翻訳と辞書 Words near each other ・ Hilbert's ninth problem ・ Hilbert's Nullstellensatz ・ Hilbert's paradox of the Grand Hotel ・ Hilbert's problems ・ Hilbert's program ・ Hilbert's second problem ・ Hilbert's seventeenth problem ・ Hilbert's seventh problem ・ Hilbert's sixteenth problem ・ Hilbert's sixth problem ・ Hilbert's syzygy theorem ・ Hilbert's tenth problem ・ Hilbert's theorem ・ Hilbert's theorem (differential geometry) ・ Hilbert's Theorem 90 ・ Hilbert's third problem ・ Hilbert's thirteenth problem ・ Hilbert's twelfth problem ・ Hilbert's twentieth problem ・ Hilbert's twenty-first problem ・ Hilbert's twenty-fourth problem ・ Hilbert's twenty-second problem ・ Hilbert's twenty-third problem ・ Hilbert, West Virginia ・ Hilbert, Wisconsin ・ Hilbert–Bernays paradox ・ Hilbert–Bernays provability conditions ・ Hilbert–Burch theorem ・ Hilbert–Huang transform ・ Hilbert–Kunz function Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hilbert's third problem ： ウィキペディア英語版 Hilbert's third problem The third on Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Gauss,〔Carl Friedrich Gauss: ''Werke'', vol. 8, pp. 241 and 244〕 Hilbert conjectured that this is not always possible. This was confirmed within the year by his student Max Dehn, who proved that the answer in general is "no" by producing a counterexample. The answer for the analogous question about polygons in 2 dimensions is "yes" and had been known for a long time; this is the Bolyai–Gerwien theorem. ==History and motivation== The formula for the volume of a pyramid, :$\backslash frac\},$ had been known to Euclid, but all proofs of it involve some form of limiting process or calculus, notably the method of exhaustion or, in more modern form, Cavalieri's principle. Similar formulas in plane geometry can be proven with more elementary means. Gauss regretted this defect in two of his letters. This was the motivation for Hilbert: is it possible to prove the equality of volume using elementary "cut-and-glue" methods? Because if not, then an elementary proof of Euclid's result is also impossible. 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「Hilbert's third problem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.