Friday, January 06, 2006

Can't square a circle

Before reading this post, it might help going through the article on irrational numbers, if the reader is not familiar with them already. Irrational numbers are important to understand when talking about circles. This is because of the irrational number pi (3.1415....). Pi defines the relationship between a circle's radius and it's circumference and area. A circle has a circumference of length pi x diameter, and an area of pi x (radius squared), so all circles have a circumference and an area which is irrational.
Mathematicians and philosophers since ancient Greek times have tried to square a circle. Squaring a circle simply means constructing a square with the same area of any circle. Now here's the weird part. Because the circle has these irrational properties, you can't construct a square out of any circle.

Consider the circle with radius of length 1 unit. The circle has an area of pi (pi x radius squared) and so the square that you would construct out of this circle would have a side length of the square root of pi. Now, since pi is irrational, the square root of pi is irrational. But the length of a square is measurable - rational. Lindemann proved the impossibility of squaring a circle in 1882, putting to rest the famous classical problem.

Food for thought:

A right-angled triangle with two side lengths of 1 unit has a hypotenuse of length square root 2. This is an irrational, straight length.

9 comments:

Anonymous said...

what is the length of a square if the square and an inserted circle are cut from the same piece of wire

Anonymous said...

"all circles have a circumference and an area which is irrational" seems evidently false. For example, whenever radius r = sqrt ( A / pi ) where A is any rational number, the circle's area is rational number A.

Anonymous said...

I like thinking about this, thanks for the food...
Actually DaVincis' Vitruvio should be examine in the light of the mentioned.
Btw, measures are always associated with errors, so pi cannot be measure in a purist way of expressing. That is why we don't know yet the 'exact' value of it...
Can we say that squares are from this world but circles belong to the world of though and we can only construct imperfect representations of them ? ;)

Anonymous said...

When did pi become 3.1605?

Anonymous said...

Lol... Oh the universe has expanded, therefore pi has shrunk. And I thought you guys were cool....

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