| Revision as of 17:48, 4 August 2005 edit66.200.223.112 (talk) →Impossibility← Previous edit | Revision as of 17:51, 4 August 2005 edit undo66.200.223.112 (talk) →ImpossibilityNext edit → | ||
| Line 4: | Line 4: | ||
| == Impossibility == | == Impossibility == | ||
| In the last millenia many mathematicians thought of the classical problems of Greek antiquity as irrational; ie they could be proven rigorously impossible in three dimensions even though the Greeks themselves could readily solve the problem in four dimensions by allowing an arc to be defined by a compass mounted on a board which was moving in time and space. The problem dates back to the invention of ] and occupied mathematicians for centuries. It was not until ] that the impossibility was proven rigorously, 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 ] and ] that makes the problem difficult. If other simple instruments, for example something which can draw an ], are allowed, then it is not difficult to draw a square and circle of equal area. | In the last millenia many mathematicians thought of the classical problems of Greek antiquity as irrational; ie they could be proven rigorously impossible in three dimensions even though the Greeks themselves could readily solve the problem in four dimensions by allowing an arc to be defined by a compass mounted on a board which was moving in time and space and there is a construction by an Egyptian architect which dates back to the 3rd millenium BC which solves the problem by the use of two different coordinate systems. The problem dates back to the invention of ] and occupied mathematicians for centuries. It was not until ] that the impossibility was proven rigorously, 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 ] and ] that makes the problem difficult. If other simple instruments, for example something which can draw an ], are allowed, then it is not difficult to draw a square and circle of equal area. | ||
| A solution demands construction of the number <math>\sqrt{\pi}</math>, and the impossibility of this undertaking follows from the fact that π is a ], i.e. it is ], and therefore a non-]. The transcendentality of ] was proven by ] in ]. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π — this is impossible. Nonetheless it is possible to construct a square with an area arbitrarily close to that of a given circle. | A solution demands construction of the number <math>\sqrt{\pi}</math>, and the impossibility of this undertaking follows from the fact that π is a ], i.e. it is ], and therefore a non-]. The transcendentality of ] was proven by ] in ]. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π — this is impossible. Nonetheless it is possible to construct a square with an area arbitrarily close to that of a given circle. | ||
Revision as of 17:51, 4 August 2005
Squaring the circle is the problem proposed by ancient Greek geometers, of using ruler-and-compass constructions, in a finite number of steps, to make a square with the same area as a given circle. In 1882, it was proven to be impossible. The term quadrature of the circle is synonymous.
Impossibility
In the last millenia many mathematicians thought of the classical problems of Greek antiquity as irrational; ie they could be proven rigorously impossible in three dimensions even though the Greeks themselves could readily solve the problem in four dimensions by allowing an arc to be defined by a compass mounted on a board which was moving in time and space and there is a construction by an Egyptian architect which dates back to the 3rd millenium BC which solves the problem by the use of two different coordinate systems. The problem dates back to the invention of geometry and occupied mathematicians for centuries. It was not until 1882 that the impossibility was proven rigorously, 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 that makes the problem difficult. If other simple instruments, for example something which can draw an Archimedean spiral, are allowed, then it is not difficult to draw a square and circle of equal area.
A solution demands construction of the number , and the impossibility of this undertaking follows from the fact that π is a transcendental number, i.e. it is non-algebraic, and therefore a non-constructible number. The transcendentality of π was proven by Ferdinand von Lindemann in 1882. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π — this is impossible. Nonetheless it is possible to construct a square with an area arbitrarily close to that of a given circle.
, and the impossibility of this undertaking follows from the fact that π is a 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 conditions and limitations 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 ruler-and-compass constructions or by performing the operations on certain non-euclidian spaces also makes squaring the circle possible.
While the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky space.
"Squaring the circle" as a metaphor
The mathematical proof that the quadrature of the circle is impossible has not proven to be a hindrance to the many "free spirits" who've invested years in this problem anyway. The futility of undertaking exercises aimed at finding the quadrature of the circle has brought this term into use in totally unrelated contexts, where it is simply used to mean a hopeless, meaningless, or vain undertaking. See also pseudomathematics.
External links
- Squaring the circle
- Math World's Article on Squaring the Circle includes information on procedures based on various approximations of Pi