What is the golden ratio?

The golden ratio, also known as the golden number, golden proportion, or divine proportion, is a ratio between two numbers that equals approximately 1.618. Usually written as the Greek letter phi, it is strongly associated with the Fibonacci sequence, a series of numbers wherein each number is added to the last. The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on, with the ratio of each number and the previous number gradually approaching 1.618, or phi.

History of the golden ratio.

The first known mention of the golden ratio is from around 300 BCE in Euclid’s Elements, the Classical Greek work on mathematics and geometry. Euclid and other early mathematicians like Pythagoras recognized the proportion, but they didn’t call it the golden ratio. It wasn’t until much later that the proportion would take on its mystique. In 1509, Italian mathematician Luca Pacioli published the book De divina proportion, which, alongside illustrations by Leonardo da Vinci, praised the ratio as representing divinely inspired simplicity and orderliness.

Because of Pacioli’s book and Leonardo’s illustrations, the golden ratio gained fame among mathematicians and artists. In the centuries since Pacioli’s book, many enthusiasts have claimed that the number is naturally pleasing to the eye, that it is a mathematical distillation of beauty, and that golden ratio line segments, golden rectangle side lengths, and golden triangles are represented throughout art history.

The golden ratio in art and graphic design.

A few artists and designers have deliberately based their work around the golden ratio. Le Corbusier, a famous mid-century modern architect, based a good deal of his architectural system around the golden ratio. Salvador Dali, the surrealist painter, intentionally used a canvas shaped like a golden rectangle for his painting The Sacrament of the Last Supper. In 2001, American prog-metal band Tool released “Lateralus,” a song with Fibonacci-inspired time signatures.

Calculating the Golden Ratio

Let’s briefly get down to some mathematics now. As with the image shown at the top of this lesson, the equation for calculating the ratio is simple. It is the relationship between two sides of a design (usually the horizontal and the vertical). It does not matter which side we choose as the longest (A) and which we choose as the shortest (B). (Although if you are trying to see whether the golden ratio has been used in another piece, you will need to follow which side is longest or shortest.)

The formula for this is:

A/B = (A+B)/A = 1.618033987 = Φ

Φ is the Greek letter Phi – how we represent the golden ratio. Why does A/B = (A+B)/A? It does because we are following the Fibonacci sequence and A and B (if expressed in whole numbers) are simply two consecutive numbers in that sequence. Fortunately, we can approximate this to 1.6 or 1.61, or 1.618 in designs without surrendering the aesthetic appeal of the golden ratio. Our eyes aren’t bothered by such slight deviations.

The Golden Ratio has also found applications in various fields such as mathematics, design, and even stock market analysis. It is believed to have certain harmonious and balanced properties that make it visually appealing and naturally occurring in many phenomena.

However, it's important to note that while the Golden Ratio has captivated many people, its significance and application are still a topic of debate among mathematicians and scientists. Some argue that its prevalence in nature and art may be coincidental, while others find deeper connections and patterns. Regardless, the Golden Ratio continues to intrigue and inspire creative minds.

If your software doesn’t calculate the golden ratio automatically, you can always use an online tool to help specify the ratio for side lengths. Here are three such tools:

• https://www.thismanslife.co.uk/projects/phiculator/
• https://atrise.com/golden-section/
• https://apps.apple.com/us/app/goldenratio-tool-for-every/id485258055?mt=12