What is the most satisfying proportion in today design? The Greeks thought they knew. Their temples were designed according to certain rules relating to “the golden section”. (Which is what we, layman, know as the Divine Proportion, the Golden Proportion, the Golden Number or even the Golden hat Mean.) In the 13th century, Fibonnaci, an Italian mathematician, put it all down on paper. He said, “‘the golden section’ or perfect proportion was 0.618034 to 1 (about 5 to 8).” The Parthenon (a temple in the Athenian Acropolis that the Greeks built, dedicated to the Greek goddess Athena) fits into Fibonnaci’s Golden rectangle. Incidentally, so do the pyramids at Giza. Does this make the Golden proportion a necessary rule to follow in design?
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In the 16th century, Leonardo Da Vinci wrote a book on geometric recreations called Divine Proportion. In 1948 Le Cobusier also wrote a book on mathematical proportioning. Others who have benefited this ratio are biologists, artists, psychologists and even mystics have pondered and debated on the basis of ubiquity and appeal. It is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other numbers in the history of mathematics.
Throughout the generations, many architects have also searched for the golden rule of design, thinking that it is that of the Golden Ratio. However, their search is far from over. This is because mathematics alone will not tell you what the most eye-pleasing proportion for a building’s structure is. Proportion must be generically correct and determined by the nature of the material. In other words, it is one thing for stone, another for concrete, and something else for steel. This, we would discuss further in another segment. Present technology has also given architects and engineers unlimited range to compose new forms of design and exciting spaces.
My stand is that the Golden Ratio is an important aspect in designing a building but it is not the most crucial. Besides having proportion in a building, functionality is also important. A creative design through the creative intuition of a designer will make the building outstanding.
The Golden Ratio is related to many things in the world today, not only during the times of Renaissance, LeCobusier and Alberti. It exists in architecture, art, music, design and even fashion.
Since Renaissance, many artists and architects have proportioned their works to the Golden Ratio, especially in the form of golden rectangle, in which the ratio of the longer side to the shorter in the GR, causing this proportion to be aesthetically pleasing. Mathematicians have studied this because of its unique and interesting properties applying it to geometry.
Since then, it has opened up doors for me how I view design and architecture and how it balances harmony to architecture design in this modern world.
Others who have benefited this ratio are biologists, artists, psychologists and even mystics have pondered and debated on the basis of ubiquity and appeal. It is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other numbers in the history of mathematics.
Presence of Golden Ratio
Contribution of the Golden Ratio in architectural designs
Le Corbusier is said to have contributed to many modern international style architecture, centering on harmony and proportion. Its faith in the mathematical order was closely bound by the GR and the Fibonacci series. He uses the GR in his modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, and others who used the proportions of the human body, to improve the appearance and function of architecture.
In addition to Golden Ratio, Le Corbusier based the system on human measurements, Fibonacci numbers and the double unit. He took Leonardo’s suggestion of the Great Ratio in human proportions to an extreme, he sectioned his model human body’s height at the navel with the two sections in the Golden Ratio, then subdivided those sections in Golden Ratio at the knees and throat; he used these Golden Ratio proportions in the Modulor system. The Villa Stein in Garches exemplified the Modular system. The Villa’s rectangular ground, elevation and inner structure closely approximate golden rectangles.
Fractal Dimensions in modern architecture
Recently, fractal dimensions have been calculated to be used frequently for Frank Lloyd Wright’s and Le Corbusier’s buildings. It can be found that both architects use the method of increasingly smaller rectangular grids. Frank Lloyd Wright’s buildings display a self-similar characteristic over a wide range of scales (far and spaced versus micro small sizes), so those buildings are intrinsically fractal. However for this specific project, Wright was following the brilliant example of his teacher, Louis Sullivan.
By contrast, Le Corbusier’s architecture displays a characteristic over only two or three of the largest scales. In more detail, Le Corbusier’s architecture is flat and straight, and therefore has no fractal qualities.
“A fractal dimension between one and two characterizes a design that has an infinite number of self-similar levels of scale, whereas the fractal dimension of Le Corbusier’s buildings immediately drops to one.” (Bovill, 1996. Salingaros, 1999.)
The Golden Ratio as seen in painting
Leonardo da Vinci’s illustrated yet another divine proportion in the infamous painting of Mona Lisa. Other equally well known painting which has made use of the Golden Ratio is The Sacrament of the Last Supper by Salvador Dali.
The Golden Ratio as seen in our natural world
The Golden Ratio is expressed in the arrangement of branches along the stems of plants and of veins in leaves and even to the skeletons of animals including their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, to the use of proportion in artistic endeavours.
From this, the Golden Ratio has become a universal law in strive to create completeness and beauty, with both nature and art, in structure, forms and proportions, organic and inorganic, in the human form.
According to Volkmar Weiss and Harold Weiss the Golden Ratio also affects the clock cycle of brain waves, known as psychometric data.
Relevance in Present Times
Modernising the Traditional Intimate Relationship Between Architecture and Mathematics
The traditional intimate relationship between architecture and mathematics has changed in the 20th century.
“Architecture students no longer need to have a mathematical background” according to the article Architecture, Patterns and Mathematics by Nikos Salingaros.
It may be promoting an anti-mathematical mindset. Mathematics is a science of patterns, the presence or absence of patterns in our surroundings influences how easily one grasp the concepts that rely on patterns. However, it has been seen that an increase in technological advances, rather especially in the area of environmental factors, has made mathematics almost redundant in architecture.
Environmental psychologists know that our surroundings influence the way we think, so if we are raised in an anti-mathematical environment, then we would deem to subscribe more human qualities. This is not an argument about preferences or styles, it concerns more about a trained functionality of the human mind!
An example to illustrate the meaning of functionality in the human mind is made by Christopher Alexander where:
“the need for lights from two sides of a room; a well-defined entrance; interaction of footpaths and car roads; hierarchy of privacy in different rooms of a house and etc. It speaks about specific building types, about building blocks that can be combined in an infinite number of ways.”
This implies a more mathematical and combinatoric approach to design in general. Alexandrine patterns represent solutions which repeat itself in time and space, thus relating to visual patterns transforming into other dimensions.
A new concept: Organic Architecture
In recent years, there has been a shift in architecture looking away from GR to other ways in which design can still have a sense of proportion by looking at nature for inspiration; the term given is Organic Architecture.
The term organic architecture was coined by the famous modern architect, Frank Lloyd Wright (1867-1959), though never well expressed by his cryptic style of writing:
“So here I stand before you preaching organic architecture: declaring organic architecture to be the modern ideal and the teaching so much needed if we are to see the whole of life, and to now serve the whole of life, holding no traditions essential to the great TRADITION. Nor cherishing any preconceived form fixing upon us either past, present or future, but instead exalting the simple laws of common sense or of super-sense if you prefer determining form by way of the nature of materials…” – Frank Lloyd Wright, written in 1939.
Rules of Organic Architecture
Architect and planner David Pearson proposed a list of rules towards the design of organic architecture. These rules are known as the Gaia Charter for organic architecture and design. It reads:
Let the Design:
be inspired by nature and be sustainable, healthy, conserving, and diverse.
unfold, like an organism, from the seed within.
exist in the “continuous present” and “begin again and again”.
follow the flows and be flexible and adaptable.
satisfy social, physical, and spiritual needs.
“grow out of the site” and be unique.
celebrate the spirit of youth, play and surprise.
express the rhythm of music and the power of dance
While Organic Architecture does describe some form of individuality, it also expresses our need to connect the designs, we create, to Nature.
Using Nature as a fundamental for design, from there a building or design must grow, as Nature grows, from the inside out. Many architects design their buildings as that similar to a shell and force their way inside. Nature grows from the idea of a seed and reaches out to its surroundings. A building thus, is akin to an organism and mirrors the beauty and complexity of Nature.
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Where the Golden Ratio Fits In
However, in the research that I have done on this topic, many of the historic scholars who devoted their entire lives to studying the GR has always studied nature for inspiration and they derived the GR from nature itself. Modern architects who claim to move away from the GR as it is too conformist and look towards nature for their inspiration for proportion instead still end up following the GR as it was from studying nature that led to the discovery of GR. Hence the continuing relevance of GR in today’s architecture.
How the Golden Ratio is evident in our everyday lives
The Golden Ratio seen in Music
Rhythm is everywhere in nature, at every scale from cosmic phenomena to the oscillations of atoms. Our every cell has its own clock, governing its own repetitive rhythms. Time itself, once measured by the motion of earth, sun and stars, is now defined, less memorably, as 9,192,631,770 oscillations of a single atom of an obscure metal. At the scale of the biosphere, the fidelity of replication in the genetic system is such that no more than about 200 errors are made in copying the 300 million bases strung into the chromosomes that hoard the design of our bodies. Without those errors, however, there could be no change and so no evolution.
With this is mind, we shall now look at how rhythm ties in with the GR.
Much of the rhythm and movement and design of our bodies and normal everyday life experiences all tie in with the Golden Ratio, how we perceive an object and whether we find it pleasing all goes back to the Golden Ratio. Because it is the one of the universal constants that allow for the interactions between all things on earth, it continues to hold relevance in our lives, regardless of the advancements in technology, which in fact is actually discovering more and more how life and design is so intimately associated with the Golden Ratio.
Architectural evidence of the Golden Ratio
Take a look at modern architecture and you will soon realize that the last decades have produced an increasing number of buildings with exotic shapes. Of course, also in earlier times the design of buildings has been influenced by mathematical ideas regarding, for instance, symmetry. Both historical and modern developments show that mathematics can play an important role, ranging from appropriate descriptions of designs to guiding the designer’s intuition.
C Case study
Case Study One: Republic Poly Technology of Singapore by Fumihiko Maki
Fumihiko Maki designed the new campus attempting to preserve the green qualities and the topography of the original site introducing landscape elements that contrast with the natural widerness and strengthen the sense of place based on Golden Ratio.
Case Study Two: Palladio’s Villa Rotunda.
The Villa Rotonda is symmetrical on all axes, including diagonals. Any architect will tell
you this is hard to do, much less sell to a client; even Palladio only did it once, probably just to see if he could. Palladio based his design on simple progressions in the Fibonacci series leading to the Golden Mean. This is also hard to do.
Case Study Three: Taj Mahai
In India, the Golden Mean was used in the construction of the Taj Mahal, which was completed in 1648. http://archgeom.blogspot.com/2010/03/golden-section-in-taj-mahal.html
Case Study Four: CN Tower in Toronto
The CN Tower in Toronto, the tallest tower and freestanding structure in the world, has contains the golden ratio in its design. The ratio of observation deck at 342 meters to the total height of 553.33 is 0.618 or phi, the reciprocal of Phi!
Case Study Five: California Polytechnic State University
The College of Engineering at the have plans for a new Engineering Plaza based on the Fibonacci numbers.
4.2. What I have perceived until this moment
In my analysis, GR forms the basis of understanding of architecture, however it is not the entirety. Because form follow function, function plays an important part of the architectural design because without understanding the functionally of form, it is not possible to develop a building of good use, for example a good architect must be able to understand the utility of function.
For example,the architect must know how many rooms a house needs, whether a swimming pool is required or a badminton court needed. After a form is selected and function must go beyond the concerns of biotechnical materialism.
The creative architects must go beyond utility & technical knowledge to an awareness of experiential associations and symbolic meanings that lies behind the visible form. Beauty in design is not guaranteed when all of the above is satisfied. Some intuition is required by the architect and an outstanding design depends also in skill and intuition with functionality.
Therefore, the great architect of age and every culture, the basis of which is mathematical.
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