翻訳と辞書 Words near each other ・ Metallate ・ Metallaxis ・ Metallaxis amandae ・ Metallaxis semiustus ・ Metallaxis teledapa ・ Metallgesellschaft ・ Metallic ・ Metallic Blues ・ Metallic bonding ・ Metallic color ・ Metallic Diseases ・ Metallic dragon ・ Metallic fiber ・ Metallic hydrogen ・ Metallic K.O. ・ Metallic mean ・ Metallic Metals Act ・ Metallic microlattice ・ Metallic paint ・ Metallic path facilities ・ Metallic pigeon ・ Metallic ringtail ・ Metallic roller bearing ・ Metallic silhouette shooting ・ Metallic Skink ・ Metallic Spheres ・ Metallic starling ・ Metallic tower of Fourvière ・ Metallic-green tanager ・ Metallic-winged sunbird 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 Metallic mean ： ウィキペディア英語版 Metallic mean The more general simple continued fraction expressions :$$ n + \cfrac }}} = () = \frac\left(n+\sqrt\right)\, } |- | 2:|| ||2.414213562 |- | 3:|| ||3.302775638 |- | 4:|| ||4.236067978 |- | 5:|| ||5.192582404 |- | 6:|| ||6.162277660 |- | 7:|| ||7.140054945 |- | 8:|| ||8.123105626 |- | 9:|| ||9.109772229 |- |   ⋮ |- | ''n'':|| |} are known as the metallic means or silver means〔Vera de Spinadel (1999). (The Family of Metallic Means ), ''Vismath'' 1(3) from Mathematical Institute of Serbian Academy of Sciences and Arts. Also: (Vera W. de Spinadel, "The Metallic Means and Design", pp. 141–157 in ''Nexus II: Architecture and Mathematics, ed. Kim Williams'', Fucecchio (Florence): Edizioni dell'Erba, 1998. ) Also: (Vera W. de Spinadel. "The Family of Metallic Means." ''Visual Mathematics'' 1.3 (1999): 0–0. )〕 (also ratios or constants) of the successive natural numbers. The golden ratio (1.618...) is the metallic mean between 1 and 2, while the silver ratio (2.414...) is the metallic mean between 2 and 3. The term "bronze ratio" (3.303...), or terms using other names of metals (copper, nickel), are occasionally used to name subsequent metallic means. The values of the first ten metallic means are shown at right.〔"(An Introduction to Continued Fractions: The Silver Means )", ''Maths.Surrey.ac.UK''.〕 Notice that each metallic mean is a root of the simple quadratic equation :$x^2\; -\; nx\; =\; 1,\backslash ,$ where ''n'' is any positive natural number. As the golden ratio is connected to the pentagon (first diagonal/side), the silver ratio is connected to the octagon (first diagonal/side). As the golden ratio is connected to the Fibonacci numbers, the silver ratio is connected to the Pell numbers, and the bronze ratio is connected to . Each Fibonacci number is the sum of the previous number times one plus the number before that, each Pell number is the sum of the previous number times two and the one before that, and each "bronze Fibonacci number" is the sum of the previous number times three plus the number before that. Taking successive Fibonacci numbers as ratios, these ratios approach the golden mean, the Pell number ratios approach the silver mean, and the "bronze Fibonacci number" ratios approach the bronze mean. == Properties == These properties are valid only for integers m, for nonintegers the properties are similar but slightly different. The above property for the powers of the silver ratio is a consequence of a property of the powers of silver means. For the silver mean ''S'' of ''m'', the property can be generalized as :$\backslash !\backslash \; S\_^n\; =\; K\_n\; S\_m\; +\; K\_$ where :$\backslash !\backslash \; K\_n\; =\; mK\_\; +\; K\_.$ Using the initial conditions and , this recurrence relation becomes :$\backslash !\backslash \; K\_n\; =\; \backslash frac\; -\; ^)\}.$ The powers of silver means have other interesting properties: :If ''n'' is a positive even integer: ::$\backslash !\backslash \; =\; 1\; -\; S\_m^.$ Additionally, ::$\backslash !\backslash$ ::$\backslash !\backslash \; .$ Also, ::$\backslash !\backslash \; S\_m^3\; =\; S\_$ ::$\backslash !\backslash \; S\_m^5\; =\; S\_$ ::$\backslash !\backslash \; S\_m^7\; =\; S\_$ ::$\backslash !\backslash \; S\_m^9\; =\; S\_$ ::$\backslash !\backslash \; S\_m^\; =\; S\_.$ In general: ::$\backslash !\backslash \; S\_m^\; =\; S\_\}.$ The silver mean ''S'' of ''m'' also has the property that :$\backslash !\backslash \; 1/S\_m\; =\; S\_m\; -\; m$ meaning that the inverse of a silver mean has the same decimal part as the corresponding silver mean. :$\backslash !\backslash \; S\_m\; =\; a\; +\; b$ where ''a'' is the integer part of ''S'' and ''b'' is the decimal part of ''S'', then the following property is true: :$\backslash !\backslash \; S\_m^2\; =\; a^2\; +\; mb\; +\; 1.$ Because (for all ''m'' greater than 0), the integer part of , . For , we then have :$\backslash !\backslash \; S\_m^2\; =\; ma\; +\; mb\; +\; 1$ :$\backslash !\backslash \; S\_m^2\; =\; m(a+b)\; +\; 1$ :$\backslash !\backslash \; S\_m^2\; =\; m(S\_m)\; +\; 1.$ Therefore the silver mean of m is a solution of the equation :$\backslash !\backslash \; x^2\; -\; mx\; -\; 1\; =\; 0.$ It may also be useful to note that the silver mean ''S'' of −''m'' is the inverse of the silver mean ''S'' of ''m'' :$\backslash !\backslash \; 1/S\_m\; =\; S\_\; =\; S\_m\; -\; m.$ Another interesting result can be obtained by slightly changing the formula of the silver mean. If we consider a number :$\backslash !\backslash \; \backslash frac\backslash left(n+\backslash sqrt\backslash right)\; =\; R$ then the following properties are true: :$\backslash !\backslash \; R\; -\; \backslash lfloor\; R\; \backslash rfloor\; =\; c/R$ if ''c'' is real, :$\backslash !\backslash \; \backslash left(\backslash right)c\; =\; R\; -\; \backslash lfloor\; \backslash operatorname(R)\; \backslash rfloor$ if ''c'' is a multiple of ''i''. The silver mean of ''m'' is also given by the integral :$S\_m\; =\; \backslash int\_0^m\; +\; \backslash right)\}\; \backslash ,\; dx.$ 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「Metallic mean」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.