| Squaring The Circle |
Website Links For Circle |
Information AboutSquaring The Circle |
| CATEGORIES ABOUT SQUARING THE CIRCLE | |
| pi | |
| euclidean plane geometry | |
|
Squaring the circle is the problem proposed by Ancient Geometers of using a finite Compass And Straightedge construction to make a Square with the same area as a given Circle . In 1882, the problem was proven to be impossible. The term '' Quadrature of the circle'' is synonymous. IMPOSSIBILITY ).]] The problem dates back to the invention of Geometry and has occupied mathematicians for millennia. It was not until 1882 that Ferdinand Von Lindemann rigorously proved its impossibility, though even the ancient geometers had a very good practical and intuitive grasp of its intractability. It should be noted that it is the limitation to just Compass And Straightedge construction that makes the problem difficult; there are other curves, chiefly the Quadratrix , which would solve the problem. The problem is to find an ''exact'' solution, in a finite number of steps. Methods to ''approximate'' a square with the area of a given circle were known already to , St Andrews University . Note that the transcendence of π implies the impossibility of exactly "circling" the square, as well as of squaring the circle. TRANSCENDENCE OF π A solution of the problem of squaring the circle by compass and straightedge demands construction of the number , and the impossibility of this undertaking follows from the fact that π (pi) is a Transcendental Number —that is, it is Non-algebraic and therefore a non- Constructible Number . If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π, which is impossible. Lambert conjectured that π was transcendental in 1768 in the same paper he proved its irrationality, even before the existence of transcendental numbers was proved. It wasn't until 1882 before Ferdinand Von Lindemann proved its transcendence. It is possible to construct a square with an area ''arbitrarily close'' to that of a given circle. If a rational number is used as an approximation of π, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations. Bending the rules by allowing an infinite number of compass and straightedge operations or by performing the operations on certain non- Euclidean Space s also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky Space . MODERN APPROXIMATIONS Though squaring the circle is an impossible problem, approximations to squaring the circle can be given by constructing lengths close to π. Among the correct approximate constructions to square the circle was one by E. W. Hobson in 1913. This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079..., which is equivalent to the 4 decimal places of π. Indian mathematician Srinivasa Ramanujan in 1913, C. D. Olds in 1963, Martin Gardner in 1966, and B. Bold in 1982 all gave geometric constructions for : which is accurate to 6 decimal places of π. Srinivasa Ramanujan in 1914 gave a ruler and compass construction which was equivalent to taking the approximate value for π to be : giving a remarkable 8 decimal places of π. R. Dixon in 1991 gave constructions for
|