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In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. ==Conditions for constructibility== Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides,〔 and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.〔Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969).〕 This led to the question being posed: is it possible to construct ''all'' regular ''n''-gons with compass and straightedge? If not, which ''n''-gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his ''Disquisitiones Arithmeticae''. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons: :A regular ''n''-gon can be constructed with compass and straightedge if ''n'' is the product of a power of 2 and any number of distinct Fermat primes (including none). (A Fermat prime is a prime number of the form ) Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem. Equivalently, a regular ''n''-gon is constructible if and only if the cosine of its common angle is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. 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「Constructible polygon」の詳細全文を読む スポンサード リンク
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