The Golden Section

13 May

The golden section (or golden ratio) has fascinated Western intellectuals of diverse interests for over two thousand years. Basically, two quantities are ‘golden’ if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. If you can get your head around this, you may start to notice how this ratio appears frequently in mathematics, architecture and the arts — especially in the form of the ‘golden rectangle’, in which the ratio of the longer side to the shorter is, in the above sense, ‘golden’. Through the ages, the golden ratio has been regarded as having unique and interesting properties, both because it is intrinsically aesthetically pleasing and because it may have a deeper meaning. Some of the greatest mathematical minds in history, from Pythagoras and Euclid in ancient Greece, to the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, have spent endless hours musing over the golden ratio and its properties. Even today, the golden ratio is used in the analysis of financial markets, in strategies such as Fibonacci retracement. But the fascination with the golden ratio is not confined just to mathematicians: biologists, artists, musicians, historians, architects, psychologists, and even mystics have for centuries pondered and debated the basis of its ubiquity and appeal. This inevitably invites the question – what is it about the golden ratio that has so intrigued thinkers of all disciplines?

 

The golden section was first studied seriously in ancient Greece by mathematicians who became aware of it through its frequent appearance in geometry. The golden rectangle is supposed to appear in many of the proportions of the famous ancient Greek temple, the Parthenon, in the Acropolis in Athens but there is no original documentary evidence that this was deliberately designed in this way. The Greeks usually attributed the discovery of this concept to Pythagoras, although it was Euclid who provided the first known definition of the term and the sculptor Phidias (in the form of the first letter of his name) who is credited with inspiring the use of the Greek letter phi to symbolize the golden ratio. During the Renaissance, Luca Pacioli (1445-1517) in his Divina proportione wrote about the golden section, which he called ‘the divine proportion’. Pacioli’s work directly inspired Albrecht Durer and even Leonardo da Vinci himself. The layout of both The Annunciation and Madonna with Child and Saints show clearly Da Vinci’s adherence to the golden section. To this day, many books on oil painting and water colour point out that it is better to position objects not in the centre of the picture but to one side or about one-third of the way across, and to use lines which divide the picture into thirds. This seems to make the picture design more pleasing to the eye and relies again on the idea of the golden section being an ideal to aspire to, as it was for the artists of old.

Throughout the ancient, medieval, Renaissance and modern worlds, the golden section (and its counterpart, the Fibonacci sequence) appear again and again. It is present in Sanskrit poetry, Eqyptian pyramids, the design of the Sistine Chapel, the Theorem of Pythagoras, the Christ The King tapestry behind the altar in Coventry Cathedral and the meter of Virgil’s Aeneid. In architecture, the golden section can be seen in the design of the Eden Project in St. Austell, the United Nations Building in New York, the Great Mosque of Kairouan and even, bizarrely, in the Turku power station in Finland. In music, Mozart appeared to use the golden mean in the construction of his sonatas, Beethoven in his Fifth Symphony and Chopin in his Etudes and Nocturnes. Bartók, Debussy, Schubert, Bach and Satie also may have deliberately used the golden section in their music. Many books produced between 1550 and 1770 show the golden ratio proportions exactly, to within half a millimetre. When the Russian Sergie Eisenstein directed the classic silent film of 1925 The Battleship Potemkin he divided it up using golden section points to start important scenes in the film, measuring these by length on the celluloid film. Returning to art, Salvador Dalí explicitly used the golden ratio in his masterpiece The Sacrament of the Last Supper and Mondrian is said to have used it extensively in his geometrical paintings. To this day it is argued (but by no means universally accepted) that the golden ratio is commonly used in the design of everything from the shapes of postcards to playing cards, posters, wide-screen televisions, photographs and light switch plates.

Most intriguingly, perhaps, it is also argued that the golden ratio seems so pleasing to the eye due to its frequent appearance throughout nature and that this explains why it has been adopted in so many man-made designs. Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavours. In all these otherwise unrelated phenomena he saw the golden ratio operating as a universal law. Several researchers have proposed connections between the golden ratio and human genome DNA. The idea that the golden ratio in art derives from nature goes as far back as Leonardo da Vinci’s Vitruvian Man painting, which suggests that the shape of the human body is directly in proportion to the golden ratio. If, having read the above, the idea of the golden section interests you, then you might enjoy the following short story by Jeffrey Ford. Click here to read The Scribble Mind – enjoy!

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3 Responses to “The Golden Section”

  1. Ng Zhi An May 13, 2012 at 1:35 pm #

    And you can see golden ratio in web designs too! 😀
    http://www.smashingmagazine.com/2010/02/09/applying-mathematics-to-web-design/

  2. almondjoycie May 13, 2012 at 3:08 pm #

    I am fascinated with the Golden Section too. I recently wrote about it for my blog Little Poets, to introduce the concept to kids and have them write a poem using its principles. I first wrote about it in my blog This Limerick Called Life as well. I found a great video through Khan Academy that just keeps you watching. Hope you like them!

    http://joycefied.wordpress.com/2009/12/02/telling-fibs/

    http://littlepoets.wordpress.com/2012/05/07/day-8-telling-fibs/

  3. wildinjapan May 15, 2012 at 9:48 pm #

    There is also the silver ratio, which plays a part in East Asian perceptions of beauty – the architecture of the Horyuji pagoa, for instance, is based on this ratio. Japanese paper sizes (the B series) also use this.

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