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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. Mathematically all three problems are related. |
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. Mathematically all three problems are related. |
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==Solvability of similar problems== |
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"The ancient Greeks, however, did not restrict themselves to attempting to find a plane solution (which we now know to be impossible), but rather developed a great variety of methods using various curves invented specially for the purpose, or devised constructions based on some mechanical method." |
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[[http://www-groups.dcs.st-and.ac.uk/~history/Indexes/Greeks.html#squarers Greek circle squarers]].[[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Trisecting_an_angle.html Trisecting the angle]]. [[http://www-groups.dcs.st and .ac.uk/~history/HistTopics/Doubling_the_cube.html Doubling the cube]]. Although squaring the circle by straightedge and compass could be rigorously proven impossible in three dimensions, Greeks such as [[Eratosthenes]] and [[Plato]] had built machines to solve the problem of doubling the cube which is mathematically related to squaring the circle. 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. |
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==Some history== |
==Some history== |
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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 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. |
The problem 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. |
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==Transcendence of π== |
==Transcendence of π== |
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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 01:07, 13 September 2005
Squaring the circle is a 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. "The Greek Oenopides may have been the first to lay down the restriction of the means permissible in constructions as the ruler and compass which became a canon of Greek geometry for all plane constructions." If a ruler is allowed instead of a straightedge the nature of the problem is changed and the problem becomes trivial rather than impossible. In 1882, it was proved to be impossible to square a circle using only a straightedge and compass. 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. Mathematically all three problems are related.
Some history
The problem 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.
Transcendence of π
A solution of the problem of squareing the circle by straightedge and compass 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 transcendence 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
- Squaring the circle
- Math World's Article on Squaring the Circle includes information on procedures based on various approximations of Pi