# 黄金分割率

（重定向自黄金分割比

 无理数√2 - φ - √3 - √5 - δS - e - π 二進制 1.1001111000110111011... 十進制 1.6180339887498948482... 十六進制 1.9E3779B97F4A7C15F39... 连续分数 ${\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}$ 代數形式 ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$ 無限級數

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

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

## 基本計算

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

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

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

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

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

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

### 替代或其他形式

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

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

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

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

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

## 黃金分割數高精度計算編程

#include <iostream>
#include <stdio.h>
using namespace std;
int main (void)
{
long b,c,d=0,e=0,f=100,i=0,j,N;
cout<<"请输入黄金分割数位数\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;
}


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## 参考文献

### 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. ISBN 0-7679-0815-5.
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. ISBN 978-0-87413-580-0.
4. ^ Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997
5. ^
6. ^
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"
9. ^

## 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.