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## Answers

This means if s = short side and l = long side:

l / s = phi (golden ratio)

s / (l-s) = phi

So:

s / (l-s) = l / s

s^2 = l(l-s)

l^2 - ls - s^2 = 0

Solving this quadratic, we get l = s*(sqrt(5) + 1)/2

So phi, the golden ratio = (sqrt(5) + 1) / 2 = 1.618...

This is also the limit of the ratios of two adjacent elements of the Fibonacci series! In other words, in the series:

1, 1, 2, 3, 5, 8, 13, ...

1 / 1 = 1.0

2 / 1 = 2.0

3 / 2 = 1.5

8 / 5 = 1.6

13 / 8 = 1.625

....

So if you construct a figure by taking two unit squares, putting a 2x2 square on the side (making a 3x2 rectangle), then adding a 3x3 square, a 5x5 square, and so on, the resulting rectangles approach this "golden ratio". And if you draw a curve through successive vertices of the squares, you get a smooth logarithmic curve, like a snail shell.