Tuesday, February 12, 2008

Squaring the circle

This is a very ancient question. To put it in a more concrete way, is there a way using only straightedge and compasses (Euclid's toolkit), to construct a square with exactly the some area as a given circle in a finite number of steps?
There is a simple answer, but lets first look at the following method to square the circle:

1. Draw a horizontal line in the middle of the paper and mark a point R at the right end.
2. Open your compasses about an inch (roughly) and mark a point T on the line.
3. With unchanged compasses, mark a point O on the line so that distance OT=2*TR.
4. Draw a circle centered at O, and label point P at the left of the diameter line. This will be the circle POR whose area we are trying to equal with a square.
5. Construct point H halfway along the line PO, (you should know how to do that).
6. Raise a perpendicular at T, intersecting the circle at Q.
7. Draw a chord RS of the same length as QT.
8. Join PS and draw MO and NT both parallel to RS.
9. Below the horizontal diameter, draw a chord PK of length PK=PM.
10. Construct a vertical tangent PL of the same length as line segment MN.
11. Join RL, RK and KL.
12. Label point C on RK so that RC has length RH.
13. Construct CD parallel to LK, meeting RL at D.
14. Construct a square on base RD. It will have the same area as the circle! You can verify this by measuring the length RD as the radius RO times the square root of (~1,7724539). So the area of the square is *r2.

This is very convincing. It is wrong, but it is not obvious were the problem is. It turns out that using this method we created a square of area (355/113)*r2. This fraction is a representation of that is good to 8 digits. However it is only an approximation. If we would try to do this construction on a large enough circle, it would be possible to see that there is a mistake. But for this we would need a circle with a one mile radius.

It is however impossible to square the circle with the restrictions listed above. It is so because is transcendental. If such a construction was possible it would mean that is an algebraic number and this is not true.

The construction shown in this post is due to Srinivasa Ramanujan. He is considered to be the greatest mathematician ever by some people.


Michael Rack said...

Quadrature. Pi being a transcendental number, how can my C++ graphics program which simulates compass and straight edge,produce an algerbraic vulue for pi that when divided by my PC pi produce one and subtraction 4.44089209850026126e -16? How accurate must the result be? One could never achieve this accuracy by hand because of minor errors. I am 75 and would love to know the error after working on the solution for so long. Can you please assist. Michael Rack michael-rack@beeb.net. http://www.geocities.com/mikerack_uk/index.html

Anatoly said...

Hello Michael,
I sent you a letter about the error, for some reason the numbers on the page you linked didn't add up.
I wrote a post about a general algorithm that can be used to get this construction - you can read it here.