Best way to discover the 'golden ratio'

Question

Let's say you live in a world where nobody ever discovered the Golden ratio. What's the most intuitive way to discover this proportion?

Wikipedia defined it this way:

$$\phi = \frac{a+b}{a} = \frac{a}{b}$$

Solving this equation, we get:

$$\phi = \frac{1+\sqrt5}{2}$$

Well, this definition is a lot clear but is it possible that someone (not a mathematician but a curious person) first discovered the Golden ratio this way? Isn't there a better intuitive way to discover the golden ratio?

I know that the golden ratio also can be a number that is equal to 1 + 1/itself. What should be the most probable way that someone discovered it? It's a pretty easy discover since it was 'rediscovered' many times in history.

Answer 1

The beautiful continued fraction $$ 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}} $$

Answer 2

The golden ratio is the diagonal of the regular pentagon.

Answer 3

The roots of the equation $x^2-x-1=0$ are $\phi$ and $-1/\phi$. You can arrive at this quadratic by simply multiplying both sides by $x$ from the identity $x-1/x=1$ and rearranging.

More generally, for every non-negative real number $y$ there exists a unique positive real number $x\leq y+1$ (with equality only for $y=0$) such that $x-1/x=y$, and after multiplying both sides by $x$ it's clear that this is the special case of the quadratic equation where $a=1,b=-y,$ and $c=-1$.

In other words, $x=\dfrac{y\pm\sqrt{y^2+4}}{2}$. The special case $y=1$ produces phi and its opposite reciprocal. For each positive integer $y$, $x$ is the value of the simple continued fraction for which every quotient is $y$, as is apparent from the property $x-1/x=y$ (this is true for any positive real $y$, but by convention the quotients of continued fractions are integers).

Another elementary appearance of $\phi$ is with the ratios of consecutive Fibonacci numbers. Let $F_0=1,F_1=1,$ and $F_{n+1}=F_n+F_{n-1}$ for every positive integer $n$. Then $$\lim_{n\to\infty}\dfrac{F_{n+1}}{F_n}=\phi$$

The reason is that $F_n\sim\dfrac{\phi^n}{\sqrt5}$, thus $\dfrac{F_{n+1}}{F_n}\sim\phi$.

Answer 4

The golden ratio can be considered the result of searching for, and finding, a certain type of equilibrium point. Suppose that x > 1. Then 1/x < 1. What about the quantity x - 1? If x is very large (> 2, at least), then x - 1 is also > 1. So, if x < 2, then x - 1 < 1. The obvious question to ask is whether x - 1 coincides with 1/x. If that equilibrium obtains, then, by definition, that value of x is the golden ratio.

Answer 5

Of course people who don't know anything about math would discover the golden ratio as the fundamental unit of the ring of integers of the number field with lowest prime discriminant possible.