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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