods.
  • #1
moe darklight
409
0
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.
 
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  • #2
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 [itex]\pi \ne 4[/itex]).
 
  • #3
D H said:
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 [itex]\pi \ne 4[/itex]).

uh oh! :rofl::rofl: I guess I did. thanks.

my uncanny ability to overlook the obvious strikes yet again! :biggrin:
 
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  • #4
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.
 
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  • #5
haha yea, I did it without realizing that it wrongfully assumes that area and perimeter are proportional.
 
  • #6
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.
 
  • #7
moe darklight said:
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! :smile:

(why has everyone got side-tracked? :confused:)

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)! :smile:

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

(:biggrin: best signature ever …)
 

What is the "Squaring The Circle" thought experiment?

The "Squaring The Circle" thought experiment is a mathematical problem that has been attempted to be solved for centuries. It involves the challenge of constructing a square with the same area as a given circle, using only a compass and straightedge.

Why is it considered impossible to square the circle?

It is considered impossible to square the circle because the mathematical constant pi (π) is a transcendental number, meaning it cannot be expressed as a finite sequence of algebraic operations. Therefore, it is impossible to construct a perfect square with an area equal to the area of a given circle.

Why has the "Squaring The Circle" thought experiment been attempted for so long?

The challenge of squaring the circle has been attempted for so long because it is a fundamental problem that highlights the limitations of geometry and the search for a perfect solution. It also has a rich history and has been attempted by many famous mathematicians throughout the centuries.

What are some proposed solutions to the "Squaring The Circle" thought experiment?

Many proposed solutions to the "Squaring The Circle" thought experiment involve using non-standard geometric constructions or making use of other mathematical techniques such as calculus. However, none of these solutions have been accepted as a valid solution to the problem.

How does the "Squaring The Circle" thought experiment relate to other mathematical concepts?

The "Squaring The Circle" thought experiment is closely related to other mathematical concepts such as the concept of pi (π), the study of transcendental numbers, and the limitations of geometric constructions. It also highlights the connection between mathematics and philosophy, as it raises questions about the nature of perfection and the pursuit of absolute truth.

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