We, as humans, have always been fascinated by the unknown and seek to explain it with
reason. We long for order and resolution, and we seem to inherently find beauty in this
concept, especially when searching for answers regarding nature. Our desire to find
structure, in the human body in particular, has led to the discovery of da Vinci’s Vitruvian
Man, Le Corbusier’s modulor system and countless other geometrical/mathematical
systems for proportion that have inspired and defined much of history’s Western
architecture. Vitruvius’ concept of an “ordered self” in his treatise De architectura defined
the ‘design’ of the human body as structured and complete and thus, beautiful (Luscombe,
2021). Subsequently, Vitruvius proposed that architecture should mirror the human body’s
‘organisation’ in order to achieve this same beauty and eloquence (Luscombe, 2021). This
Classical idea was adapted in the Renaissance period and continues to be a point of
influence for contemporary architecture today (Luscombe, 2021). The concept of order in
architecture is ever-prevalent throughout Western architectural history and is underpinned
by the use of geometries, mathematics, proportions and ratios – all comprised under the
term ‘orientation’ (Luscombe, 2021).

Da Vinci’s Vitruvian Man sort to highlight the natural, apparently inherent proportions
within the human body in relation to the circle and the square (considered the ‘most
perfect’ shapes in the Renaissance) (Luscombe, 2021). His drawing depicts the human
body perfectly fitting within a square and a circle with the navel (belly button) being the
centre of both. This idea came from Vitruvius’ belief the navel was the centre of the human
body. Da Vinci’s Vitruvian Man brought to light an apparent universal system of
proportions in the human body. This relationship between geometry and man became
idealised as its ‘perfect’ ratios produced harmony. Architecture, especially during the
Renaissance, sought to mirror this sense of structure and proportion of geometry as a way
to create beauty.

Similar to the Vitruvian Man, Le Corbusier’s Modular system became a key focal point for
architectural orientation – also being based on the proportions of the human body. However, both Le Corbusier’s and da Vinci’s proportional systems centre around a very
generalised human body and does not account for the “subtle variations” – as Theodore
Cook mentions in The Curves of Life) – “…which are essential to life, and, as I think, to
beauty.” (T.A. Cook as cited in Padovan, 1999, pp. 323-332). Architecture of the past was
defined by ‘order’ as a result of the “ordered self”, yet we are inherently not perfect as we
are all different. The proportional systems idealised in historical architecture seem somewhat flawed in my perspective as a result of this need for the mathematical study to
be anthropocentric and thus based on the form/shape of the human body. The belief that
we, as humans, are the centre of the universe and thus can define the space around us
seems to be a very speculative statement. My own research surrounding the Vitruvian Man (using my own body proportions) expressed how the assumption of man fitting perfectly within a circle and square is highly idealised (see report diary). It becomes almost ironic that beauty in architecture came from apparent beauty in the body (in being perfect). If we are not, does this then mean all of history’s architecture suddenly becomes not beautiful? Does harmony and eloquence of geometry/proportion necessarily have to be linked to mankind? With these questions in mind, I will be using the analysis of the Acropolis (Parthenon and surrounding buildings) and the Campidoglio (piazza and surrounding buildings) to explore how geometry and proportion has underpinned architecture throughout history and whether man-based proportionality holds any significance in a modern day perspective.

When looking at the Acropolis, Athens (400 BCE) and its surrounding buildings, I will be
looking at the plan of the Acropolis III (after 450 B.C.) as it has the most extensive and
relevant geometries to that time period and is the most similar to the modern day remains
of the site.

This site was derived from what Aristotle describes as a “traditional system”, in which
“they give the impression of having no comprehensive plan.” (Doxiadis, 1972, pp. 20-38)
This system was a reflection of their culture and philosophical intellect – just as the Greek’s brought order by answering questions of the universe, the layout of buildings was brought to order by this system (Doxiadis, 1972, pp. 20-38). The twelve-part system in particular was utilised for the Acropolis and can be seen in the radiating 30° angles from the entrance point in the propylaea (Point A on Model 1) (Doxiadis, 1972, pp. 20-38). This system was used to symbolise the divine – emphasising this site as one built for ‘the gods’ (specifically Athena) (Fleming, 2014).

The use of the propylaea (entrance) as a means of spatial definition was a result of orientation and proportion based on the human body. Doxiadis states that the ancient Greeks believed “man was “the measure of all things.””, so the spatial planning of the Acropolis was all based on the sight lines of the viewer (Doxiadis, 1972, pp. 20-38). When standing at Point A in the propylaea, one is in the optimal position to view all of the Acropolis’ buildings (Fleming, 2014). Following Doxaidis’ study of geometry in the Acropolis, each 30° angle line radiating from Point A hits all of the main buildings’ corners to give the viewer the best possible view of their façades. For example, the 30° sector, angle FAH, encompasses three corners of the Parthenon, thus allowing the viewer to see “the most” of the building itself from a single position. Furthermore, two of the 30° sectors form a equilateral triangle (ΔAMN) that defines most of the negative space throughout the Acropolis. The midpoint of this triangle also intersects with the radiating Line AF that hits the edge of the Parthenon. A smaller equilateral triangle (ΔLOG) can also be found when drawing lines from the midpoint Line AK towards Point R (Erechthieon) and Point G (Parthenon). The equilateral triangle is particularly symbolic as it is associated with Athena and often used in ancient Greek architecture in partnership with the twelve-part system (Fleming, 2014; Doxiadis, 1972). The Pythagorean’s also saw this triangle as the basis of the universe (Doxiadis, 1972, pp. 20-38).

Moreover, Point L becomes a point of interest as radiating 30° sector lines intersect the main buildings’ corners and produces another ‘optimal viewing position’ for the audience (Doxiadis, 1972, pp. 20-38). Doxiadis discusses how “relations between buildings had to be as simple as possible so that there would be the fewest possible lines in man’s angle of vision“, highlighting how proportion, scale and spatiality was determined by the body – linking this geometrical study to that of human-based proportional systems (Doxiadis, 1972, pp. 20-38). There has been a careful consideration of the Acropolis’ site plan to not overlap sight lines throughout the composition so each building is distinct, but also can be seen as a unified space (Doxiadis, 1972, pp. 20-38). This specific planning type could be considered a form of “architectural scenography, the mathematical organisation of the perceived environment in the manner of larger-scale perspective scene-painting” (Pont, 2015). This concept of relationality between buildings and its overall environment as a way to unify and create harmony is continued into the Renaissance as well.

To better understand the use of geometry and proportion in ancient Greece I will also look
at the Parthenon in both plan and elevation. This building utilises many different
proportions and ratios within its design and it is greatly debated as to whether this was an
intentional or unintentional decision since there were relatively no architectural plans at the time. As a result of this ambiguity, various sources have attempted to ‘find’ its geometries and proportions. “No building has been subjected to so many and such conflicting interpretations of its proportions as the Parthenon,” says architect Richard Padovan (Padovan, 1999, pp. 75-95).

What I have found to be consistent throughout my research was the Parthenon’s plan being at a ratio of √5 : 1 (Fleming, 2014). Irrational numbers fascinated the ancient Greeks as it could never be accurately represented and thus created a limit to their own knowledge and understanding (Fleming, 2014). This ratio was considered a ‘divine proportion’ due to its seeming impossibility and thus creates symbolic continuity with the overall Acropolis site plan (Luscombe, 2021). Furthermore, there is also a consistent appearance of the golden ratio and rectangle. The golden ratio was said to be a formula that allowed one to ‘calculate’ a shape’s “natural area of greatest beauty” (Fleming, 2014), so by using it within the Parthenon the building would thus be considered aesthetically pleasing to the eye. Most of the internal columns and structures are consistent with the golden ratio and the inner area formed from the interior walls fits a golden rectangle (see Model 2) (Meisner, 2020). The heavy use of a particular geometry leads me to believe in its intentional inclusion within the Parthenon’s design and this is further demonstrated in the continued use of the golden ratio and rectangle in the elevation of the Parthenon.

The Parthenon’s elevation can seemingly be constructed from geometry alone – starting with the creation of a golden rectangle that spans the 8 columns in width and goes from ground to rooftop in length (see Model 3) (Taoumi, 2019). The placement of the columns can also be generated from overlapping golden rectangles (Taoumi, 2019). The use of 8 columns on the façade (octastyle) broke the traditional 6 column temple design, possibly due to their desire to express their ‘revolutionary society’ and appeal to the symbolism of the divine (that of which is above them) (Luscombe, 2021; Fleming, 2014). Not only the columns, but the stylobate below and the entablature above can be derived from intersecting diagonal lines within the golden rectangle (Taoumi, 2019). For example, by drawing a circle within the square of the golden rectangle and intersecting it with Line AB, one can find the stylobate (seed Model 3). Further symmetry (aside from the middle axis CD) can be seen in the equal area of the pediment to the entablature, further harmonising the space as ordered and thus beautiful (Taoumi, 2019).

Looking now at the Campidoglio, Rome (1538 – present day), I will be analysing its architecture as it is built today. This site has received many revisions since Michelangelo first designed the piazza, but its main geometries and proportions still remain. In many ways one can draw similarities between this piazza and the Acropolis in characteristics and also architectural design choices. Both sites are placed on higher ground so the viewer cannot see the site until they have reached a certain point (the entrance). This forces the audience to having only one main entry point into the space and thus influences their initial lines of sight. Like the Acropolis, the Campidoglio utilises the notion of spatial planning – specifically axial planning – in order to move the viewer through the piazza. The unique trapezoidal shape of the Campidoglio’s piazza is due to fact Michelangelo had to work around the pre-existing buildings, one of which sat at an 80˚ angle (previous Piazza dei Conservatori) (ArcheoRoma, 2021). Working with the circle and square was no longer possible due to the awkward angle and thus Michelangelo sort to modify the piazza and its buildings to create a cohesive space. By creating a new building (Palazzo Nuovo) that mirrored the pre-exisiting Palazzo dei Conservatori he was able to create a geometric shape that would unite the piazza and allow for symmetry about the main building, the Palazzo Senatorio (see Model 4) (ArcheoRoma, 2021). Much like the organisation of the Acropolis having a centre relationality, the palazzos of the Campidolgio work in conjunction with each other and the piazza in order to function as a whole. The trapezoid allows for a sense of convergence and divergence from its directional lines along the façade and emphasises the dynamic nature of the site (Ackerman, 1991). This is contrasted with the organic elliptical design of the pavement – its inner detailing inspired from cosmological designs of the times. Its overlapping, organic and geometric lines could be likened to a circular rosette pattern. “The oval combined in one form the principles of centrality and axiality”, as stated by James S. Ackerman (Ackerman, 1991). Like the trapezoid, the ellipse ‘solved’ the problem perfect shapes such as the circle and square could not and this started the Baroque’s movement away from Classical ideas of perfect geometry and proportions (Padovan, 1999). “Whether the proportions of elliptical plans are mathematically determined or not, it is virtually impossible to grasp them exactly, and one of the effects of the adoption of elliptical geometry is to make the use of precise ratios an irrelevance,” says Padovan (Padovan, 1999). This clear move towards less obvious proportionality can be seen the modified façades of the Palazzo Senatorio and the Palazzo dei Conservatori.

The Palazzo Senatorio can be divided by the central axis of symmetry AB (see Model 5). Within the raised pilaster sections a number of geometries can be found that are not immediately apparent. A golden rectangle and golden ratio aligns with the outermost pilaster section, cutting the bottom window’s rounded tympanum. The overall rectangular ratio of this section when measuring from the centre of the pilaster is also the irrational ratio √2 : 1. Furthermore, the inner sections (when measured from the pilasters centre) can be defined by the ratio √3 : 1 (see Model 5). The √5 rectangle that encompasses most of the façade (excluding the roof detail and its tower) can also define the general triangular shape of the building’s large staircase through diagonally intersecting lines. The lower section of the building comprising this staircase also has the ratio phi : 1. Finally, linking this building to the Palazzo dei Conservatori’s façade, one can also derive the inner section tympanum from a square whose edges touch the outside pilaster edge on either side. All these geometries are not obvious at initial study as a result of the prominence at the time of architecture that kept its ratios and geometries hidden by symmetry, regular façades with uniform horizontal and vertical lines (Padovan, 1999). Having such uniform, yet unique façade designs allowed for each building to stand alone but work together by “blurring the articulation of parts within the whole so that they appeared to melt into each other” (Padovan, 1999).

The Palazzo dei Conservatori is a key example of this push for extreme continuity as its façade is particularly regular and uniform. With a central axis of symmetry (Line AB), the elevation is able to create a sense of rhythm from its repetition in geometric design. The open porticos on the bottom align to an exact square that can be mirrored above with a rectangle of ratio √2 : 1 in between (see Model 6). The façade can also be divided into a tripartite from top to bottom, isolating each row of windows/openings in a separate section. When sectioning the building by pilasters there is a ratio of √6 : 1 apparent (measuring from the inside of the pilaster) (see Model 6). Much like the Palazzo Senatorio’s elevation, this building has a clear use of geometries and proportions that is concealed by the regularity of the façade. We seem to find this phenomenon visually pleasing, if not beautiful, so yet again we have beauty being created from the use of proportions.

The concept of “built by man for man” that Doxiadis mentions, I feel, can sum up the findings of this report (Doxiadis, 1972). We, as humans, can define as ‘beauty’ as we are the ones that view and define it, even if the premise of the definition might be flawed it does not remove the inherent aesthetics in certain shapes and forms we produce. Perhaps it then becomes a question as to what it is we see as beautiful in architecture if not the “ordered self” proposed by Vitruvius – if we even need to define beauty to begin with. With that being said, there is certainly a link between ourselves and the orientation we impose on design for aesthetic pleasure. We are the ones who experience what we design, so why shouldn’t it be interconnected?

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