[Math]PHI, the golden ratio

PHI, the golden ratio 黃金分割比

轉載自 http://paulbourke.net/miscellaneous/miscnumbers/

1. Definition

將一個線段分紅兩段，那麼長的部分與短的那部分的比率等於整個線段與長的部分的比率時，

      　　　　　　

這個條件可被解釋爲 $\frac{a}{1-a}=\frac{1}{a}$.即以下的二項式： $a^2+a-1=0$,方程有兩個解, $-\phi$,和$\phi-1$。

$$\therefore \phi = \frac{\sqrt{5}+1}{2} \approx 1.618$$
這是古希臘數學中初始定義，咱們通常用$\phi-1$
$$\phi-1 = \frac{\sqrt{5}-1}{2} \approx 0.618$$

2. 常見關係式

\begin{align*}
& \phi^2=1+\phi \qquad \phi^3 = 1+2\phi\\
& \frac{1}{\phi}=\phi-1 \qquad \frac{1}{\phi^2} = 2-\phi\\
& \sin(18)=\frac{\phi-1}{2} \qquad \cos(36)=\frac{\phi}{2}\\
& \phi^{x+1}=\phi^{x}+\phi^{x-1}
\end{align*}

3. Continued_fraction 連分式

關於一些常見連分式，參見Wiki之Continued_fraction . 

　　　　　　　　

phi = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + .....))))

4. Relationship to the Fibonnaci series

(1).Ratio

當斐波那契數列趨向$\infty$時，$a_{n-1}/a_{n}$趨近於$\phi-1$

\begin{align*}
&1\quad 1\quad 2\quad 3\quad 5\quad 8\quad 13\quad 21\quad 34\quad 55\quad 89\cdots\\
&1\quad 0.5 \quad 0.67 \quad 0.6\quad 0.625 \quad 0.6154 \quad 0.619 \quad 0.6176\quad 0.6182\cdots
\end{align*}

(2).Phi Fibonnaci series

數列知足下面兩個條件：

\begin{align*}
&(a).u_{n+1}=u_{n}+u_{n-1}\\
&(b).\frac{u_{n+1}}{u_{n}}=constant\\
\end{align*}

 驗證可知，這樣的數列有且僅有一個： 
$$1,phi,1+phi,2+3phi,3+5phi,5+6phi,\cdots$$

5. 2 dimensional golden ratio 二維黃金分割比

由原來的一維線段概括推導出來的定義爲: "find a rectangle such that when a square is removed the remaining rectangle has the same proportions as the original". The solution to this is a rectangle with the ratio of its sides being phi.

　　　　　　　　

These rectangles can be inscribed in a so called logarithmic(對數的) spiral(螺旋) also known as equiangular(等角) spirals. Such spirals and occur frequently in nature, for example: shells(貝殼), sunflowers, and pine cones(松果). The limit point of the spiral is called the "eye of God".

6.Phi Pyramid