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I was playing around with Desmos Graphing Calculator and found that the two graphs below intersect at (-1,0), (0,-1), (1.618, 1.618), and (-0.618, -0.618). The latter two points are the golden ratio and the negative of the golden ratio plus 1.

Graphs:

  • $y = x^2 - 1$
  • $x = y^2 - 1$

Why do the graphs intersect at these points?

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You have $x = (x^2-1)^2 -1 \iff x+1 = (x^2-1)^2 \iff x^4 - 2x^2 - x = 0$ so $$x(x^3 - 2x - 1) = x(x+1)(x^2-x-1) =0$$ so the roots are $x=0,-1, \phi, -\phi^{-1}$ by using the quadratic formula on the last term.

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    Is the last term ($−ϕ^−1$) is -0.618...? The properties of the golden ratio keep surprising me! – 2017-02-26
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    @TimK It's the negative of the inverse of $\phi$ – 2017-02-26
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    @Omry, yes I understand the notation. I was just surprised. – 2017-02-26
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    @TimK $1+\frac{1}{\phi}=\phi$ – 2017-02-26
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    @Omry, do people know why that this number has so many of these kind of properties? – 2017-02-26
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    @TimK That last property is just what you get from taking $x^2-x-1=0$, transferring the $-x-1$ to the other side of the equation, and dividing by x. Nothing special about $\phi$ in this one. – 2017-02-26