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In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith〔Henry J.S. Smith (1874) “On the integration of discontinuous functions.” ''Proceedings of the London Mathematical Society'', Series 1, vol. 6, pages 140–153.〕〔The “Cantor set” was also discovered by Paul du Bois-Reymond (1831–1889). See footnote on page 128 of: Paul du Bois-Reymond (1880) “(Der Beweis des Fundamentalsatzes der Integralrechnung ),” ''Mathematische Annalen'', vol. 16, pages 115–128. The “Cantor set” was also discovered in 1881 by Vito Volterra (1860–1940). See: Vito Volterra (1881) “Alcune osservazioni sulle funzioni punteggiate discontinue” (observations on point-wise discontinuous functions ), ''Giornale di Matematiche'', vol. 19, pages 76–86.〕〔José Ferreirós, ''Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics'' (Basel, Switzerland: Birkhäuser Verlag, 1999), pages 162–165.〕〔Ian Stewart, ''Does God Play Dice?: The New Mathematics of Chaos''〕 and introduced by German mathematician Georg Cantor in 1883.〔Georg Cantor (1883) "(Über unendliche, lineare Punktmannigfaltigkeiten V )" (infinite, linear point-manifolds (sets) ), ''Mathematische Annalen'', vol. 21, pages 545–591.〕〔H.-O. Peitgen, H. Jürgens, and D. Saupe, ''Chaos and Fractals: New Frontiers of Science 2nd ed.'' (N.Y., N.Y.: Springer Verlag, 2004), page 65.〕 Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment. Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense. ==Construction and formula of the ternary set== The Cantor ternary set is created by deleting the open middle third from each of a set of line segments repeatedly. One starts by deleting the open middle third (, ) from the interval (), leaving two line segments: () ∪ (). Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: () ∪ () ∪ () ∪ (). This process is continued ad infinitum, where the ''n''th set is : and The Cantor ternary set contains all points in the interval () that are not deleted at any step in this infinite process. The first six steps of this process are illustrated below. An explicit closed formula for the Cantor set is : or : The proof of the formula above as the special case of two family of Cantor sets is done by the idea of self-similarity transformations and can be found in detail.〔Mohsen Soltanifar, ''On A sequence of cantor Fractals'', Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1, paper 9, 2006.〕〔Mohsen Soltanifar, ''A Different Description of A Family of Middle-a Cantor Sets'', American Journal of Undergraduate Research, Vol 5, No 2, pp 9–12, 2006.〕 This process of removing middle thirds is a simple example of a finite subdivision rule. It is perhaps most intuitive to think about the Cantor set as the set of real numbers between zero and one whose ternary expansion in base three doesn't contain the digit 1. This ternary digit expansion description has been more of interest for researchers to explore fractal and topological properties of the Cantor set. 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「Cantor set」の詳細全文を読む スポンサード リンク
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