What is the golden ratio?

The golden ratio is a special number approximately equal to 1.618, which is appreciated for its aesthetic appeal and is utilized in art, architecture, and design to create visually pleasing compositions. Its significance is attributed to its presence in natural forms, mathematical patterns like the Fibonacci sequence, and its ability to evoke a sense of balance and harmony in various aspects of human creativity and perception.

When we look around, the golden ratio appears in nature. It appears in the spirals of seashells, the center of sunflowers, the branching of trees, and even the proportions of the human body. For example, each chamber’s size ratio approximates the golden ratio in the nautilus shell. The frequent occurrence of this ratio in nature fascinates scientists, mathematicians, and artists.

Let’s discuss a few scenarios where the golden ratio applies.

The rule of thirds in photography

The rule of thirds is a technique that derives from the golden ratio. If we divide a picture into nine equal parts using two horizontal and two vertical lines. The key idea behind this rule is to place important elements of the composition along these lines or at their intersections. This approach is believed to create more balanced, visually appealing, and dynamic compositions compared to centering the subject.
The following image demonstrates this concept.

Leonardo da Vinci's painting

Leonardo da Vinci’s painting, the Mona Lisa, is often cited in discussions about the golden ratio due to its composition.

While it’s not definitively proven that Leonardo intentionally used the golden ratio in the painting, many art historians and enthusiasts have analyzed its elements and found intriguing connections, including:

• Facial proportions: The subject’s facial features align with the concept of the golden ratio. For example, the distance from her hairline to her chin is divided by the distance from her lips to her chin. This distance is close to the value of the golden ratio.

• Composition: The subject’s placement within the frame and the use of negative space contribute to a sense of balance and proportion.

• Spiral composition: The composition of the Mona Lisa follows a logarithmic spiral, which be observed in the positioning of her head and the landscape in the background.

• Geometry: Leonardo had a deep interest in mathematics and geometry. It’s possible that he incorporated geometric principles, including the golden ratio, into his artistic work.

Music

Using the golden ratio in music is more of an artistic choice. Composers and musicians may interpret music in various ways to create aesthetically pleasing compositions.

• Song structure: Some composers and musicians use the golden ratio to determine the structure and timing of different sections within a musical piece.

• Rhythmic patterns: The golden ratio can be used to create rhythmic patterns and time signatures that follow the proportion of the golden ratio.

• Melodic development: Composers can use the golden ratio to guide the development of melodies and musical themes within a piece.

• Instrumentation: The golden ratio can influence the arrangement and distribution of different instruments or voices in a musical ensemble.

Financial and mathematics

The golden ratio and Fibonacci sequence have interesting mathematical properties. These properties are observed in various aspects of nature and finance. Their predictive or prescriptive power in financial markets is a subject of debate and interpretation among analysts and investors.

• Fibonacci sequence in financial markets: The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, etc.) is related to the golden ratio. In finance, traders and analysts use Fibonacci retracement levels to identify potential areas of support or resistance in price movements.

• Price patterns: Financial analysts sometimes observe price patterns that align with the golden ratio. For example, price increases or decreases following the golden ratio indicate potential turning points in stock prices.

• Investment analysis: Investors and analysts use the golden ratio in technical analysis to determine investment entry and exit points. They use phi-based levels to set stop-loss or take-profit orders.

• Portfolio allocation: The golden ratio is applied in portfolio allocation strategies. Investors allocate a certain percentage of their portfolio based on the golden ratio, believing it can optimize returns and risk.