The Golden Ratio

The golden ratio $\varphi$ shows up in geometry, art, and nature—earning a reputation as the “most pleasing” proportion.

Definition

Golden ratio

Split a segment into $a$ (long) and $b$ (short). If $\dfrac{a+b}{a} = \dfrac{a}{b} = \varphi$, the common value is the golden ratio.

Let $x = \dfrac{a}{b}$. From $\dfrac{x+1}{x} = x$ we obtain $x^2 = x + 1$, so

$$\varphi = \frac{1+\sqrt{5}}{2}.$$

Algebraic Properties

• $\varphi$ is irrational.
• Reciprocal: $\dfrac{1}{\varphi} = \varphi - 1$.
• Square: $\varphi^2 = \varphi + 1$.

Fibonacci Connection

The Fibonacci sequence $1,1,2,3,5,8,\dots$ satisfies

$$F_n = \frac{\varphi^n - (1-\varphi)^n}{\sqrt{5}}, \quad \frac{F_{n+1}}{F_n} \to \varphi.$$

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练习题

练习 1

Write the exact value of $\varphi$ and its defining equation.

参考答案

$\varphi = \dfrac{1+\sqrt{5}}{2}$, it satisfies $x^2 = x + 1$.

练习 2

Show that $\dfrac{1}{\varphi} = \varphi - 1$.

参考答案

From $\varphi^2 = \varphi + 1$, divide both sides by $\varphi$: $\varphi = 1 + \dfrac{1}{\varphi}$ ⇒ $\dfrac{1}{\varphi} = \varphi - 1$.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\varphi$	希腊字母	Phi（费/菲）	黄金比例
$F_n$	数学符号	F sub n	斐波那契数列的第 $n$ 项
$\sqrt{5}$	数学符号	square root of five	出现在精确表达式中的根号

中英对照

中文术语	英文术语	音标	说明
黄金比例	golden ratio	/ˈɡəʊldən ˈreɪʃiəʊ/	满足 $\dfrac{a+b}{a} = \dfrac{a}{b}$ 的比例
黄金分割	golden section	/ˈɡəʊldən ˈsekʃən/	与黄金比例同义
斐波那契数列	Fibonacci sequence	/fɪbəˈnɑːtʃi ˈsiːkwəns/	邻项比趋向黄金比例的数列
无理数	irrational number	/ɪˈræʃənəl ˈnʌmbə/	不能表示为整数比的实数

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