THEORETICAL STUDY:
Geometry, Numbers and Proportions: Beauty and the Golden Section INTRODUCTION Also known as the Divine Proportion, the Sacred Cut or Phi, the Golden Section has a long tradition in western culture. It can be found in plant seed patterns, in bees’ family trees, in the Pyramids, in Gothic Cathedrals, in Renaissance paintings, in the human body or in shells: these examples are part of an infinite series. The history of the Golden Section began with calculations on Babylonian clay tablets and emerges today in the digital age.1 Even though the Golden Section is found in many aspects of culture and science, a visible experience of the ratio is best seen in structures of ancient and modern architecture. Proportions have been fundamental to architecture’s claims on the beautiful since the time of Pythagoras (c 570 – 490 BC) when the art of building was first described by a theory. He identified proportion as the instrument for establishing the mean between extremes and blending opposites into a relationship.2
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Hemenway, P. (2008). The Secret Code The mysterious formula that rules art, nature, and science. EVERGREEN GmbH, Cologne, Germany. Westfall, C.W. (2013). Beauty and Proportionality in Architecture. [online] Newlington-Cropsey Cultural Studies Center. Available at: http://nccsc.net/essays/beauty-and-proportionality-architecture.
J. Douglas-Hill
University of Adelaide
The Greeks continued to incorporate the “pleasing aesthetic effects” of the golden ratio, which provided a sense of balance and equilibrium. Believed to have been constructed around 4,600 years ago, the Great Pyramids of Giza were built around the golden ratio, with the largest pyramid containing the use of phi and the golden ratio. Similarly, Notre Dame built beginning in 1163 incorporates phi and the golden ratio in its design, as does (with some controversy as to intent) the Parthenon of the Acropolis in ancient Athens.3 More modern representations, which are said to incorporate the golden ratio, are the UN Building in New York, the CNN Tower in Canada, the works of Frank Lloyd Wright and Le Corbusier.4 The Golden Section has been utilized for millennia, utilized in buildings and equally in the design of urban landscapes, notably landscape architecture.
WHAT IS THE GOLDEN SECTION? The Golden Section is also referred to as the Golden Ratio, Golden proportion, Golden Triangle, the Divine Proportion, Divine Section, and Phi. It has also been referred to as the golden rectangle.3 First described in Euclid’s Elements 2,300 years ago, the established definition for the Golden Ratio is this: two objects are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.5 This value to which this works out is usually written as 1.6180. The Fibonacci sequence is related to the golden ratio: by dividing a value in this sequence by the previous value the result is almost phi: the higher the number in the sequence, the closer the ratio is to phi.4 Conceptually, the golden ratio draws from the mathematical term ‘proportion’, that is, the equality of two ratios. Two pairs of quantities are in proportion if the equation a/b = c/d holds true. These quantities refer to lengths of line segments when considering the golden ratio.
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Meisner, G. (2015). Golden Ratio Overview. [online] The Golden Number. Available at: https://www.goldennumber.net/golden-ratio. Nathe, C. and Hobgood, K. (2013). The Golden Ratio in Architecture. [online] Available at: http://jwilson.coe.uga.edu/EMAT6680Fa06/Hobgood/Kate_files/Golden%20Ratio/GR%20Arch.html. Brownlee, J. (2015). The Golden Ratio: Design’s Biggest Myth. [online] Co.Design. Available at: https://www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth.
WHAT IS BEAUTY IN ARCHITECTURE? It is often contended that the highest honour a traditional building can receive is to be called beautiful. Those architects form a traditional perspective, and sometimes more modernist architects tend to support their view which, for example, generally includes reference to “well-proportioned”. Statements to support this often advance to ratios between whole numbers, geometric constructions that yield the golden section, diagrams which show parallel lines falling on parts of its composition, and other numerical and geometric propositions.6
Le Corbusier, Ronchamp chapel, 1954
It is often said that “beauty is in the eye of the beholder”, however there is a collection of architectural spaces that never fail to evoke a sense of emotion in the observer. Studies have shown that these “neuroaesthetic” responses in the brain can be demonstrated with consistency. Beauty has been defined as “the quality present in a thing or person that gives intense pleasure or deep satisfaction to the mind”.7 Beauty in architecture is perceived in a building’s visual qualities. The aesthetic is often linked with order and balance: rhythm or example is often desired within architectural compositions. While the aesthetic sense assesses the beautiful, the moral sense, assesses the good. In the design of a building, proportionality is required between a building’s use and its appearance: it enables the fittingness of a building in the context it inhabits, be that an urban or rural environment. Proportionality guards against falling into chaos, or disorder. Langhein (2010) contends that a knowledge, or theory of architectural aesthetics, proportions and other objective and systemic parameters of aesthetics is still in its infancy6 and that proportion is, for beauty, a necessity, though not a sufficient pre-condition. His view is that beauty in architecture complies with visual patterns, “wit and workmanship” but will tolerate some contrasts, “tensions and vagueness.” He further proposes that beauty in architecture should be relevant to all conditions and settings, relate to the senses and inspire both the cognitive and emotional aspects in “man.” In his view, the Golden Section, as one of a number of (geometrical) proportion systems, fulfills those functions and others which include clarity and unity of the whole bodies of buildings, balance
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Westfall, C.W. (2013). Beauty and Proportionality in Architecture. [online] Newlington-Cropsey Cultural Studies Center. Available at: http://nccsc.net/essays/beauty-and-proportionality-architecture. Archbeauty. (n.d.). What is Beauty in Architecture?. [online] Wordpress. Available at: https://archbeauty.wordpress.com/ [Accessed April-May 2017].
J. Douglas-Hill
University of Adelaide
between visual unity/order, and clarity and readability, important components of the formal functions of perception. In the quest for beauty, one of the most immutable traditions in Architecture and Design is the golden section. Modernist architect Le Corbusier used the relationship to create what he considered “humane” buildings – structures which better reflected the human form.
Frank Lloyd Wright, Guggenheim Museum, 1959
ADVANTAGES OF THE GOLDEN RATIO THAT ARCHITECTS BRING TO THEIR WORK
Le Corbusier’s Modulor scale
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A sense of history and heritage Bringing balance and height: generally, we gravitate towards buildings that appear balanced. By applying the principles of the golden rectangle in design, a building’s proportions can be altered according to a client’s needs, at the same time maintaining the ratio’s proportions. Varying Shapes: Architects need to accommodate an array of shapes in their design dictated by the natural landscape, existing site boundaries, and client preferences. The ratio can be easily amended with application to any shape, for example: - Golden Triangle: Connotes an isosceles triangle in which the smaller side is in golden ratio with its adjacent side - Logarithmic Spiral: Can be created by taking an existing golden triangle and bisecting the angles to make another golden triangle and continuing indefinitely8 It makes buildings aesthetically pleasing: as well as form and function, the aesthetic of a building in its environment should guide design.
In landscape design, the Golden Ratio and its variants are often applied to garden design and landscaping, notably invoking the proportions in selection and layout of plantings and the use of the spiral. Historically, the Golden ratio “spiral” first observed in the Nautilus and the human inner ear, guided the “top down” approach to garden layout first in the Mogul empire. Contemporary adaptions of this approach are evident in the works of Paul Bangay and Edna Walling.9,10
Mastroeni, T. (2014). How Architects Take Advantage of The Golden Ratio. [online] Freshome. Available at: http://freshome.com/2014/10/29/how-architects-take-advantage-of-the-golden-ratio. Bangay, P. (2003) The Balanced Garden Town, Country and Courtyard. The Penguin Group. Australia. Dixon, T and Churchill, J (2004) The Vision of Edna Walling. Blooming’s Books Pty Ltd. Melbourne, Australia.
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As with any discipline, architecture has been informed by the architectural lessons of the past. The Golden ratio is among the few aesthetic traditions which has lasted more-or-less continuously throughout the history of western architecture. The sizing or relating windows or rooms according to the golden ratio is not uncommon. Its application to landscape architecture, buildings and the built environment is well documented. Architecture without rules or order defeats the notion of good and documented design - and this stipulates the visual, historical and mathematical values of architectural beauty and the golden section.
REFERENCES
Archbeauty. (n.d.). What is Beauty in Architecture?. [online] Available at: https://archbeauty.wordpress.com/ [Accessed April-May 2017]. Bangay, P. (2003) The Balanced Garden Town, Country and Courtyard. The Penguin Group. Australia. Brownlee, J. (2015). The Golden Ratio: Design’s Biggest Myth. [online] Co.Design. Available at: https://www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth. Dixon, T and Churchill, J (2004) The Vision of Edna Walling. Blooming’s Books Pty Ltd. Melbourne, Australia. En.wikipedia.org. (2017). List of works designed with the golden ratio. [online] Available at: https://en.wikipedia.org/wiki/List_of_works_designed_with_the_golden_ratio [Accessed AprilMay 2017]. Mastroeni, T. (2014). How Architects Take Advantage of The Golden Ratio. [online] Freshome. Available at: http://freshome.com/2014/10/29/how-architects-take-advantage-of-the-goldenratio. Mihai. (2012). 10 Great Architectural Lessons from Frank Lloyd Wright. [online] Freshome. Available at: http://freshome.com/2012/09/03/10-great-architectural-lessons-from-frank-lloydwright. Hague, M. (2014). In search of the golden ratio in architecture. [online] The Globe and Mail. Available at: http://www.theglobeandmail.com/life/home-and-garden/architecture/in-searchof-the-golden-ratio-in-architecture/article20040240. Hemenway, P. (2008). The Secret Code The mysterious formula that rules art, nature, and science. EVERGREEN GmbH, Cologne, Germany. Knott, R. (2010). Fibonacci Numbers and The Golden Section in Art, Architecture and Music. [online] University of Surrey. Available at: http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibInArt.html.
J. Douglas-Hill
Langhein, J. (2010). On the Path to a Proportion Science in Architecture and the Golden Section. 1st ed. Gmuend, Austria. Lehman, M. (2008-2017). Understanding Beauty in Architecture: Guiding Neuroaesthetics. [online] Available at: http://marialorenalehman.com/post/understanding-beauty-inarchitecture-guiding-neuroaesthetics [Accessed April-May 2017]. Meisner, G. (2013). Architectural design and the Golden Ratio using PhiMatrix software. [online] PhiMatrix. Available at: http://www.phimatrix.com/architectural-design-golden-ratio. Meisner, G. (2015). Golden Ratio Overview. [online] The Golden Number. Available at: https://www.goldennumber.net/golden-ratio. Meisner, G. (2016). Michelangelo and the Art of the Golden Ratio in Design and Composition. [online] The Golden Number. Available at: https://www.goldennumber.net/michelangelosistine-chapel-golden-ratio-art-design. Nathe, C. and Hobgood, K. (2013). The Golden Ratio in Architecture. [online] Available at: http://jwilson.coe.uga.edu/EMAT6680Fa06/Hobgood/Kate_files/Golden%20Ratio/GR%20Arch. html. Phil Kean Design Group. (2015). Contemporary Residential Architecture with Proportion and the Golden Ratio. [online] Phil Kean. Available at: http://philkeandesigns.com/blog/contemporaryresidential-architecture-golden-ratio [Accessed April-May 2017]. Scribd. (2017). The Golden Section and Architecture. [online] Available at: https://www.scribd.com/document/7242379/The-Golden-Section-and-Architecture. Westfall, C.W. (2013). Beauty and Proportionality in Architecture. [online] Newlington-Cropsey Cultural Studies Center. Available at: http://nccsc.net/essays/beauty-and-proportionalityarchitecture.
University of Adelaide