The origin of this number can be traced back to Euclid, who mentions it as the “extreme and mean ratio” in the Elements. It is closely associated with the Fibonacci sequence, where each term is the sum of the two preceding terms, e.g. 1, 1, 2, 3, 5, 8, 13, etc. As the sequence progresses, the ratio between successive terms approaches ϕ.

In ancient Greece, the Golden Mean is said to have played a key role in both philosophy and aesthetics. Pythagoras and Plato viewed the ratio as a representation of balance, harmony, and beauty. It is believed to have influenced Greek architecture, including the design of the Parthenon, although its intentional use remains hotly debated. Philosophically, Aristotle referenced a similar concept in ethics, where the "golden mean" described a virtuous middle ground between extremes, such as courage lying between recklessness and cowardice.

The Golden Ratio gained renewed importance during the Renaissance when artists and architects embraced its principles to achieve visual and spatial perfection. Although once again debated, artists like Leonardo da Vinci are said to have applied the ratio to their works, as seen in Vitruvian Man, which explored the proportions of the human body, and in the composition of the Mona Lisa.

In the natural world, the Golden Ratio does in fact appear in a variety of phenomena, reflecting patterns of optimisation and efficiency. Spiral arrangements of leaves, seeds in sunflower heads and pinecone scales often align with the Fibonacci sequence and the Golden Ratio. Similarly, the proportions of certain shells, like the nautilus, and structures in animals can conform to this ratio. In astronomy, the spiral arms of galaxies sometimes exhibit patterns related to ϕ, reinforcing the idea of a universal order. These natural occurrences have led to the mystical belief that the Golden Ratio symbolises divine design.

The Golden Ratio also left a profound mark on 20th-century art and design. Influential figures like Le Corbusier, a modernist architect, developed the “Modulor” system of proportions based on ϕ to create visually harmonious, human-centred designs.

Symbolically, the Golden Ratio has represented balance and harmony across various cultures and contexts. It aligns with humanity's innate desire to find order in art, science and life. While its significance is widely recognised, the ratio's actual use in historical works is overstated. Scholars debate whether its presence in certain artworks or buildings was intentional or merely coincidental. Some have argued that it is in fact a self-perpetuating myth, the persistence of which is a result of our human tendency to have our biases confirmed by selectively perceiving patterns that comply with our beliefs.

Despite this criticism, the Golden Mean’s enduring allure lies in its intersection of mathematics, nature and art … embodying humanity's quest for harmony, balance and beauty.

Story Idea: Damien O'Brien
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References

wikipedia.org/wiki/Golden_ratio
prestigeonline.com/th/lifestyle/art-plus-design/golden-mean-proportions
plus.maths.org/content/myths-maths-golden-ratio
skeptoid.com/episodes/4325

Images

1. Golden Mean: Sunflower Seeds and the Mona Lisa
2. Euclid by Jusepe de Ribera, 1630–1635. Credit: J. Paul Getty Museum
3. A papyrus fragment of Euclid's Elements dated to c. 75–125 CE
4. Many of the proportions of the Parthenon are alleged to exhibit the golden ratio, but this has largely been discredited.
5. Golden ratio superimposed on The Great Wave Off Kanagawa by Katsushika Hokusai, 1831
6. The logarithmic spiral of the nautilus shell
7. Modulor by Le Corbusier
8. The flag of Togo aspect ratio uses the golden ratio
9. Chris Budd OBE, Professor of Applied Mathematics at the University of Bath, is a Golden Mean sceptic