Removed the usual rktekt self-made theories about everything originating from Egypt |
reverted to previous author. Egil should discuss on talk page before making major changes to a well established article |
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== Impossibility == |
== Impossibility == |
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The problem dates back to the invention of [[geometry]] and has occupied mathematicians for millennia. It was not until [[1882]] that the impossibility was proved 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 (drafting)|compass]] and [[straightedge]] that makes the problem difficult. If the straightedge is allowed to be a ruler or 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 or to trisect an angle or double a cube. |
The problem dates back to the invention of [[geometry]] and has occupied mathematicians for millennia. It was not until [[1882]] that the impossibility was proved 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 (drafting)|compass]] and [[straightedge]] that makes the problem difficult. If the straightedge is allowed to be a ruler or 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 or to trisect an angle or double a cube. |
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In the last millennium many mathematicians thought of the classical problems of Greek antiquity as irrational. The argument that they could be rigorously proven impossible in three dimensions was all that they found interesting even though Greeks such as Eratosthenes and Plato had built machines to solve the problem. Those who 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 were on no interest to mathematicians because the problem is not mathematically challenging in more than three dimensions. |
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In recent years the analytic geometry of the Egyptians unit fraction algorythms have become interesting. The mathematics underlying the solutions of the Egyptians who first attacked the Problem in the Rhind Papyrus are once again being studied because of their implications for continued fractions. |
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There is a construction by an Egyptian architect which dates back to the 3rd millennium BC which solves the problem by the use of two different coordinate systems. The problem really dates back to the invention of [[geometry]] and has occupied mathematicians for millennia. 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 (drafting)|compass]] and [[straightedge]] that makes the problem difficult. If the straightedge is allowed to be a ruler or 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 or to trisect an angle or double a cube. |
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[[Image:egyptian circle.jpg|right|320px|The square and the circle have the same area]] |
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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 [[transcendental number]], i.e. it is [[algebraic number|non-algebraic]], and therefore a non-[[constructible number]]. The transcendentality of [[pi|π]] was proved 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. |
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 [[transcendental number]], i.e. it is [[algebraic number|non-algebraic]], and therefore a non-[[constructible number]]. The transcendentality of [[pi|π]] was proved 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. |
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==External articles== |
==External articles== |
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* [http://members.optusnet.com.au/fmet/main/intro.html Egyptian analytic geometry] |
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* [http://www.cut-the-knot.org/impossible/sq_circle.shtml Squaring the circle] |
* [http://www.cut-the-knot.org/impossible/sq_circle.shtml Squaring the circle] |
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* [http://mathworld.wolfram.com/CircleSquaring.html Math World's Article on Squaring the Circle] includes information on procedures based on various approximations of Pi |
* [http://mathworld.wolfram.com/CircleSquaring.html Math World's Article on Squaring the Circle] includes information on procedures based on various approximations of Pi |
Revision as of 20:08, 11 September 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 proved 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 the impossibility was proved 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 the straightedge is allowed to be a ruler or 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 or to trisect an angle or double a cube.
In the last millennium many mathematicians thought of the classical problems of Greek antiquity as irrational. The argument that they could be rigorously proven impossible in three dimensions was all that they found interesting even though Greeks such as Eratosthenes and Plato had built machines to solve the problem. Those who 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 were on no interest to mathematicians because the problem is not mathematically challenging in more than three dimensions.
In recent years the analytic geometry of the Egyptians unit fraction algorythms have become interesting. The mathematics underlying the solutions of the Egyptians who first attacked the Problem in the Rhind Papyrus are once again being studied because of their implications for continued fractions.
There is a construction by an Egyptian architect which dates back to the 3rd millennium BC which solves the problem by the use of two different coordinate systems. The problem really dates back to the invention of geometry and has occupied mathematicians for millennia. 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 the straightedge is allowed to be a ruler or 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 or to trisect an angle or double a cube.
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 proved 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.
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-Euclidean 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 proved to be a hindrance to the many "free spirits" who have 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.
See also
External articles
- Egyptian analytic geometry
- Squaring the circle
- Math World's Article on Squaring the Circle includes information on procedures based on various approximations of Pi