# 黄金分割率

φ
 數表—无理数√2 - φ - √3 - √5 - δS - e - π

${\displaystyle \varphi \approx }$1.61803...

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

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

## 基本計算

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

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

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

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

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

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

### 替代或其他形式

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

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

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

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

${\displaystyle \varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }}$
${\displaystyle \varphi ={1 \over 2}\csc(\pi /10)={1 \over 2}\csc 18^{\circ }}$
${\displaystyle \varphi =2\cos(\pi /5)=2\cos 36^{\circ }}$
${\displaystyle \varphi =2\sin(3\pi /10)=2\sin 54^{\circ }.}$
• 黃金分割率的乘冪與費氏數列的關係
${\displaystyle \varphi ^{n}=F_{n-1}+F_{n}\times \varphi }$ ${\displaystyle (1-\varphi )^{n}=F_{n+1}-F_{n}\times \varphi }$ ，其中${\displaystyle n}$ 為任意整數，${\displaystyle F_{n}}$ 費氏數列的第${\displaystyle n}$ [註 1]

## 黃金分割數高精度計算程式碼

### 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;
}


[9]

## 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 [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. ^
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. ^

### 註釋

1. ^ 這可以透過${\displaystyle \varphi ^{2}=1+\varphi }$ ${\displaystyle {\frac {1}{\varphi }}=\varphi -1}$ ${\displaystyle {\frac {1}{1-\varphi }}=-\varphi }$ 此三個等式，以及費氏數列的的遞歸定義，以數學歸納法證明。