Proportion and Geometry Within Architecture

Geometry along with music, arithmetic and astronomy, made up the Quadrivium, the four quantitative areas of learning in the ancient Greek world and advocated in the middle ages as essential for the education of all human beings. Even though the educated person would usually have learnt to master a musical instrument, it was the mathematical and proportional aspect of music which was held to be of most relevance. One could say the importance of proportion and geometry within architecture and life is a phenomenon that can be seen within nature and also be applied to a range of disciplines.

The ancient Greek were one of the first to apply these concepts, the design of Greek temples was based upon strict rules of proportion ruling all parts of the structure. Roman architecture also shared many simple characteristics with the Greek including the portico, the use of orders and stepped podium however it tended to be more ornate and elaborate overall. The main ambition behind the ancient Greeks using the orders was to discover eternally valid rules of form and proportion, to erect buildings that had human scale and yet were suited to the divinity of their gods, and finally to create a classically ideal architecture. Doric temples were a mainly artificial construct rather than a form based on functionality. They were a bright presence of right angles and sharp geometries. It stood apart in the landscape and were monuments of vital abstraction.

The orders in Architecture are a specific assembly of parts subject uniform already established proportions. Throughout time during the ancient Greek period the orders went through major adjustments and changes as well as which ones were more in or out of fashion. The Hellenistic temple of Apollo at Didyma is a place where there was a great amount of change, especially with the Ionic order which had already developed a lot through the 3rd and 4th centuries. A rejuvenation of proportions made the ionic column taller, more slender and the spans wider which was influenced by the architects Hermogenes of Albanda and Pythios of Priene. In Didyma the exterior columns which have a height of 20 meters, being almost 10 times their lower diameter, are the tallest and slimmest of any Greek temple. However it was the decorative possibilities that made the Ionic order so popular not its proportions. The stricter and more fixed Doric order was less adaptable meaning it eventually lost its popularity after its proportions were diminished, in an attempt to make it more modernised. The contrast of the natural and the devised is at the heart of Greek religious Architecture. It heralds both the separateness of human achievement from the dark forces of the land and the appeasing of these divinely controlled forces through the act of building.

One of the earliest mathematical teachings in the form of a movement was Pythagoreanism during 6th century BC, for the Pythagoreans number was not only the first principle of heaven but it exhibited its power. Throughout Egyptian and Greek geometric methods of composition there was the idea that there was a divine relationship between numbers and human form. The Tetraktys was a triangular figure that was used as a geometrical representation of the fourth triangular number. It was very important to Pythagoreanism and influential with planetary motions. Pythagoras’s theorem states that in a right angled triangle, the sum of the squares of two sides is equal to the square of the hypotenuse. When an architect is designing a square building or structure they can split the square into two triangles and if they know the length of 2 sides the third side can easily be worked out. Using Pythagoras’s theorem can make the life of an architect easier when determining unknown lengths and shows how mathematics and geometry is important within the world of designing.

In terms of individuals Vitruvius is the man most commonly known for his discussions of perfect proportion in architecture as well as the human body. He stated 3 conditions of a good building to be its usefulness, strength and beauty. Proportion plays a part in all 3 as it provides guidelines for laying out useful spaces, designing structural systems, and creating an aesthetically pleasing environment. Through this we can learn how the aspect of ratios and distances in-between space create the basis for all structures. The proportions of a space can dramatically change how people feel in it, and proportions of a façade design can affect whether a building appears welcoming, imposing or simply neither.

The Golden ratio was used within ancient Greek architecture to determine aesthetically pleasing relationships between the width of a building and its height in terms of its dimensions. This would also include things such as the size of the portico and even the position of the columns supporting the main structure. The final result of this is a building that feels completely in proportion. The second temple of Hera in Paestum Italy is a good example of how the golden ratio is applied. A person famous for centring his design philosophy on systems of harmony and proportion as well as the golden ratio was the Swiss architect Le Corbusier. His trust in the mathematical order of the universe was closely linked to the Fibonacci series and the Golden ratio. He described them as 'rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned.”

The golden ratio was specifically used in his system for the scale of architectural proportion. The system was seen as a continuation of the principles of Vitruvius, the work of Leon Battista Alberti, and many others who used the proportions of the human body to improve the appearance and function of architecture. He took the concept of the golden ratio in human proportions to an extreme through sectioning his model human body's height in different subdivided sections along many body parts. According to these principles buildings should scale down to dimensions humans can relate to in order for them to be more appreciated, this theory that we can relate to things within our grasp is for the most part true as distant things usually seem less important us. It also provides an insight of our relationship with each other, and the universe as scale can greatly change how we perceive things. There was the belief that all buildings should be developed from a standard module meaning the dimensions should be proportional to said set module, this rigid structure would enable almost all buildings to have a sense of uniformity and make determining dimensions much easier for architects. For some this might be a limiting factor in terms of design restriction however it is widely accepted for the most part that proportion is required in buildings for them to look good.

In 15th century Florence during the Italian renaissance period, a change in attitude saw people become more open to new ideas. The current work had a relationship with the forms and rules of antiquity to the extent that they were reused and reinvented when necessary. The Mathematics perspective was at a point where it was necessary as a way of representing things in space with visual accuracy and therefore enabled an ideal level of proportion that generates beauty. Italy was heavily influential for the rest of Europe during that period. The main Renaissance principles were symmetry, proportional relationships, the balance of parts, all these things were the basis of the construction and contributed to the innate quality of the architecture, this is what was known to Alberti as “beauty”. Beauty and ornament are seen as the two main elements that bring about aesthetic pleasure, with the former being natural as it resides in proportions due to “the harmony and concord of all the parts in such a manner that nothing could be added or taken away or altered except for the worst”. The renewal of the arts saw a boom of Italian architects and designers referring to old Roman and Greek texts, inscriptions and artworks.

During this time it was aided by a form of mathematical humanism. An increase of research into optics and geometry propelled linear perspective as the main inspirational force at that period. Brunelleschi’s use of these studies brought about his substitution for gothic design, which at the time worked according to an abstract system of proportions. They were only loosely applied, individual elements of the building had no fixed ratios within themselves or with respect to the overall measurements. As well this a lot of details were improvised or changed on the spot during the building process. These Gothic habits were eventually thrown out. Raphael Bramante and Leonardo da Vinci carried on the work of Alberti and Brunelleschi and aimed to stabilise their research by moving towards an abstract ideal perfection. They preferred centrally composed buildings, uniform squares, city-forms of immaculate geometry and facades strictly balanced on either side. This shows how proportion and geometry have remained an essential staple throughout most design periods of architecture but their importance and use have always shifted slightly depending on the values or styles that were most influential at the time. “We shall therefore borrow all our rules for the finishing of our proportions, from the musicians who are the greatest masters of this sort numbers, and from those things wherein nature shows herself most excellent and compleat” –Leon Battista Alberti. Geometry and beauty in nature can be seen as subjective or in the eyes of the beholder but there are also scientific principles behind it in the form of proportions which can allude to the conclusion that nature is science.

The ancient Greeks developed astronomy as a branch of mathematics to a highly sophisticated level, the first geometrical three dimensional models were made to explain the apparent motion of the planets. However during the 17th century is where significant changes within the field of cosmology and astronomy took place. The discovery that planetary orbits were elliptical not circular by Johannes Kepler revolutionised astronomy at the time. Celestial physics that linked the shapes to God showed how significant geometry was in terms of creation and understanding the universe. Kepler quite often stressed his philosophical approach to the questions he faced. Expressing the concept that God manifests himself not only through the scriptures but also in the arrangement of the universe and in its compliance with our human intellect. Therefore, astronomy represents for Kepler, if done philosophically and correctly, the best path to God. Kepler re-established the traditional ideas about creation, giving to the ancient beliefs a more systematic and a quantitative character responsible for creation. Even the idea of the Trinity can be geometrically represented, using the centre for the Father, the spherical surface for the Son, and the in-between space for the Holy Spirit. In Kepler's model we’re able to reduce all appearances to straightness and curvature which form the foundation of the geometrical structure of the world's vast creations. The first system through which God envisioned himself into his creations through is quantity. In which quantity was also introduced to us intellectually for the purpose that this fundamental symmetry could be understood and known scientifically.

Inigo Jones who was an Architect during the Stuart and Baroque period introduced a design style not just based upon geometry and shapes but also the theories behind Architecture which were pivotal throughout ancient time. He was the first English architect of great significance to display rules from Vitruvius of proportion and symmetry in his buildings, as well as introduce architecture from the Italian renaissance into Britain. This influence along with Palladianism created buildings which were uniform and beautifully proportioned.

Villa Farnese at Caprarola was a massive renaissance and mannerist construction which was one of the finest pieces of architecture during that period. Ornament decoration was used scarcely on the building to achieve proportion and harmony which makes it aesthetically pleasing. Therefore while the villa dominates the surroundings, its extreme design still compliments the site. Mannerism style used during that time was developed as a reaction to the earlier ornate high renaissance designs of 20 years prior. The plans were conceived for a pentagon constructed around a circular collimated courtyard. Another fine example of architecture during that time was Dome of St. Carlo, Rome by Francesco Borromini. It has a circular shape with a centre of light opening to represent God and the applied distortion makes it seem larger and taller than it actually is. Chiswick house inspired by Jones takes the Palladian/Italian idea of sequencing rooms where different geometrically shaped rooms are all connected together to provide a range of experiences from different perspectives. Palladian windows were a key feature of the design style and easily recognisable. They were a large, three-section window where the centre section is arched and larger than the two side sections. This symmetry within them is what makes them distinct and added more order within the buildings themselves.

Venetian Carlo Lodoli (1690-1761) was an advocate of a radical approach back to how Architecture was prior, his extreme stand was that architecture should be thought of as a science and not a field of artistic imagination governed by rules of beauty. He disagreed with the fixed proportions of the orders. Design was not dependent on ideal models but on (1) function, by which he understood how the structure of a building performs in accordance with the use of the building, and (2) representation, “the individual and total expression which results from the way the material is disposed according to geometrical-arithmetic-optical rules to achieve the purpose intended”. Ornament could be added once the main functional requirements were met first but to Lodoli ornament was not just a decorative topping it had to be based upon scientific laws that govern the essential building materials. Whereas Laugier on a similar note didn’t completely rule out the orders but urged that they behave as working parts and not applied as decoration. Mainly meaning that the construction and decoration of buildings such as columns, pediments and entablatures must do the actual work of holding up the buildings and not just seem to do so. This shows how serious these principles were taken and the systematic approach in which they were enforced in both of their works.

The field of anthropometrics is the measurement of human individuals which can play a big role in ergonomics and architecture. Humans engage with their environments based on their physical dimensions, limits and capabilities. Buildings scaled to human physical capabilities have steps, doorways, railings, work surfaces, seating, shelves, fixtures, walking distances, and other features that fit well to the average person or 90th percentile. We as humans also interact with our environments based on our senses. The fields of human perception systems aren’t quite exact sciences because the way humans process information is not a purely physical act, and because perception is affected by cultural factors, personal preferences, experiences, and expectations. So human scale in architecture can also describe buildings with sightlines, acoustic properties, task lighting, ambient lighting, and spatial grammar that fit well with human senses. However the main thing to consider is that human perceptions are always going to be less predictable and less measurable than physical dimensions which is only natural.

For a greater effect certain Buildings, statues, and memorials are constructed in a scale larger than life as a social and cultural statement that the subject matter is also larger than life. One extreme example is the Rodina (Motherland) statue in Volgograd standing at 85 metres tall. This is essentially opposite to the beliefs of Vitruvius and Corbusier who believed in smaller proportions we can relate to, but this disparity shows how proportion and scale can mainly be perceptive, depending on what you’re trying to achieve

We as humans often associate proportion, numbers and geometry purely with mathematics but they are proven to be more than that due to how instrumental they are in nature, making calculations, music, and the human form. It sometimes makes it easier to see proportion, geometry, and number as scientific laws which govern how buildings should be designed, they can’t be altered and are simply forces in which we can use to interpret certain aspects of life. When there are clear structures and rules to follow it can make creative processes easier and provide a basis on to which to start. They are principles that run through us humans in the form of our physical bodies, symmetry and shapes within buildings and geometry within cosmology and creation itself. Throughout ancient works there was always said to be a divine relationship between numbers and the human. This spiritual and to a certain extent even religious connection makes them appear pure in a beautiful way. Natural beauty lies in its proportions and there are ratios and in essentially everything we physically perceive. One could say the importance of proportion and geometry within architecture and life is a phenomenon that can be seen within nature and also be applied to a range of disciplines. The reinvention and recycling of ideas is a common theme within Architecture as principles from early Architects such as Vitruvius and Palladio have been used for inspiration and the basis of many design movements throughout time, further showing how significant they are within the world of Architecture and that they always have to be considered.