# 黄金比例本文重定向自 黄金分割

 无理数√2 - φ - √3 - √5 - δS - e - π 二进制 1.1001111000110111011... 十进制 1.6180339887498948482... 十六进制 1.9E3779B97F4A7C15F39... 连续分数 ${\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}$ 代数形式 ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$ 无限级数

${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\,{\stackrel {\text{def}}{=}}\,\varphi \quad (a>b>0)}$

${\displaystyle \varphi =1.61803398874989484820\ldots }$

## 基本计算

${\displaystyle {\frac {a+b}{a}}={\frac {a}{a}}+{\frac {b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}}$

${\displaystyle 1+{\frac {1}{\varphi }}=\varphi }$

${\displaystyle \varphi +1=\varphi ^{2}}$

${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots }$

${\displaystyle {\frac {1}{\varphi }}=\varphi -1}$

${\displaystyle \Phi ={1 \over \varphi }={1 \over 1.61803\,39887\ldots }=0.6180339887\ldots }$，亦可表达为：
${\displaystyle \Phi =\varphi -1=1.6180339887\ldots -1=0.6180339887\ldots }$

### 替代或其他形式

${\displaystyle \varphi =[1;1,1,1,\dots ]=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}$

${\displaystyle \varphi ^{-1}=[0;1,1,1,\dots ]=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}$

${\displaystyle \varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+...}}}}}}}}}$

${\displaystyle \varphi ={\frac {13}{8}}+\sum _{n=0}^{\infty }{\frac {(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}}.}$

${\displaystyle \varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }}$
${\displaystyle \varphi ={1 \over 2}\csc(\pi /10)={1 \over 2}\csc 18^{\circ }}$
${\displaystyle \varphi =2\cos(\pi /5)=2\cos 36^{\circ }}$
${\displaystyle \varphi =2\sin(3\pi /10)=2\sin 54^{\circ }.}$

## 黄金分割数高精度计算编程

### C++

#include <iostream>
#include <stdio.h>

using namespace std;

int main() {
long b, c, d = 0, e = 0, f = 100, i = 0, j, N;
cout << "請輸入黃金分割數位數\n";
cin >> N;
N = N * 3 / 2 + 6;
long* a = new long[N + 1];
while (i <= N) a[i++] = 1;
for (; --i > 0;
i == N - 6 ? printf("\r0.61") : printf("%02ld", e += (d += b / f) / f),
e = d % f, d = b % f, i -= 2)
for (j = i, b = 0; j; b = b / c * (j-- * 2 - 1))
a[j] = (b += a[j] * f) % (c = j * 10);
delete[] a;
cin.ignore();
cin.ignore();
return 0;
}

## 贵金属分割

### 引用

1. ^ Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) p. 37. "And the same applies in architecture, to the rectangles representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design."
2. ^ Livio, Mario. The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books. 2002 [2016-07-12]. ISBN 0-7679-0815-5. （原始内容存档于2016-07-07）.
3. ^ Piotr Sadowski. The knight on his quest: symbolic patterns of transition in Sir Gawain and the Green Knight. University of Delaware Press. 1996: 124 [2016-07-12]. ISBN 978-0-87413-580-0. （原始内容存档于2016-07-07）.
4. ^ Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997
5. ^ Strogatz, Steven. Me, Myself, and Math: Proportion Control. New York Times. 2012-09-24 [2016-07-12]. （原始内容存档于2016-02-12）.
6. ^ 埃里克·韦斯坦因. Golden Ratio Conjugate. MathWorld.
7. ^ Max. Hailperin; Barbara K. Kaiser; Karl W. Knight. Concrete Abstractions: An Introduction to Computer Science Using Scheme. Brooks/Cole Pub. Co. 1998. ISBN 0-534-95211-9.
8. ^ Brian Roselle, "Golden Mean Series"页面存档备份，存于互联网档案馆
9. ^ "黄金分割数高精度计算.pdf"[永久失效链接]

### 来源

• 《黄金比例》;远流出版公司;2004年;ISBN 957-32-5270-8.

## 延伸读物

• Doczi, György. The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala Publications. 2005 [1981]. ISBN 1-59030-259-1.
• Huntley, H. E. The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications. 1970. ISBN 0-486-22254-3.
• Joseph, George G. The Crest of the Peacock: The Non-European Roots of Mathematics New. Princeton, NJ: Princeton University Press. 2000 [1991]. ISBN 0-691-00659-8.
• Livio, Mario. The Golden Ratio: The Story of PHI, the World's Most Astonishing Number Hardback. NYC: Broadway (Random House). 2002 [2002]. ISBN 0-7679-0815-5.
• Sahlqvist, Leif. Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design 3rd Rev. Charleston, SC: BookSurge. 2008. ISBN 1-4196-2157-2.
• Schneider, Michael S. A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperCollins. 1994. ISBN 0-06-016939-7.
• Scimone, Aldo. La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo: Sigma Edizioni. 1997. ISBN 978-88-7231-025-0.
• Stakhov, A. P. The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. Singapore: World Scientific Publishing. 2009. ISBN 978-981-277-582-5.
• Walser, Hans. The Golden Section. Peter Hilton trans. Washington, DC: The Mathematical Association of America. 2001 [Der Goldene Schnitt 1993]. ISBN 0-88385-534-8.