How to get the Golden Ratio Formula, Explained

Here we will see how to get the golden ratio formula out of thin air. We take a basic geometrical relationship and solve for the golden ratio without adding in any extra numbers or physical constraints. This process really shows how pure and fundamental this fractal number is. Besides basic algebra, we will also use the quadratic formula, but that’s the whole extent of the skills we need to do this.

The Geometrical Premise of How to get Golden Ratio

The relationship we want to describe is “the whole to the part is the same as the part to the other part”, where the other part is related to the whole and the part by being the actual difference between them. Put another way, “the whole to the part is the same as the part to the difference between the whole and the part.” If that was word salad to you or you’re more visual, you can see the relationship in the diagram.

This relationship found in pentagonal geometry and the equality of ratios was first verified by direct measurements. Then some mathematicians wanted to generalize it and see if this ratio could be broken down to a pure number, and it can. We just take a proportion and a hunch and you’ll see how to get the golden ratio formula.

So to start out getting φ down to a number, we start with the “part to whole” relationships described. Let L be the long part, and s be the shorter part.

L / s = s / (L-s) = φ

“Long to short is the same ratio as short to the difference between long and short.”

Recursive, self-similar properties guide the Derivation

The recursive property of the Golden Ratio was first found by direct measurement. The φ² segments in particular were indicative of a deeper pattern. This led mathematicians to play around looking for ways to algebraically eliminate the “long” and “short” from the problem and just get a number for φ.

One of the cool things about fractal numbers and geometries is it condenses more information into a smaller amount of relationships. Usually when we take a random number like 2, 7, or 100, or fractions like .5, .22, or .99, for example, the square of the number is one thing, and the number plus one is something different, there is no relationship. If the square and the number plus one were the same thing, it actually implies dozens and dozens of relationships that can be unveiled with algebra, for example, the reciprocal being the same as the complement (1-φ), and more.

Breaking it down to Where you Can Solve for the Golden Ratio

So let’s say we just have this hunch that φ² = φ+1. We can easily show this agrees with the proportions (equality of ratios) we set up for the pentagram line segments.

φ² = (L/s) * (s/[L-s]) = L/(L-s)

φ+1 = (s/[L-s]) + ([L-s]/[L-s]) = L/(L-s)

In the first one, the s’s cancel because s/s is 1, and one times anything is itself. In the second one, the s’s cancel because there is +s and -s added in the numerator.

Now we proved that φ² = φ+1. This relationship only has ONE variable, so we can solve for it! Since this is a quadratic equation, we can solve for φ using the quadratic formula (which can also be derived from thin air, but we won’t be doing that here because it has a lot of steps. Each individual step is simple though so if interested you can find people doing it online.)

Final Result Deriving the Golden Ratio

Let’s rearrange the quadratic φ² = φ+1 to give:

φ² – φ – 1 = 0

The quadratic equation is:

For a quadratic of the form ax²+bx+c=0, x is:

x = [-b + √(b² – 4ac) ] / 2a

Normally the quadratic formula gives two roots, and in this case the two roots are distinct. We only are going to care about the positive root because that’s what turns out to be the form of the golden ratio most people are used to. The negative root has its own fun stuff we can explore in a later post. In our φ quadratic we have a=1, b=-1, and c=-1. So we can simply plug those into the quadratic formula and the result is the EXACT version of φ.

φ = (1+√5)/2

If you put this in a calculator, you get 1.618…. and ONLY the form of the golden ratio with this SQUARE ROOT OF 5 is exact. The square root of five is what makes this whole number irrational and interesting. Otherwise, it would be no more interesting than 1/2. Notice the golden ratio’s irrationality (and it’s VERY irrational) all comes from the root 5, and we happen to find it from pentagonal geometry. Read more about FIVE-ness here.

Hope you had fun learning how to get the golden ratio formula. Follow me on socials and stuff if you want to see my future posts further extrapolating on this to apply to spirals and energy flow through crystalline organs in the body. <3