Divide a line in two, such that the ratio of the small part to the large part is equal to the ratio of the large part to the whole line.
For the sake of simplicity, let’s say that the small part is 1 foot long. If the small part is 1 foot long, and the large part is x feet long, then the length of the whole line is obviously 1 + x feet long. The ratio of the small part to the large part is 1/x while the ratio of the large part to the whole thing is x/(1+x)
The Golden Ratio is where the ratio of the small to the large is equal to the ratio of the large to the whole, so we set the two ratios equal to each other: x/(1+x) = 1/x
Multiply both sides by x to get: x2/(1+x) = 1
Then multiply both sides by (1+x) to get: x2 = 1 + x
Subtract 1 + x from both sides to get: x2 – x – 1 = 0
This is a quadratic equation. The value of x which meets this equation is: (1 + √5)/2
This is about 1.618
Thus, take a line of length 2.618 feet and divide it up into two parts, the short part is 1 foot in length, the second part is 1.618 feet. The two parts have a Golden Ratio.
— Zero, The Biography of a Dangerous Idea by Charles Seife