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Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with collapsing compass; see compass equivalence theorem.) More formally, the only permissible constructions are those granted by Euclid's first three postulates. Every point constructible using straightedge and compass may be constructed using compass alone. The ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems. Many of these problems are easily solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone. In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots. ==Compass and straightedge tools== The "compass" and "straightedge" of compass and straightedge constructions are idealizations of rulers and compasses in the real world: *The compass can be opened arbitrarily wide, but (unlike some real compasses) it has no markings on it. Circles can only be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse when it's not drawing a circle. *The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a line segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a long and checkered history.〔Godfried Toussaint, "A new look at Euclid’s second proposition," ''The Mathematical Intelligencer'', Vol. 15, No. 3, (1993), pp. 12-24.〕 Each construction must be ''exact''. "Eyeballing" it (essentially looking at the construction and guessing at its accuracy, or using some form of measurement, such as the units of measure on a ruler) and getting close does not count as a solution. Each construction must ''terminate''. That is, it must have a finite number of steps, and not be the limit of ever closer approximations. Stated this way, compass and straightedge constructions appear to be a parlour game, rather than a serious practical problem; but the purpose of the restriction is to ensure that constructions can be ''proven'' to be ''exactly'' correct, and is thus important to both drafting (design by both CAD software and traditional drafting with pencil, paper, straight-edge and compass) and the science of weights and measures, in which exact synthesis from reference bodies or materials is extremely important. One of the chief purposes of Greek mathematics was to find exact constructions for various lengths; for example, the side of a pentagon inscribed in a given circle. The Greeks could not find constructions for these three problems, among others: * Squaring the circle: Drawing a square the same area as a given circle. * Doubling the cube: Drawing a cube with twice the volume of a given cube. * Trisecting the angle: Dividing a given angle into three smaller angles all of the same size. For 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be impossible generally (angles with certain values can be trisected, but not all possible angles). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Compass-and-straightedge construction」の詳細全文を読む スポンサード リンク
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