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The GR is a mathematical relationship that exists in art, shape, nature and patterns and is also popularly known as Golden Section, Golden Mean, Divine Proportion, Golden Proportion or even the Golden Number. It exists in architecture, art, music, design and even fashion. Since Renaissance, many artists and architects have proportioned their works to GR, 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. 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.
Vitruvius discussed proportions where it can be expressed in whole numbers, as opposed to irrational proportions. Le Corbusier is said to have contributed to many modern international style architecture, centering on harmony and proportion. Leonardo da Vinci's illustrated yet another divine proportion in the infamous painting of Mona Lisa.
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 which uses Nature as our basis for design, a building or design must grow, as Nature grows, from the inside out.
My stand is that GR is important however it is not everything. Beside the proportion of the building, functionality is important also. A creative design through creative intuition will make the project outstanding.
The Golden Ratio
The GR is a mathematical relationship that exists in art, shape, nature and patterns. This ratio is aesthetically pleasing and beneficial to the objects that possess them. The GR is represented by the Greek letter phi ( ) states the division of a line segment into two, creates a ratio of the shorter part to the longer equal to that of the longer to the whole. It works out to approximately 1.61803 and is derived from the Fibonacci sequence.
The GR is also popularly known as Golden Section, Golden hat Mean, Divine Proportion, Golden Proportion or even the Golden Number.
1.2.1 Renaissance Period
The GR 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.
When a GR is subdivided into smaller golden rectangles, a pattern is obtained. From this, a spiral can be drawn which grows logarithmically, where the radius of the spiral, at any given point, is the length of the corresponding square to a golden rectangle. The golden rectangles are whose side lengths are in the GR.
Since Renaissance, many artists and architects have proportioned their works to GR, 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 GR has inspired thinkers of all disciplines like no other numbers in the history of mathematics.
Presence of Golden Ratio
The first and most influential to illustrate this is the project De Divina Proportione by Luca Pacioli, a mathematician with a keen interest in art. He explored the theory of GR, yielding pleasing, harmonious proportions, until Livio points out that the interpretation has been traced to an error in 1799, and Pacioli used the Vitruvian system of rational proportions. With Leonardo Da Vinci, De Divina Proportione was a major influence in subsequent generations.
The Parthenon's facade is a clear example of proportioned GR and design, with it being circumscribed by golden rectangles, or could it be just pure sense of good proportion by the architects at that time? Unlikely I feel, as it is seen from the pictures, the measurements and the superimpose golden rectangles, these choices are so well made that there must be some work of the mathematical calculations to derive such proportioned structure of a building.
Some scholars however denied that the Greeks had any aesthetic association with GR. They feel that it was not until Euclid that mathematical properties were studied. Before Elements (308BC) the Greek merely regarded the number merely as an interesting irrational numbers, with regular pentagons and decagons and dodecahedron (a regular polyhedron) and regular pentagons. But one thing for sure, it was the Euclid where it is showed how to the calculate the value. Vitruvius discussed proportions where it can be expressed in whole numbers, as opposed to irrational proportions.
Are modern designers concern with the issue of GR to architectural design? Whether they still apply GR?
This sequence is named after the Italian mathematician who lived during the 12th century. It occurs in nature, modeling the population growth in rabbits, and also the development of the spiral in a snail's shell. Fibonacci numbers can also be seen in the arrangement of seeds on flower heads. the ratios when you take a Fibonacci number divided by the previous Fibonacci number, and make a list:
1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ...
Da Vinci and Le Corbusier
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, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to GR, Le Corbusier based the system on human measurements, Fibonacci numbers and the double unit. He took Leonardo's suggestion of GR in human proportions to an extreme, he sectioned his model human body's height at the navel with the two sections in GR, then subdivided those sections in GR at the knees and throat; he used these GR 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.
Recently, fractal dimensions have been calculated for Frank Lloyd Wright's and Le Corbusier's buildings, using the method of increasingly smaller rectangular grids. The results show that (at least some of) Frank Lloyd Wright's buildings display a self-similar characteristic over a wide range of scales, from a distant view to finger-tip size detail, so those buildings are intrinsically fractal. In this, Wright was following the brilliant example of his teacher, Louis Sullivan. By contrast, Le Corbusier's architecture displays a self-similar characteristic over only two or three of the largest scales; namely, those corresponding to a distant view. Up close, 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.)
Leonardo da Vinci's illustrated yet another divine proportion in the infamous painting of Mona Lisa. Other equally well known GR painting is The Sacrament of the Last Supper by Salvador Dali.
The GR is 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 including 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 this, he saw GR as 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.
GR also affect the clock cycle of brain waves according to Volkmar Weiss and Harold Weiss known as psychometric data.
In 2010, the journal science reported GR is present at the atomic scale in magnetic resonance of spins and cobalt niobate crystals. I am also almost certain that the MRI diagnostic machine also operates using GR in this case.[
2.1 Relevance in Present Times
2.1.1 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 required to have a mathematical background according to the article Architecture, Patterns and Mathematics by Nikos Salingaros . On the contrary, it may be promoting an anti-mathematical mind set. 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. They have however seen an increase in technological advances rather especially in the area of environmental factors.
2.1.2 Promotion of Anti-Mathematical Mindset
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 repeated in time and space, thus akin to visual patterns transposed into other dimensions.
2.2 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 architect, Frank Lloyd Wright (1867-1959), though never well articulated 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.
2.2.1 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 an expression of individuality, it also explores our need to connect to Nature.
Using Nature as our basis for design, a building or design must grow, as Nature grows, from the inside out. Most architects design their buildings as 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.
2.2.2 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.
2.3 How it Affects Our Lives
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 GR, how we perceive an object and whether we find it pleasing all goes back to the GR. 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 GR.
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
3.1 Republic Poly Technology by maki
The Library is divided up into 6 sections and each section is named in Greek as an extension to the Republic's naming convention. For example, Tekhnë (Greek= art, craft i.e. Technology) spans the Information and Communications Technology. Biôtikos (Greek = of or pertaining to life i.e. Applied Sciences) with its focus on Applied Sciences. Mechanikos (Greek = resourceful, inventive i.e. Engineering) that concentrates on Engineering. Or even Sunthesis (Greek = putting together, composition, combination i.e. Synergy) - where all disciplines find common ground and are challenged to synergise!
The 'campus within the community' is also planned into the existing rustic Regional Park contours. It integrates well with the surrounding greenery, thus promoting a 'campus in the park' feel. A meandering jogging track extends from the regional park into the campus, establishing a soft yet defined boundary between the campus and the park. The design therefore complements and creates synergies between the Park and the campus, thus further enhancing the learning experience at the polytechnic.
Of course the importance of the GR will have its detractors, among whom Fischler in 1981, Markowsky in 1992, Steinbach in 1997, and Fowler in 1982 are the more popular ones. They argue that the claim that the golden ratio has a special place among refuted that this ratio is somehow exceptionally pleasing and that it occurs frequently in art and architecture.
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.