Squaring the circle
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'''Squaring the circle''' is the attempt to construct, using [[Compass and straightedge constructions|straightedge and compass]], a square with an area equal to the area of a given circle. The word "attempt" is used above because the task has been [[Proof|proven]] impossible.<ref>By the [[wp:Lindemann–Weierstrass theorem|determination]] that π is a transcendental number.</ref> This has been known for over 100 years, but it had been suspected for much longer. | '''Squaring the circle''' is the attempt to construct, using [[Compass and straightedge constructions|straightedge and compass]], a square with an area equal to the area of a given circle. The word "attempt" is used above because the task has been [[Proof|proven]] impossible.<ref>By the [[wp:Lindemann–Weierstrass theorem|determination]] that π is a transcendental number.</ref> This has been known for over 100 years, but it had been suspected for much longer. | ||
Revision as of 19:16, 11 November 2012
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Squaring the circle is the attempt to construct, using straightedge and compass, a square with an area equal to the area of a given circle. The word "attempt" is used above because the task has been proven impossible.[1] This has been known for over 100 years, but it had been suspected for much longer.
Naturally, such a minor obstacle as impossibility has not stopped people from making attempts to square the circle. A person who attempts to square the circle is called a moron circle-squarer, and the term, by metaphorical extension, may be applied to any practitioner of similar recreational impossibilities.
So how can you do it?
Contents |
Cheat
A common way to square the circle is to cheat. Recall that the problem statement is to construct a square of the same area as a circle using straightedge and compass. Any of the terms in italics should be considered merely optional.
For example, given a circle, it is simple to construct a square having an area equal to 3.2 times the square of the radius of the given circle. This square does not have the same area of the circle, but it will look awfully close. That should be good enough for the mathematicians[2].
Or, instead of starting with a circle, we could start with a polygon with, say, 96 sides. That's close enough to a circle -- right, everyone?. It is possible to "square the polygon" (as was known to the Greeks), so it's basically possible to square the circle. Alternatively, you could show how to square a polygon with 96 sides, a polygon with 192 sides, a polygon with 384 sides, and so on. Therefore, passing to the limit, we can square the circle.
Cheating in multiple ways at the same time
The following process involves a calculator. It's not exact, but can be refined up to the accuracy of the tools you have[3].
- First, calculate the area of the circle.
- Then, take the square root of the area, to get the length of the edge of the square.
- If you got good drawing tools, you can even draw the square now that you have the length of the edge.
Cheating with a physical aid
- Create a wheel of the same size as the circle and which is half as wide as the circle's radius.
- Cover the side in wet paint and make it revolve over a flat surface exactly once.
- This leaves a painted rectangle with the same surface as the circle.
- Finish up by squaring this rectangle (this step can be done even with straightedge and compass).
Origami
We actually mean cutting up the pieces of the circle and put them back together as a square like this one.
Either you end up with ~1050 non-constructible pieces of the circle, or pulverize the material you make the circle with (which is probably way less than ~1050) into individual atoms and work from there.
Alternatively, if a square with a different area is desired, this method will be needed.
Why would you want to square the circle?
Squaring the circle is a problem that has not been solved since the time of the ancient Greeks. Therefore, it must be true that if you can solve it, you are smarter than anyone since the time of the ancient Greeks. Also, you will probably get widespread recognition for knocking off such a long-standing (and therefore, extremely important) problem. Maybe you'll win a Fields medal!
Of course, there are downsides to working on a problem like this. The mathematical community won't listen to you; they have an ideological bias against circle-squarers because they threaten the comfortable orthodoxy of Western deconstructionist mathematics. Or maybe they don't want to waste their time.
Sketch of the proof
In a compass and straightedge construction one is free to define the unit length from any pair of given points. Hence it suffices to square a circle with radius 1. Additionally, only points that are given and intersections of previously constructed circles and lines may be considered, and lines and circles may only be constructed from previously defined points.
Finding the intersection of a line/circle and another line/circle involves simultaneously solving a system of two equations each of which is either quadratic or linear. These lines and circles in turn depend on the points which define them, therefore, with a little algebra, it can be seen that defining a point from some given ones is equivalent to solving a quadratic equation whose coefficients are either integers, or are the result of repeated applications of this method.
Numbers which are the root of some polynomial with integer coefficients are what is known as algebraic numbers. Moreover they form what is known as an algebraically closed field, that is, all roots of polynomials with algebraic coefficients are themselves algebraic numbers. Therefore, all numbers that it is possible to construct with compass and straightedge must be algebraic, which pi (and therefore its square root) are not[4]. Thus the construction is impossible.
Warning
If you develop an urge to talk with or debate circle-squarers, you should immediately seek medical attention. Circle-squarers are not, for the most part, interested in having their ideas critiqued. They are not convinced by "proof" -- if they were, they wouldn't have started on the problem. See Keith Devlin's take on this for more.
See also
Footnotes
- ↑ By the determination that π is a transcendental number.
- ↑ So good they actually tried to make π = 3.2 so they could do this.
- ↑ An ordinary compass and straightedge, both available at Dollar Store.
- ↑ http://planetmath.org/encyclopedia/ProofOfLindemannWeierstrassTheoremAndThatEAndPiAreTranscendental2.html