5 min read

Every day, we experience sights and sounds that strike us as pleasant or calming. In some cases, there's no particular reason why some images in nature or art resonate more profoundly than others. But in certain cases, we're actually responding unconsciously to a powerful mathematical attribute: the Golden Ratio, or Phi.

This ratio is found all around us — in nature, in architecture, in paintings, and even in music. In mathematical terms, it refers to a number that's approximately equal to 1.618. This Golden Ratio is intimately related to the Fibonacci sequence, which in turn helps us map the beautiful Golden Spiral. But what is Phi? When was it discovered? And where do we see Phi, the Golden Ratio, and the Golden Spiral in our everyday lives?

Breaking Down the Math Magic: What is Phi?

As we said, Phi refers to the (approximate) number 1.618. Like Pi, a measurement used to describe the circumference of a circle, Phi is an irrational number. Irrational numbers don't round neatly — the digits go on for infinity, without repeating.

As a concept, Phi describes the ratio between two parts. Those well-versed in mathematics will understand it as the following equation

"a/b = (a+b)/a = 1.6180339887498948420 …"

For those of us not quite as comfortable with equations, there are other ways to visualize this number. If you used the principles of the Golden Ratio to draw a line, then divided the entire line into two sections — one long, one short — then a strange mathematical truth would emerge.

The number you'd get from dividing the longer section by the shorter section would be perfectly equal to the number you'd get from dividing the entire line by the longer section. And that number you'd get in both division problems? That's right: it would be Phi, or approximately 1.618. 

What is the Fibonacci Sequence and the Golden Spiral?

Let's break this down one step further and talk about Phi's relationship to the Fibonacci sequence of numbers. In this special sequence, each increasing number is made by adding up the two numbers before it:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

These Fibonacci numbers are intimately related to the Golden Ratio. By taking the ratio of two successive Fibonacci numbers, you get a number close to Phi. The bigger the numbers, the closer the approximation to Phi. By mapping this sequence, you create the Golden Spiral — a spiral based on the Golden Ratio that's ubiquitous in nature. 

Who Discovered the Golden Ratio?


Many records trace Phi back to Phidias (500 BC - 432 BC) who is thought to have used the tenets of Phi to design sculptures for the Parthenon. A little later on (428 BC - 357 BC), the famous philosopher Plato celebrated Phi as a key mathematical concept on earth. Between 365 BC - 300 BC, Euclid highlighted how the Golden Ratio could be applied to the shape of a pentagram. 

The Fibonacci Sequence was described by a mathematician named Leonardo Fibonacci in the year 1200. Then, in 1509 Leonardo DaVinci provided illustrations for Luca Pacioli's book, which described Phi as the "Divine Proportion." Even without its being consciously discovered, the rules of the Golden Ratio seem to permeate many aspects of our everyday lives.

Where Do You Find the Golden Ratio and the Golden Spiral?

Once you know the proportions of the Golden Spiral and the Golden Ratio, you start to see how incredibly widespread this mathematical truth is in art, nature, and the galaxy. Let's look at some surprising places you might see it pop up.

The Golden Ratio in Art & Architecture

Humans have been working with the power of Phi for thousands of years. For example, the structure of the Great Pyramid of Egypt at Giza has been examined for its relation to several advanced mathematical formulas, the Golden Ratio among them. As we discussed, you can see the Golden Ratio at work in the beautiful architecture of the Parthenon. In more modern designs, The United Nations Building contains Phi ratios.

When it comes to fine art, Michelangelo's famous paintings on the Sistine Chapel contain elements of the Golden Ratio, as do Leonardo Da Vinci's The Last Supper and his famous Mona Lisa.

Music & the Fibonacci Sequence

Though it's a bit more difficult to visualize, musical compositions also follow the rules of the Golden Ratio. That's because the relationships between notes echo the Fibonacci Sequence. And the way compositions are divided can reflect the divine proportion of Phi.

For example, Mozart's Sonata n. 1 in C Major is composed of 100 total measures, but divided into sections of 38 (exposition) measure and 62 (recapitulation and development) measure pieces, creating a golden proportion. This has also been found in the music of other talented composers like Beethoven and Bartok.

The Golden Spiral in Nature


The Golden Ratio and the Golden Spiral are all around us in nature, creating intuitively pleasing shapes. Here are just a few examples:

  • Flower petals. Remember the Fibonacci Sequence from above? The number of flower petals on flowers seems to match these numbers. For example, buttercups have 5 petals, chicories have 21 petals, and daisies have 34. In addition, petals are placed at a 0.618034 turn from one another.
  • Seed heads & Pine cones. Ever looked at the center of a sunflower? The seed heads are arranged in a perfect Golden Spiral pattern. The same goes for the arrangement of seed pods on a pinecone.
  • Galaxies. Incredibly, the spiral arms of the galaxy itself are arranged in the familiar Golden Spiral pattern.
  • Human faces. Artists know that human faces adhere to some of the rules of Phi. In fact, it's been said that the closer people's features conform to the Golden Ratio, the more attractive they will appear.
  • DNA. The measurements of our DNA are 34 angstroms long by 21 angstroms long. Both of those numbers are parts of the Fibonacci sequence, and the ratio between them is close to Phi.

One of the most incredible aspects of Phi is how unifying it is. From the microcosm of a strand of DNA to the macrocosm of the galaxy, these repetitive patterns seem to unify seemingly separate aspects of our everyday lives.

Random Occurrences of the Golden Ratio

The closer you look, the more you'll find this magic number in seemingly random places:

  • Stocks.The Fibonacci Sequence has been used to predict the behavior of the stock market.
  • Theology. In the Bible, the Ark of the Covenant is described as having measurements that reflect the Phi ratio.
  • Credit cards. If you take the measurements of a credit card, you'll find it's a golden rectangle in proportion to Phi.

The Divine Proportion & You

Once you recognize the Golden Ratio, you start seeing it everywhere: in the spiral of a seashell, in the facial features of a friend, in your favorite piece of art. Even the bones of your hands are in a Golden Ratio. At Phidle, we love celebrating the spiritual and mathematical significance of the Golden Ratio and other aspects of Sacred Geometry. If you're interested, you can check out our shop here or contact us today!

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