# The Golden Ratio Consider the following process: start with a [[rectangle]], inscribe a square so that the the square shares a side with one of the shorter sides of the rectangle, then consider the remaining region of the rectangle. This is a new rectangle (though if the original shape was itself a square, this rectangle will be empty). A rectangle is a **golden rectangle** if the new rectangle is [[similar]] to the original one. All golden rectangles are [[similar]] to each other, and the **golden ratio** is the ratio of its side lengths. It is commonly denoted by $\phi$. Numerically, the golden ratio is the positive solution to the quadratic $x^2 = x + 1$, which is: $$ \phi = \frac{1 + \sqrt{5}}{2} \simeq 1.618 $$ category: definition