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Golden Ratio

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The golden ratio a renowned proportion in mathematics and design, traces its origins to the ancient Greeks.

What is the golden ratio?

Often referred to as the golden number, golden proportion, or divine proportion, is a mathematical ratio of approximately 1.618. Represented by the Greek letter phi, this ratio is closely linked to the Fibonacci sequence, where each number is the sum of the two preceding ones. The sequence starts with 0, 1, 1, 2, 3, 5, 8, 13, 21, and continues indefinitely, with the ratio of successive Fibonacci numbers converging towards the golden ratio, phi.

History

The earliest recorded reference, dates back to around 300 BCE in Euclid’s “Elements,” a seminal work in Classical Greek mathematics and geometry. While Euclid and other ancient mathematicians such as Pythagoras acknowledged this proportion, they did not refer to it as the golden ratio. Its enigmatic reputation emerged much later. In 1509, the Italian mathematician Luca Pacioli released “De Divina proportione,” a book that, with illustrations by Leonardo da Vinci, extolled the ratio for its divine simplicity and order.

The golden ratio owes its fame among mathematicians and artists to Pacioli’s book and Leonardo’s illustrations. Over the centuries, numerous enthusiasts have asserted that this number is inherently pleasing to the eye, a mathematical embodiment of beauty, and that elements such as golden ratio line segments, golden rectangle side lengths, and golden triangles are prevalent throughout the history of art.

The golden ratio in oysters
Nature

Proponents argue that it is aesthetically pleasing due to its prevalence in nature. Examples often cited include the proportions of nautilus shells and human bodies, which are said to exhibit the golden ratio, although individual variations are significant. While some seashells grow in accordance with the golden ratio, forming a pattern called the spiral, this is not universal. Nautiluses, for instance, retain consistent shell proportions throughout their lives, but these proportions typically form a logarithmic spiral rather than an exact expression of phi.

Phi manifests in various natural phenomena. The growth patterns of tree leaves and pine cone seeds often resemble the golden ratio, while sunflower spirals and other seed arrangements tend to align closely with phi.

In art and graphic design

Several artists and designers have intentionally incorporated the into their work. Le Corbusier, the renowned mid-century modern architect, significantly integrated the into his architectural designs. Salvador Dali, the surrealist artist, purposefully chose a golden rectangle-shaped canvas for his masterpiece “The Sacrament of the Last Supper.” In 2001, the American progressive metal band Tool unveiled “Lateralus,” a track featuring time signatures inspired by the Fibonacci sequence.

Art historians have identified instances of the in works like the Mona Lisa, the Parthenon in Athens, and the Great Pyramid of Giza. Yet, often there is no concrete proof that artists deliberately applied the ratio as Le Corbusier, Dali, or Tool did. In the absence of design documents or specifications for the pyramids, it remains uncertain whether ancient engineers consciously utilized phi.

References: Adobe / fstoppers / zekagraphic

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