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Compass and straightedge (view source)
Revision as of 03:10, September 29, 2010
, 03:10, September 29, 2010Solution to the three classical problems
In modern times, compass-and-straightedge constructions were rediscovered by [[Euler]], who resurrected them from the pages of his namesake's ''Elements'' and found that their simple geometric [[truth]]s were a pleasant diversion from the [[calculus wars]] then raging between the {{hs|Isaac|Newton}}ian British and the [[Leibniz]]ian German mathematical communities. In the twentieth century, compass-and-straightedge methods have become a popular form of [[recreational mathematics]], and the technique is taught as a one-week [[unit]] in many [[middle school]] [[math]] [[class]]es.
In modern times, compass-and-straightedge constructions were rediscovered by [[Euler]], who resurrected them from the pages of his namesake's ''Elements'' and found that their simple geometric [[truth]]s were a pleasant diversion from the [[calculus wars]] then raging between the {{hs|Isaac|Newton}}ian British and the [[Leibniz]]ian German mathematical communities. In the twentieth century, compass-and-straightedge methods have become a popular form of [[recreational mathematics]], and the technique is taught as a one-week [[unit]] in many [[middle school]] [[math]] [[class]]es.
==Unsolved problems==
==Classical unsolved problems==
The Ancident Greeks solved all their problems with compass and straightedge. There were, however, three problems which they could not solve: [[cube|doubling the cube]], [[square|completing the square]], and [[angle|trisecting the angle]]. [[Pythagoras]] himself was said to have remarked that trisecting the angle was the hardest thing he had ever attempted. [[Archimedes]] attempted a "[[rationalism|rationalistic]]" approach to trisecting the angle, in which he repeatedly ''bisected'' the angle until the number of divisions became a multiple of 3; however, this does not work for most angles, and in any event it is only physically possible to bisect most angles seven or eight times, so Archimedes' approach does not [[scale]].
The Ancident Greeks solved all their problems with compass and straightedge. There were, however, three problems which they could not solve: [[cube|doubling the cube]], [[squaring the circle]], and [[angle|trisecting the angle]]. [[Pythagoras]] himself was said to have remarked that trisecting the angle was the hardest thing he had ever attempted. [[Archimedes]] attempted a "[[rationalism|rationalistic]]" approach to trisecting the angle, in which he repeatedly ''bisected'' the angle until the number of divisions became a multiple of 3; however, this does not work for most angles, and in any event it is only physically possible to bisect most angles seven or eight times, so Archimedes' approach does not [[scale]].
These problems can't be solved with compass and straightedge; the analysis can be done with the help of [[Abstract Algebra]]. In short, starting with a fixed length ''L'', any two points that can be drawn by intersecting cicles and lines based on already existing points must lie at a distance that can be expressed as ''x L'', where ''x'' is a number whose expression uses only basic arithmetic operations and square roots - reciprocally, for every number ''x'' that can be thus expressed, it's possible to construct a line segment of length ''x L''.
''Squaring the circle'' is equivalent to drawing a line segment of length <math>\pi L\,</math>, which can't be done, because <math>\pi\,</math> is not an [[algebraic number]].
''Doubling the cube'' and ''trisecting an angle'' are equivalent to solving irreducible [[cubic equations]], whose solutions can't be expressed with only square roots - cubic roots (of the real number 2, in the doubling of the cube, or of complex numbers, in the case of the trisection) are required.
[[Category:Plane Geometry]]
[[Category:Plane Geometry]]