The Golden Ratio

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

$\varphi$ （phi，费/菲）：黄金比例常用的希腊字母，读作“菲”或“费”，数值约 $1.6180339\ldots$ 。

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.

符号说明

Symbol	Type	Pronunciation/Note	Meaning in this article
$\varphi$	Greek letter	Phi (fee/fye)	Golden ratio, approximately 1.618

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

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

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.

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

Answer and Analysis (3 个标签)

golden ratio exact value defining equation

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

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

Answer and Analysis (3 个标签)

golden ratio algebraic properties reciprocal

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

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

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