Transcription
This is the golden ratio. It equals approximately 1.618 and it comes from a strangely perfect way of dividing a line.
Imagine cutting a line into two pieces, a longer piece called A and a shorter piece called B. Normally, these pieces could have any lengths, but the golden ratio happens when the proportions match in a very specific way. The entire line A + B compared to the longer plus A must equal the longer piece A compared to the shorter piece B.
In symbols, A + B over A equals A over B, which sounds complicated, so let's use some real numbers. Let's say the shorter piece is one unit long. We'll call that B. Now, let the longer piece be X. Our equation becomes X + 1 / X = X. Multiply both sides by X and we get X + 1 = X². Then, rearrange it. X² - X - 1 = 0.
Now, we use the quadratic formula and that gives us X = 1 + or - the square root of 5 all divided by 2. The positive solution is approximately 1.6180339887. That is the golden ratio represented by the Greek letter phi. Technically, the negative solution is not phi itself. Phi is only the positive solution.
But, here's where things get interesting. If you subtract one from phi, you get approximately 0.618. And if you divide one by phi, you also get approximately 0.618. So, phi - 1 = 1 / phi. And very few numbers behave this neatly.
The golden ratio is also connected to the Fibonacci sequence. 1, 1, 2, 3, 5, 8, 13, 21. Each number is created by adding the previous two numbers. Now, divide one Fibonacci by the number before it. 13 / 8 is 1.625. 21 / 13 is 1.615. As the numbers grow, these ratios move closer and closer to 1.618. And that connection helps explain why golden ratio-like spirals appear in sunflower seeds, pine cones, leaves, and other natural growth patterns.
But people do exaggerate this. Not every building, face, painting, or galaxy follows the golden ratio. A lot of the claims are just forced onto the fact. The real math is already cool enough without the extra mythology going on.
The golden ratio is just a proportion that recreates itself. The relationship between the whole and the larger piece is identical to the relationship between the larger piece and the smaller one.