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Compass and straightedge (view source)
Revision as of 03:04, February 14, 2009
, 03:04, February 14, 2009New page: '''Compass and straightedge constructions''' played an important role in the history of mathematics. Some constructions accomplished by the ancients led to revolutionary developmen...
'''Compass and straightedge constructions''' played an important role in the [[history]] of [[mathematics]]. Some constructions accomplished by the ancients led to revolutionary developments in abstract mathematics. Other construction problems posed in antiquity remain unsolved even today.
A compass-and-straightedge construction is a [[diagram]] drawn freehand with only the aid of a [[compass]] and a [[ruler]] from which the markings have been erased, also known as a "straight edge". Part of the finished diagram should display the solution to a particular given problem. For example, if the problem is "Trisect a [[line]]," the solution might consist of the given line with three Xs constructed over it at even [[interval]]s.
The art of compass-and-straightedge construction was invented by the [[Ancient Greeks]] in the centuries before [[Christ]]. The Greeks were not only excellent [[mathematician]]s, but also accomplished [[navigation|navigators]] — so it perhaps seemed natural to them that their mathematical drawings should involve the use of compasses. The straightedge was a later embellishment; straight edges were plentiful in the ancient world (for example: sword edges, [[oar]]s, and the foundations of [[building]]s), and the concept of a straight edge was pleasing and [[aesthetic]] to the ancient Greek mind. Many popular constructions were collected in the ''[[Element]]s'' of [[Euclid]].
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 [[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==
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]].
[[Category:Plane Geometry]]
A compass-and-straightedge construction is a [[diagram]] drawn freehand with only the aid of a [[compass]] and a [[ruler]] from which the markings have been erased, also known as a "straight edge". Part of the finished diagram should display the solution to a particular given problem. For example, if the problem is "Trisect a [[line]]," the solution might consist of the given line with three Xs constructed over it at even [[interval]]s.
The art of compass-and-straightedge construction was invented by the [[Ancient Greeks]] in the centuries before [[Christ]]. The Greeks were not only excellent [[mathematician]]s, but also accomplished [[navigation|navigators]] — so it perhaps seemed natural to them that their mathematical drawings should involve the use of compasses. The straightedge was a later embellishment; straight edges were plentiful in the ancient world (for example: sword edges, [[oar]]s, and the foundations of [[building]]s), and the concept of a straight edge was pleasing and [[aesthetic]] to the ancient Greek mind. Many popular constructions were collected in the ''[[Element]]s'' of [[Euclid]].
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 [[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==
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]].
[[Category:Plane Geometry]]