The story behind this is quite silly, I was messing around with functions in geogebra, inputting several functions to see their properties, until I was amazed by something i don't know about.
Apparently, the intersections (unfortunately I don't know how the're named in english) between $f(x)=x^{2}+x^{-2}-3$ and the x-axis resembles $\varphi$, in the screenshot i've attached you can see it reveals the first 8 digits of $\varphi$ correctly. Can someone explain to me why this happens? I don't believe it's just a coincidence and I'm definitely missing something.
The intersection points are the solutions to the equation $$x^2+x^{-2}-3=0.$$ The solutions are precisely the (nonzero) solutions to the equation $$x^2(x^2+x^{-2}-3)=0,$$ or equivalently $$x^4-3x^2+1=0.$$ It turns out that this polynomial factors as the product of two quadratic polynomials: $$x^4-3x^2+1=(x^2-x-1)(x^2+x-1).$$ And of course the golden ratio $\varphi$ and its conjugate $\varphi^{-1}=\varphi-1$ are precisely the roots of $$x^2+x-1=0.$$ And similarly their negatives $-\varphi$ and $-\varphi^{-1}$ are precisely the roots of $$(-x)^2+(-x)-1=0,$$ or equivalently $$x^2-x-1=0.$$ So the four solutions to the original equation are precisely $\pm\varphi$ and $\pm\varphi^{-1}$.