What is the Golden Ratio?

Brett Berry
Jun 30, 2017 · 5 min read

You know you’re truly geeking out when you’re gushing about how beautiful a number is, but hey this number is pretty special ;)

The Golden Ratio has been heralded as the most beautiful ratio in art and architecture for centuries. From the Parthenon to Salvador Dali’s The Sacrament of the Last Supper the Golden Ratio has been found lurking in some of the world’s most celebrated creations.

Now whether you believe this divine proportion is truly a mark of beauty or simply selection bias is up to you, but without a doubt it is one of the most intriguing numbers in existence.

Phi is Golden

Represented by the greek letter phi (φ), the Golden Ratio is the irrational value:

Image for post
The Golden Ratio

Euclid and the Golden Ratio

In book 6 of The Elements, Euclid gives us the definition of the Golden Ratio.

He instructs us to take a line segment and divide it into two smaller segments such that the ratio of the whole line segment (a+b) to segment a is the same as the ratio of segment a to segment b, like this:

Image for post

Or equivalently as a proportion:

Image for post
Also known as the Golden Proportion or Golden Mean

(You geometry buffs out there probably recognize that a is the geometric mean of a+b and b 🙌🏼 )

The Golden Rectangle

The Golden Ratio is most commonly represented as the Golden Rectangle, a rectangle with side-length ratio of 1.618:1.

Golden Rectangles also have the property that if you cut off a square, you’ll be left with another Golden Rectangle.

Image for post
Golden Rectangles

Solving the Golden Proportion

To find where the value 1.618034… comes from we must solve the proportion. For simplicity assume b=1 and a=x so that you may solve for x.

Image for post
a=x and b=1

Step 1

Take the cross products.

Image for post

Step 2

Subtract x+1 to set the equation equal to zero.

Image for post

We now have a standard quadratic with a=1, b=-1, and c=-1.

Step 3

Plug these values into the quadratic formula and solve.

Image for post

Since we’re working with lengths, we need only the positive solution.

Image for post

And there it is! The Golden Ratio, as promised!

For good measure, plug in a=1.618 and b=1 to confirm the proportion holds.

Image for post

Notice anything interesting about (1.618 + 1)/1.618 = 1.618?

We can write the Golden Ratio in terms of itself! Which is totally awesome.

Image for post
Rewritten with φ in place of 1.618.

Or equivalently,

Image for post

Now let’s get crazy. Substitute φ=1 + 1/φ for φ in the denominator.

Image for post

Whoa, that’s cool! Let’s do it again!

Image for post

We could keep on doing this forever. Which is pretty spectacular. Turns out the Golden Ratio can be written as an infinite continued fraction.

Finding Fibonacci

We can use the continued fraction to approximate the Golden Ratio and uncover an interesting relationship with the Fibonacci Sequence.

Step 1

To start out we’re going to alter our continued fraction a little.

Instead of writing the formula nested in itself, we’ll add subscripts to indicate that the next value (φ_n+1) can be generated from the previous value (φ_n).

Image for post

Since this is an infinite continued fraction, as n increases, the approximation gets closer to the true value of φ.

Step 2

Define φ_0 = 1. To find φ_1 plug in n=0.

Image for post

Step 3

Repeat the process to find φ_2 with n=1, since φ_2 = φ_1+1. Use the result from Step 2 for φ_1.

Image for post

Step 4

Keep on repeating this process.

Image for post
Image for post

Step 5

Check it out. There’s the Fibonacci Sequence! Each approximation is the ratio of two adjacent Fibonacci numbers. We no longer need to go through the hassle of plugging values into the continued fraction, we can simply divide successive terms of the Fibonacci Sequence.

Image for post

As we move forward with each calculation, we find that our approximation of the Golden Ratio is getting closer and closer to its true value.

In fact, the limit of the F(n+1)/F(n) as n → ∞ (where F(n) and F(n+1) represents the nth and nth plus 1 terms in the Fibonacci sequence) converges to φ.

Visually, we can see how the Fibonacci Sequence generates rectangles closer and closer to the coveted Golden Rectangle.

Image for post
Fibonacci Squares

While the design world may be arguing over whether the Golden Ratio is folklore or not, I think it’s safe to say that the Golden Ratio is mathematically intriguing nonetheless.

Thanks for reading!


Stay up-to-date with everything Math Hacks is up to!

Instagram | Facebook | Twitter

More from Math Hacks →

Math Hacks

Tutorials with a fresh perspective.

Medium is an open platform where 170 million readers come to find insightful and dynamic thinking. Here, expert and undiscovered voices alike dive into the heart of any topic and bring new ideas to the surface. Learn more

Follow the writers, publications, and topics that matter to you, and you’ll see them on your homepage and in your inbox. Explore

If you have a story to tell, knowledge to share, or a perspective to offer — welcome home. It’s easy and free to post your thinking on any topic. Write on Medium

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store