Squaring the circle

From RationalWiki
(Difference between revisions)
Jump to: navigation, search
(Why would you want to square the circle?)
(Why would you want to square the circle?: bible)
Line 10: Line 10:
 
Squaring the circle (in a finite number of steps) is a problem that has not been solved since the time of the ancient Greeks.  So it follows that if you can solve it, you must be 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!
 
Squaring the circle (in a finite number of steps) is a problem that has not been solved since the time of the ancient Greeks.  So it follows that if you can solve it, you must be 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!
  
On a more serious note, squaring the circle would require constructing the length <math>\scriptstyle \sqrt{\pi}</math>. (A circle with radius <math>\scriptstyle r</math> has area <math>\scriptstyle \pi r^2</math>. Hence a square with the same area must have a side of <math>r \scriptstyle \sqrt{\pi} </math>.) If this number could be constructed, that would prove that <math>\pi</math> is an algebraic number, meaning there is some possible set of rational numbers you can use to calculate it. For some (essentially subjective) reason, the very thought that <math>\pi</math> was somehow inaccessible through "normal" numbers really seems to bother some people.
+
On a more serious note, squaring the circle would require constructing the length <math>\scriptstyle \sqrt{\pi}</math>. (A circle with radius <math>\scriptstyle r</math> has area <math>\scriptstyle \pi r^2</math>. Hence a square with the same area must have a side of <math>r \scriptstyle \sqrt{\pi} </math>.) If this number could be constructed, that would prove that <math>\pi</math> is an algebraic number, meaning there is some possible set of rational numbers you can use to calculate it.  
 +
 
 +
For a variety of (essentially subjective) reasons, the very thought that <math>\pi</math> was somehow inaccessible through "normal" numbers really seems to bother some people. One objection is based on passages in the Bible, as 1 Kings 7:23-26 certainly implies that <math>\scriptstyle \sqrt{\pi}</math> is rational.
  
 
Also for no good reason, during the 1700s the belief arose that squaring the circle would somehow solve the "Longitude" problem (the inability of sea vessels to determine where they were on the east-west axis). As there were some enormous cash prizes on offer (in 1714 the British government offered a prize of £20,000), this fired up every amateur mathematician in Europe. Circle squaring is actually irrelevant, all that was needed to solve the longitude problem was the ability to observe the sun and a really good clock.  
 
Also for no good reason, during the 1700s the belief arose that squaring the circle would somehow solve the "Longitude" problem (the inability of sea vessels to determine where they were on the east-west axis). As there were some enormous cash prizes on offer (in 1714 the British government offered a prize of £20,000), this fired up every amateur mathematician in Europe. Circle squaring is actually irrelevant, all that was needed to solve the longitude problem was the ability to observe the sun and a really good clock.  

Revision as of 00:19, 15 March 2013

Part of a convergent series on

Mathematics

link=:category:
2+2=4

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

Why would you want to square the circle?

Squaring the circle (in a finite number of steps) is a problem that has not been solved since the time of the ancient Greeks. So it follows that if you can solve it, you must be 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!

On a more serious note, squaring the circle would require constructing the length \scriptstyle \sqrt{\pi}. (A circle with radius \scriptstyle r has area \scriptstyle \pi r^2. Hence a square with the same area must have a side of r \scriptstyle \sqrt{\pi} .) If this number could be constructed, that would prove that \pi is an algebraic number, meaning there is some possible set of rational numbers you can use to calculate it.

For a variety of (essentially subjective) reasons, the very thought that \pi was somehow inaccessible through "normal" numbers really seems to bother some people. One objection is based on passages in the Bible, as 1 Kings 7:23-26 certainly implies that \scriptstyle \sqrt{\pi} is rational.

Also for no good reason, during the 1700s the belief arose that squaring the circle would somehow solve the "Longitude" problem (the inability of sea vessels to determine where they were on the east-west axis). As there were some enormous cash prizes on offer (in 1714 the British government offered a prize of £20,000), this fired up every amateur mathematician in Europe. Circle squaring is actually irrelevant, all that was needed to solve the longitude problem was the ability to observe the sun and a really good clock.

In the mathematics world the question was put to bed in 1882 when Ferdinand von Lindemann proved that \pi is not algebraic (in technical jargon, it is "transcendental"). Because there are definitely no rational numbers that can calculate \pi, it is impossible to construct \scriptstyle \sqrt{\pi} in Euclidean space.

However true believers won't be deterred by anything as flimsy as "proof". They persist because they believe that there is an ideological bias against circle-squarers whose brave investigations threaten the comfortable orthodoxy of Western deconstructionist mathematics.

In actuality, real mathematicians just can't be bothered to waste their time with cranks.

Sketch of the proof

y = \sin(x)\,

\frac{dy}{dx}=\ ?

This article/section deals with mathematical concepts appropriate for a student in late high school or early university.


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.[2] Thus the construction is impossible.

Cheat

3x-5y=6

2x+4y=2

x=? y=?

This article/section deals with mathematical concepts appropriate for a student in mid to late high school.


You can cheat it easily, but can you do it with a compass and straightedge?

A common way to square the circle is to cheat. (Mathematicians call this approximation). 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.[3]

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.[4]

  • 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).

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

  1. By the determination that π is a transcendental number.
  2. Roger Lipsett and Warren Buck. "proof of Lindemann-Weierstrass theorem and that e and π are transcendental." PlanetMath.org. 2012 November 19.
  3. So good they actually tried to make π = 3.2 so they could do this.
  4. An ordinary compass and straightedge, both available at the dollar store.
Personal tools
Namespaces

Variants
Actions
Navigation
Community
Tools
support