The Expected Solution

One solution giives Phi = 1.618...

y = x + 1 y = x * x Phi = 1.618... Phi = -0.618...?

Derive Phi Graphically


Phi = 1.618...

This was the answer we expected to find, since we already know this is the approximate value of phi from the Fibonacci Series earlier.

One solution giives Phi = 1.618...

Naturally enough, if you consider the rectangle formed by the yellow lines, the x axis, and the y axis, you'll see it's a Golden Rectangle with sides of phi (horizontally) and both phi * phi and phi + 1 (vertically).

One solution giives Phi = 1.618...

But it's not the only one. Where the green line crosses the y axis it reveals the next smaller rectangle, with sides of 1 and phi.

One solution giives Phi = 1.618...

Then, the red line shows yet the next smaller rectangle. What are the lengths of its sides?


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