Where They Meet

Graph of the function y = x * x

[y_=_x_+_1] [y_=_x_*_x] [Phi_=_1.618...] [Phi_=_-0.618?]

[Derive_Phi_Graphically]

x + 1 = x * x

On the left, we have arithmetic proportion, on the right, geometric proportion.

All the values that will ever appear on the 'x + 1' side are on the straight green line, the line of Arithmetic Proportion.

All the values that will ever appear on the 'x * x' side are on the curved red line, the line of Geometric Proportion.

Only a value that's on both lines can satisfy both sides of the equation, so only such a value can be phi.

But what is its value, exactly, and why are there two???

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