The golden ratio is a mathematical ratio that can also be expressed as a number. It has been discerned throughout the natural and artificial world. It is sometimes called the 'divine proportion' or the 'divine number' due to its ubiquity and links to beauty and aesthetics.
The astrophysicist Mario Livio once said of the golden ratio, "some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece […] to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties."
So it is not an understatement to say that this mysterious number that permeates maths, science, nature, art, music, physics, economics, and just about any other domain of life you can think of could perhaps lie at the heart of either how the fabric of nature is constructed, or how we humans perceive the construction of the fabric of nature.
We will explore the golden ratio, what it is, how it is calculated, what has been written about it, and where you can find it.
The golden ratio approximately equals 1.618 and is found when a line is divided into two unequal parts so that the length divided by the long part equals the long part divided by the short part. It has fascinated mathematicians and philosophers for millennia and appears in nature, geometry, art, architecture, and many other areas of the world and the universe.
The golden ratio shows up in so many areas of life without us knowing it. Some people argue that it doesn't exist and is just an example of humans finding patterns where there are none.
So let's jump in and find out the exact formula behind the golden ratio.
What is the golden ratio formula?
The golden ratio formula can be conceived in many ways. The best way to formulate it is to consider separating a line into two smaller lines. The longer of the two lines is a, and the smaller of the two lines is b.
- (a + b) / a = a / b
When the dissection of a line can be written as the above formula, it is split by the golden ratio.
The ratio can also be decimalised to an infinitely long number. It is approximately equal to the following:
So let's look at a numerical example in practice.
If you have a piece of string that is 100 cm long and you want to split it using the golden ratio, you would need your pieces of string to be:
- 61.8 cm
- 38.2 cm
- 100 / 61.8 = 1.618
- 61.8 / 38.2 = 1.618
What are golden shapes?
Golden shapes are geometric shapes that adhere to the principles of the golden ratio. A golden rectangle is a shape that is measured using the golden ratio. This means that the height and width of the rectangle are:
- height = a
- width = a + b
The values of a and b must be such that their relationship can be formulated into the golden ratio formula.
Artists often use the golden rectangle to create the frame size of their paintings, and some buildings fit perfectly into golden rectangles. For example, the Parthenon in Greece is a golden rectangle, but it is unknown whether that was intentional or not.
Golden circles can be drawn within the squares of a Fibonacci Spiral, which we will explore later on. A perfect circle in each square of the Spiral follows the 1:1.618 ratio to create a sequence of golden circles. Both Pepsi and Twitter use golden circles within their branding.
Golden triangles-also called 'sublime triangles'-are isosceles triangles whereby the size of the duplicated side is in the golden ratio to the base.
What is the history of the golden ratio?
Ancient Greek mathematicians and philosophers first wrote about the golden ratio. Euclid called it the 'extreme and mean ratio.' Pythagoras and his school were also fascinated by the geometric implications of the golden ratio and the fact that it was an irrational number.
However, a few thousand years before this, the Great Pyramid in Giza was constructed, and it, too, is said to adhere to the geometric principles of the golden ratio. Therefore, the golden ratio may have been understood long before any surviving documents were written. However, there is also the argument that humans may have built things to emulate the golden ratio's proportions without knowing what they were doing.
Mathematicians in the Islamic world continued the study of the golden ratio, and many centuries after the first writings on it, the Italian mathematician Fibonacci provided perhaps the most well-known study of the golden ratio to this day, which we will explore shortly.
Over the last century, fascination surrounding the golden ratio has not waned. Inventor Steve Baer created the zome construction system based on the golden ratio. And the mathematical physicist Roger Penrose developed a pattern called 'Penrose tiling' that uses the golden ratio.
The golden ratio has had many different names, and the golden ratio was not used as a term until the 1800s. It has also been named:
- the golden section
- the golden proportion
- the divine proportion
- the golden mean
- or denoted by the Greek letter 'Phi.'
The golden ratio and the Fibonacci Sequence
The Fibonacci Sequence is a sequence of numbers in which each number is the sum of the two preceding numbers. Starting from 1, it looks like this:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
It was named after the Italian mathematician Fibonacci, but the sequence appears in ancient Sanskrit writings dating back thousands of years before Fibonacci was born.
So how does this sequence relate to the golden ratio?
If you take one number as a and the next number in the sequence as b and then perform the golden ratio formula outlined above (a / b =golden ratio), you get answers that are increasingly close to the golden ratio number (1.618).
- 2 / 3 = 1.5
- 3 / 5 = 1.666…
- 5 / 8 = 1.6
- 8 / 13 = 1.625
- 233 / 377 = 1.618025751
So the Fibonacci Sequence is directly related to the golden ratio, and as we continue through the sequence, we get closer and closer to the exact golden ratio number.
When expressed as square boxes (one by one, two by two, three by three, five by five, etc), the Fibonacci Sequence creates a spiral known as a "golden spiral."
Where can we find the golden ratio?
The golden ratio can be discerned in architecture, art, human anatomy, physics, nature, marketing, music, financial markets, and many other places.
The golden ratio is present in the following:
- The florets of the head of a sunflower. The patterns in the florets and the seeds within each floret follow the Fibonacci Sequence and create a spiral within the sunflower head.
- The number of petals of field daisies. Field daisies can have different numbers of petals. However, the number of petals is always within the Fibonacci Sequence.
- Leonardo da Vinci's The Last Supper. The golden ratio is used throughout da Vinci's masterpiece. It frames groups of disciples with one another, sets up the table, and is used within the adornments that surround Jesus and his Apostles while they eat.
- Salvador Dali's The Sacrament of the Last Supper. This was drawn in response to da Vinci's painting. Dali framed the painting within a golden rectangle. And, like da Vinci, Dali positioned the table at the golden section of the height of the painting. He positioned the disciples at the side of Jesus at the golden sections of the width. And the windows are formed by a large dodecahedron consisting of 12 pentagons, which exhibit the golden ratio in their proportions.
- The Great Pyramid. This is a contentious one. Some have suggested that the Great Pyramid uses the golden ratio from its midpoint to its apex. However, due to the degradation of the Pyramid, it is impossible to verify this with any exactitude.
- The Taj Mahal. The Taj Mahal in India uses the golden ratio in the width of its central arch to the width of the whole building. The golden ratio is also present in the height of windows within the arch to the height of the section beneath the domes.
- The Parthenon. Another contentious one. The Parthenon looks to have been constructed within the proportions of a golden rectangle. However, as with the Great Pyramid, age and ruin have changed the Parthenon, which makes it impossible to judge accurately.
- Human skull dimensions. Research by John Hopkins University showed that the golden ratio appeared in the distances between different human brain sections.
- Logos for Pepsi, Apple, Twitter, National Geographic, and many others. Marketers and advertisers often use it to subconsciously attract the human eye and capture our attention.
And countless other natural and artificial bodies.
The 'Divine Proportion'
During the Italian Renaissance, the golden ratio's ubiquity and association with beauty, perfection, and aesthetics came to be known as the 'divine proportion.' Many people saw it (and still see it) as a key to opening the door to a deeper understanding of true beauty and spirituality. Astonishingly, a single number can profoundly impact human thought, but it has played a huge role in human history and the natural world.
So, throughout the years, people have come to associate the golden ratio with divinity, spirituality, and God. This is represented in art most notably. The Renaissance painters used the golden ratio to achieve balance and beauty and to connect their works with higher powers.
The golden ratio has fascinated and bemused thinkers, artists, mathematicians, scientists, and philosophers for years. Is the fact that we find the golden ratio in nature proof that it is an inherent formula for beauty and order? Or does our desire for beauty and order mean that our eyes and brains are attuned to finding patterns that aren't really there?
While we may never know the answers to the many questions that the presence of the golden ratio elicits, something is comforting in the notion that there is a historical thread that ties us to ancient people who also found beauty in this strange and mysterious number.