Golden Ratio

The golden ratio (φ) is a mathematical constant regarded as one of the most aesthetically harmonious proportions observed in architecture and nature. Approximately equal to 1.618, this ratio expresses a special relationship between two quantities. When a line or shape is divided according to the golden ratio, the ratio of the larger part to the smaller part equals the ratio of the whole to the larger part. This proportion creates a sense of visual harmony and appeal.

Historical Development of the Golden Ratio

The origins of the golden ratio extend back to Ancient Greece. In Euclid’s work “Elements,” it is referred to as the “extreme and mean ratio.” The Ancient Greeks recognized its aesthetic balance and applied it in their architectural works. During the Renaissance, the golden ratio was rediscovered and began to be used in artistic compositions. Leonardo da Vinci frequently employed the golden ratio, which he called the “Divine Proportion,” in his works.

Mathematical Properties of the Golden Ratio

The golden ratio is closely related to the Fibonacci sequence. The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones (1, 1, 2, 3, 5, 8, 13, 21, ...). The ratios of consecutive numbers in this sequence approach the golden ratio as the sequence progresses. The golden ratio can also be expressed by the following equation:

φ = (1 + √5) / 2 ≈ 1.618

Another interesting property of the golden ratio is that its square is equal to itself plus one: φ² = φ + 1.

Presence of the Golden Ratio in Nature

The golden ratio appears in many forms in nature. It can be observed in the arrangement of sunflower seeds, the spirals of snail shells, the scales of pinecones, and in the human body. For example, the ratio of the length of the human face to its jawline is close to the golden ratio. Additionally, the lengths of the fingers on the human hand are arranged in accordance with the golden ratio.

(image generated by artificial intelligence)

• Presence of the Golden Ratio in Nature: This image is a stylized illustration emphasizing the presence of the golden ratio in nature, particularly in the spiral structure of the nautilus shell. The image is designed in a minimalist geometric style that unites mathematical proportions with natural beauty.

Use of the Golden Ratio in Art and Architecture

The golden ratio is used in art and architecture to achieve aesthetically pleasing balance. Historical structures such as the Parthenon in Ancient Greece and the pyramids of Egypt incorporated the golden ratio in their design. Many artistic masterpieces, including Leonardo da Vinci’s “Mona Lisa” and Salvador Dalí’s “The Last Supper,” also employ the golden ratio. Today, architects and designers continue to apply the golden ratio in the design of buildings and objects.

(image generated by artificial intelligence)

• Use of the Golden Ratio in Architecture: This architectural diagram is a detailed and informative visual illustrating how the golden ratio is applied in building proportions. The diagram highlights the golden spiral within building design to clearly demonstrate the relationships between different ratios.

Use of the Golden Ratio in Photography

In photography, the golden ratio is used when framing and composing images. It provides a more balanced and aesthetically pleasing arrangement of elements within the photograph. Photographers use the rule of the golden ratio to guide the viewer’s attention to key focal points in the image.

The Golden Ratio in Contemporary Context

The golden ratio continues to be applied today in numerous fields including design, art, architecture, and photography. Researchers are continuing to investigate the effects of the golden ratio on the human brain. Studies have shown that designs aligned with the golden ratio tend to evoke greater appreciation in people.