First, let's calculate it.
We begin with:
$\frac {\phi} {1-\phi} = \frac 1\phi$
We can infer the above equation based on this diagram from Wolfram Alpha:
This is known as a golden rectangle. The golden ratio is denoted by the Greek letter $\phi$
Then we simplify getting:
$\frac {\phi^2} {1-\phi} = 1$
$\to \phi^2 = 1-\phi$
$\to \phi^2 + \phi - 1 = 0$
We then use the quadratic formula:
$\phi = \frac {-(-1) \pm \sqrt {(-1)^2 - 4(1)(-1)}}{2(1)}$
Obtaining:
$x = \frac {1 \pm \sqrt {5}}{2}$
We take the positive version because $\phi$ is defined that $\phi \gt 1$
$x = \frac {1+\sqrt {5}}{2}$ $\to \frac {1+\sqrt {5}}{2} = 1.6180339887 \ldots$
Why do we care about this ratio?
The Golden Ratio has much historical significance. It has been used not only in mathematics but art, architecture, music, and some have even proposed it has a connection to nature and the human genome. Some of the examples below from Wikipedia illustrate how the golden ratio has been used in these different fields.
From Wikipedia
The Parthenon has features suggesting elements of its façade were circumscribed by golden rectangles.
From Wikipedia
Salvador Dalí's painting, The Sacrament of the Last Supper, intentionally includes the golden ratio. The dimensions of the canvas are a golden rectangle and the dodecahedron above Jesus has angles which are in a golden ratio to one another.
From Wikipedia
This image of the Fibonacci Spiral shows how it approximates the golden ratio. This description from Wikipedia provides an explanation:
"Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of one square divided by that of the next smaller square is the golden ratio."
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