PHI, the golden ratio 黄金分割比
转载自 http://paulbourke.net/miscellaneous/miscnumbers/
1. Definition
将一个线段分成两段，那么长的部分与短的那部分的比率等于整个线段与长的部分的比率时，
这个条件可被解释为 $frac{a}{1-a}=frac{1}{a}$.即如下的二项式： $a^2+a-1=0$,方程有两个解, $-phi$,和$phi-1$。
$$ herefore phi = frac{sqrt{5}+1}{2} approx 1.618$$
这是古希腊数学中初始定义，我们一般用$phi-1$
$$phi-1 = frac{sqrt{5}-1}{2} approx 0.618$$
2. 常见关系式
egin{align*}
& phi^2=1+phi qquad phi^3 = 1+2phi\
& 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 连分式
关于一些常见连分式，参见Wiki之Continued_fraction .
phi = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + .....))))
4. Relationship to the Fibonnaci series
(1).Ratio
当斐波那契数列趋向$infty$时，$a_{n-1}/a_{n}$趋近于$phi-1$
egin{align*}
&1quad 1quad 2quad 3quad 5quad 8quad 13quad 21quad 34quad 55quad 89cdots\
&1quad 0.5 quad 0.67 quad 0.6quad 0.625 quad 0.6154 quad 0.619 quad 0.6176quad 0.6182cdots
end{align*}
(2).Phi Fibonnaci series
数列满足下面两个条件：
egin{align*}
&(a).u_{n+1}=u_{n}+u_{n-1}\
&(b).frac{u_{n+1}}{u_{n}}=constant\
end{align*}
验证可知，这样的数列有且仅有一个：
$$1,phi,1+phi,2+3phi,3+5phi,5+6phi,cdots$$
5. 2 dimensional golden ratio 二维黄金分割比
由原来的一维线段归纳推导出来的定义为: "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(对数的) spiral(螺旋) also known as equiangular(等角) spirals. Such spirals and occur frequently in nature, for example: shells(贝壳), sunflowers, and pine cones(松果). The limit point of the spiral is called the "eye of God".
6.Phi Pyramid
