What is Compass-and-straightedge construction?, Explain Compass-and-straightedge construction

~~~ Compass-and-straightedge construction ~~~

Title: What is Compass-and-straightedge construction?, Explain Compass-and-straightedge construction
Created on: 2018-08-12

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Description: 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 with only one edge. The compass is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates. It turns out to be the case that every point constructible using straightedge and compass may also 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.

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