What gives the golden ratio its unusual numerical properties?

Question

Why does the golden ratio (and by extension the other metallic means) have such unusual numerical properties?

For those who don't know, the golden ratio ($\varphi$) is the positive root of the quadratic $\varphi^2-\varphi-1$. It's equal to $\frac{1+\sqrt{5}}{2}$, and appears in nature. The limiting ratio of the Fibonacci numbers is the golden ratio. The golden ratio is a common object of interest of amateur mathematicians.

And in terms of strange properties, I'm talking about the following identities with the golden ratio:

$\varphi^2=\varphi+1$

$\varphi-1=\frac{1}{\varphi}$

$\varphi = 1+\frac{1}{\varphi}$ (follows from the one above it)

$\varphi = \sqrt{1 + \varphi}$

All of these follow from the golden ratio's quadratic, and the golden ratio conjugate $\Phi = \frac{1-\sqrt{5}}{2}$ displays the same properties. Is there a reason for these properties? Or am I just unfamiliar with how this works?

Edit: do all quadratic irrational numbers behave like this? There's actually an infinite set of numbers like the golden ratio called the metallic means, which are the positive roots of the quadratic $x^2-Nx-1$ for some natural number $N$. Metallic means take the form $\frac{N+\sqrt{N^2+4}}{2}$. They all display similar properties. So why does this class of numbers in general behave this way?

Answer 1

All quadratic integers have similar properties. Say that $\psi$ satisfies the equation $x^2=ax+b$, with $a,b\in \mathbb{Z}$.

Then:

• $\psi-a = \frac{b}{\psi}$
• $\psi = a + \frac{b}{\psi}$
• $\psi = \sqrt{a\psi+b}$.

It's just that the equation for $\varphi$ corresponds to $a=b=1$ so the formulas look prettier.

Answer 2

All those equations you listed above are equivalent to each other, but I guess what makes them "unusual" is that two numbers $a$ and $b$ are said to be in golden ratio when

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

The reason it's special is because it can be visualized almost anywhere in the world and in mathematics.