The Quadralectic City is an apotheosis of the previous three cities of the mind – the ideal, future and virtual. It is here, in the fourth field of unknown architecture, that the (double) past, present and future get a meaning and meet each other in the quadralectic concept of visibility. This very point of departure means that there is not one quadralectic city, but at least four. Therefore, to present the ‘quadralectic city’ is an impossible assignment, in the same way as bringing down the initial division of a communication to one.
Nevertheless, it is fair to give my own interpretation of a quadralectic city. I am not an architect or city designer, so the city can only be of a very sketchy nature (like Costa’s plan of Brasilia). Maybe it is easier to say what a quadralectic city is not. It is not an idealistic political statement, nor a requirement for the future. There is also no need to exist as virtual reality. The quadralectic city is, in essence, the expression of the possibilities (to build a city) in a four-fold environment, born in the creativity of a quadralectic mind.
The quadralectic city should have a reference to the four-fold. Tetradic visibility has to be understood in four ways. That is: as a complete invisible ‘creed’ (I), as an ‘idea’, which can be expressed in words or graphic signs (II), as a visible entity (III) and finally as a complex application in real life, communicating with the senses (IV). The third option is probably most relevant for empirical minds, and those people who believe that every architecture should be visible architecture. However, what happens when the fourth option is put first and the visible invisibilities of the senses are chosen as the leading principle? The best way to approach this question is an inventory of the various efforts, which were made by the writer over the years to find an answer.
1. An initial idea was developed to follow a ‘biological’ approach, taking the shape of the four chambers of the human heart as the ‘genius locus’. This means that the city would be neither square nor cyclic, but follows the natural oval outline of a heart. This oval shape, consisting of four chambers of the heart, is then the (subjective-human) visualization of the ultimate unity. There is probably no further place we can go as human beings without losing ourselves in nothingness. Paul Klee’s study of the heart is an impressive effort to reach into the depth of understanding (KLEE, 1956/1961). However, this essential organ was already properly studied by Wiliam Harvey (1578 – 1657) in his fundamental work ‘De Motu Cordis’, which was published in 1628 (NEIL, 1975).
The four compartments of the city are characterized by the typification of the various quadrants, but can also be compared with the functions of the heart chambers. Two chambers receive the blood from the system, transport it to two other chambers, which pump the blood into the system. A two-division can be distinguished, in which the receiving chambers are the ‘static’ parts and the pumping chambers the ‘dynamic’ parts. A blood sequence is therefore:
——————————— 1. Receive
——————————— 2. Pump
——————————— 3. Receive
——————————— 4. Pump
A cycle through a communication can be seen in the same dynamism: information is received (First Quadrant), pumped in the system (Second Quadrant), received with new information (Third Quadrant) and pumped into the system again (Fourth Quadrant).
This sequence was – in different terms – already used by the Greek philosopher Empedocles (c. 494 – 434 BC). He divided Nature in four elements (fire, air, earth and water), and envisaged an alternating ‘life-flow’ of attraction and repulsion between Love (Philia) and Strife (Neikos). Loves aims at unity and happiness, whereas strife leads to pluriformity and misery. The ideas about ‘isomoiria’ (equal division) were drawn from the medical world of Alcmaeon and others, who recognized four humours in the human body: choleric, sanguine, phlegmatic and melancholic.
Mobility is a key word in the imaginary ‘city of the human body’. The heart is – in a biological metaphor – the central business district, which is connected with the green space of the lungs. Arteries and veins can be seen as the communication routes to mining areas (stomach) and heavy industry (bowels). The liver functioned as pollution control supported by the waste plants in the kidneys. The bladder is the water treatment plant, while the genitals are involved in (re) production.
These metaphors are historically used to visualize the living city, starting in the Italian Renaissance by architects like Francesco di Giorgio Martini in his book ‘Trattati sull’architetture civile e militare’ (1474 – 1482). He made (city) plans proportioned according to the human body. Wolfgang Lorenz of the Vienna University of Technology is wondering if these metaphors cannot be replaced by modern ones in which fractal geometry provides a new vocabulary to explain the complexities of a city in a better way (LORENZ, 2003).
The Islamic city model, as it developed in the heartland of oppositional thinking (Iran), might be the closest example of a ‘city of the heart’. Masoud KHEIRABADI (1991) gave a model city with an oval shape, four entrances and the Friday Mosque in the center (fig. 692). The City, or al-Medina, is the frame of reference of the Islam, both as a religion and as a way of life. The initial model was changed in its contact with different cultures, but the notion of ‘al-Medina’ never lost its appeal.
Fig. 692 – The Islamic city model, as given by Masoud Kheiradabi, has an oval shape and four entrances in the wall. Citadel and Friday Mosque are situated in the cross roads in the center. The quarters are divided according to monotheistic, religious beliefs.
Further tetradic designs in the world of Nature are many and varied, and might be used as inspiration for a Quadralectic City. An excellent example of tetradic beauty is contained in the so-called calcareous nanno-fossils (fig. 693). These are the remains of algae, which make calcite platelets inside their cells.
Fig. 693 – These tetradic shaped nanno fossils (calcite platelets from algae) were found in Palaeocene sediments just above (upper duplet) and below (lower duplet) of a clay-layer, which divides the Cretaceous from the Palaeocene at the locality of Stevens Klint, south of Copenhagen (Denmark). This famous section of the Cretaceous-Tertiary (K-T) boundary shows the transition from white coccolith chalks of the Maastrichtian to the bryozoan-rich mounds of the Lower Danian sequence. The discovery of a large iridium anomaly in this clay layer in places as far apart as Gubbio (Italy), Stevens Klint (Denmark) and Woodside Creek (New Zealand) led the American geologist Walter Alvarez and his collaborators in the early 1980s to the conclusion that a massive extinction of plant- and animal life (including the Dinosaurs) took place some sixty-five million years ago when an asteroid struck the earth.
2. Later ideas by the writer on the Quadralectic City were developed in a flash of creativity in only two days (21/22 August 2003). They had a distinct quadrilateral setting with four circular areas with the city boundaries (fig. 694/695). The influence of (Third Quadrant) ‘Renaissance’ elements and the idea of an Ideal City were even more obvious in this proposal. The Quadralectic City was called Quadrilo and its four quarters followed the intentions (visibilities) of the four quadrants:
Fig. 694 – The basic division within an imaginary Quadralectic City called Quadrilo follows a general classification of the quadrants. The actual concretisation of the various quarters in terms of urban design is left open in this scheme, but it gives the main charac-teristics of the four-division.
The Quadralectic City has a First Quadrant (I) – of invisible invisibility – that is visualized as an area without a minimum of human organization, left alone to form a deliberate wasteland. The First Quadrant is called Quadrilo 1 and consists of an open space with only a glass triangular pyramid (tetrahedron) in the middle. This barren area symbolizes the innumerable possibilities without actually expressing them. The space might be surrounded by a narrow ‘green zone’ of natural, indigenous vegetation to soften the view from the empty space towards the city.
The Second Quadrant (II) – of invisible visibility – carries the idea of the four fold to its first fruitation. This part will be the ‘City of Thoughts’ and is called Quadrilo 2. It comprises schools and universities, head offices and banks. Cyclic geometric patterns should be designed in such a way that the institutions of learning and commerce alternate and can interact. The banks and offices should be in direct contact with the schools and universities, making an intellectual cross fertilisation possible.
The Third Quadrant (III) – of visible visibility – is the city-proper with living quarters. Quadrilo 3 is the actual living quarters of the city with organic aligned neighborhoods. Houses comprise of a ‘communication-room’ as a central feature, consisting of a number of large screens, which allow the inhabitants to communicate with the world – either related to working or research, or as a form of leisure. The neighborhood-as-a-whole also has a communal communication center, designed to bring people together, known as the CF-Center.
The Fourth Quadrant (IV) or Quadrilo 4 is the leisure part of town, with libraries, sport facilities, a stadium and swimming pool. A labyrinth is situated in the center, with an observational building (observatory) overlooking the empty space of the Quadrilo 1 and the cosmic world.
Fig. 695 – The quadralectic city (Quadrilo) consists of a subdivision of four circular cities, called Quadrilo 1 – 4. These interconnected parts have their own specific functions in the communication between the city and its inhabitants.
Quadrilo 1 has no buildings at all, except a glass tetrahedron, possibly with subterranean cross-shaped extensions, shrouded in mystery (or in use as a mausoleum or communal burial place. Quadrilo 2 is called the ‘City of Thoughts’ and consists of institutions of learning, museums and working (offices) in an organic setup. Quadrilo 3 is the heart of the city mainly used for living and communicating. Quadrilo 4 has a labyrinthine nature with leisure facilities, like a stadium, swimming pool, concert hall, a restaurant, etc., as its main buildings. However, the most prominent building in this quarter is an observatory, symbolizing the search for visibility in general.
3. Another option of building a quadralectic city was found in the building of a settlement in the shape of the CF-graph, taking a literary interpretation of an arithmetical reality. The linear city of Don Arturo Soria y Mata (see fig. 644) and the Russian de-urbanists was taken one step further into a Graph City (fig. 696). The building process is guided by a changing visibility, as it is reflected in the various quadrants. The characterization of the different quadrants in terms of psychological units can be used to emphasize the trajectory of the city as a ‘living’ entity in time and space.
Fig. 696 – An example of the street pattern in ‘Graph City’, which is based on the form of the visible visibility area X of the CF-graph (see fig. 852 for nomenclature).
The Communication Factor (CF) is known in the quadralectic philosophy as the universal expression of a value, which is derived in a shift of two four-divisions interacting in a cyclic environment. This literal solution does not offer an advantage over the established linear cities except perhaps for its philosophical meaning.
4. A grid-design would seem to be the most logical approach to a city in a quadralectic environment. It was stated earlier – in relation to the urban development of grid cities (Ch. 220.127.116.11.) – that the grid is the collective expression of the cross, the square and the multiplicity in one. Experimental work on this subject in an artistic context was carried out by the Dutch painter Piet Mondriaan (1872 – 1944), who ventured in 1919 with his ‘Komposition’ (fig. 697 left) into a phase of grid-based paintings.
Fig. 697 – A comparison between the matrix approach of a ‘Komposition’ (1919) by Piet Mondriaan (left) and an automaton of the name ‘Quadrilo’ (2003) (right). The diagram is produced by a Data-matrix Generator provided by the Swiss firm Kaywa in Zürich. http://datamatrix.kaywa.com).
Russian contemporaries, like Malevitch (1878 – 1935), Kandinsky (1866 – 1944), El Lissitzky (1890 – 1941) and Rodtchenko (1891 – 1956), explored the same imperceptible terrain of the ‘abstract’. This distinct approach to reality was never given such a powerful expression in art before. Art – in its historical context – was a move from visibility to invisibility. It might have been a tribute to the gods, in its bare essentials. However, this sequence was broken and reversed in the early twentieth century. The invisibilities of the sentiments came first and the expression in the visible followed later. The search of the ‘avant garde’ artists and their subsequent discoveries can be recapitulated as the glorious victory in the realm of the invisible sentiments – which happens to be the dominant source of visibility in the Fourth Quadrant of a communication.
This breakthrough is now some hundred years ago. A more contemporary approach to a Quadralectic City is expected to use the advances, which were made since this avant-garde period. The grid-designs of the Roman-Renaissance-Jeffersonian stock are now passé and substituted by modern methods of mathematically generated grids. An example is the use of the data matrix code, which is a two-dimensional bar code consisting of black and white cells or modules arranged in either a square or rectangular pattern. The data matrix offers a tool to design a city on the base of chosen parameters – like the name of ‘Quadrilo’ (fig. 697 right).
David PEAK & Michael FRAME (1994) described an interesting mathematical approach to the state of chaos, which is relevant for our present study of tetradic building. In their book ‘Chaos Under Control’ they visualized a cellular automaton, a mathematical mind-construction, which gives a message to a ‘cell’ to behave in a certain way. A distinct configuration appears when the order is carried out (fig. 698).
Fig. 698 – These two cell patterns emerge from identical rules and similar starting conditions, but different limitations. The left pattern fits into a 100 x 100 grid, while the right pattern emerged from a 99 x 99 grid (as given by PEAK & FRAME, 1994).
The definition of an automaton activates a self-designing architecture. It leads to distinct patterns, which have references to a tetradic division. These patterns can be used to design a quadralectic city. The French mathematician Benoit Mandelbrot revived the importance of dynamic systems and particular the recursive character of some of these systems (MANDELBROT, 1983). He became intrigued by the behavior of equations that undergo iterations (the repetition of a process within a computer program) and ‘discovered’ the fractal. The fractal is defined as a fragmented geometric shape, which can be split into parts and have a self-similarity with the whole. Four important characteristics in the genetic process are recognized:
———– 1. The presence of a dynamic space (motion escaping to infinity)
———– 2. Periodic cycling
———– 3. A limiting agent to control space (motion represented in values)
———– 4. Chaotic behavior
Self-organization is a process of attraction and repulsion in which the internal organization of a system increases in complexity without being guided or managed by an outside source – not even by natural selection. Self-organization has four basic elements:
- Positive feedback or ‘cumulative causation’ is a feedback loop system in which the system responds to perturbation in the same direction as the perturbation.
- Negative feedback responds to the perturbation in the opposite direction.
Norbert Wiener pioneered these concepts in his seminal work on ‘Cybernetics’ (1948), which became in due course an interdisciplinary study of the structure of regulatory systems.
3. Balance of exploitation and exploration
4. Multiple interactions
The Fourth Quadrant is connected with multiplicity, which is given a distinct place in the communication. Its position in a cyclic setting is encapsulated between the solidity (part) of the Third Quadrant and the traditional notion of chaos as ‘the formless void of primordial matter, the great deep or abyss out of which the cosmos or order of the universe was evolved’ (Oxford Dictionary). This void is followed by the First Quadrant of the next cycle, providing the continuity in the communication.
This essentially oppositional view got an overhaul in the Chaos Theory. The theory can be defined as a discourse seeking to explain the behavior of non-linear dynamical systems, which exhibit the phenomenon referred to as chaos (BRAUN, 2007). Chaos is – in the modern view – the product of patterns of mathematically deterministic behavior resulting in a complexity, which is unobservable at a localized scale or time frame.
Complexity can be broken down in four distinct types of complex systems determined by the type of attractor. 1. A single point attractor system; 2. A cyclic or periodic attractor system; 3. A strange attractor system; 4. A complex adaptive system. This division fits smoothly in the four quadrants of a quadralectic setting (fig. 699):
Fig. 699 – This overview gives the four types of complex systems, which are part of the complexity theory (after Bradley BRAUN, 2007) and fit them into the quadralectic division of quadrants.
Charles JENCKS (1995) is one of the modern architects, who expressed the interest in the complexity theory as a tool for further development in architecture. He ended his book with the recommendation that architecture must look to science, especially contemporary sciences, for disclosures of the Cosmic Code. Whatever that mystic entity may be is another matter, which is not discussed. He starts with a frontal attack on Modernism – a loose term for the creative mood of the period between 1890 to 1970, which rebelled against nineteenth century artistic and historicist traditions. Progress and fragmentation of experiences were the catchwords of the movement.
A reaction took place in the mid seventies of the twentieth century resulting in Post Modernism, rebelling against the machine aesthetic of Modernism and its reductionism (Minimalism). Jencks sees himself and the contemporary society as part of the Post Modern movement, although he prefers terms related with the word ‘cosmic’ or ‘cosmogenic’. However, that still leaves the question of the chosen style. Any building makes a (Third Quadrant) statement, which can hardly be erased in the short run. It should be realized, that the opposition between simplicity and complexity is part of such vocabulary.
Robert VENTURI (1966/2002) – in his important publication ‘Complexity and Contradiction’ – spearheaded one of the first attacks of the Postmodernists against simplicity and pure functionalism. He employed lessons from the complexity sciences with the archetypal ‘Butterfly Effect’ – standing for the sensitivity to the initial conditions – as its banner. Venturi wanted an architecture rich in symbolism and history, complexity and contradiction. A choice (of division thinking), taken somewhere in the communication process, will influence the whole (of the visibility) of that communication. Or, written as a quadralectic sequence, with the approximate positions of the various entities and the elements of sciences (fig. 700):
Fig. 700 – The positions of various entities of an ‘architecture of complexity’ – as envisaged by Robert Venturi – are interpreted here within the quadrants of a quadralectic communication.
The American Bruce Goff (1904 – 1982) was an architect, who took notice of the widening of formal space. He started as an Art Deco architect, but had pioneered after the war in a non-conventional style. The imaginative spiral Bavinger House in Norman (Oklahoma) and the Joe D. Price House and Studio in Bartlesville (Oklahoma; burned in 1996) were examples of his intentions to break new ground. Jencks called him a ‘natural cosmic architect’, who followed nature’s grammar and processes. Recycling was part of his expressional language.
The developments in the field of automatons and fractals – with the computer as its calculating center – do not necessary lead to ‘better’ cities. However, the acceptance of raggedness of Nature, rather than the idealized smooth shapes of Euclidean geometry, is a step forwards. The relation between the city and its inhabitants will, in the end, be ruled by nature and a proper understanding of the latter will have its effects on the well-being in the city.
Furthermore, it has to be understood that complex forms do not necessarily represent a complexity – in the same way as quadralectic architecture is not always expressed in tetradic forms. Complexity points to the inter-connection of elements that produce the whole (VON MEISS, 1990). This particular understanding is the root of modern quadralectic architecture. The observer should penetrate through the superficial façade of symbolic or numerological appearances in order to enter in the cognitive space behind the visible visibility.
One of the disturbing conclusions of this short survey is the fact that the quadralectic city will probably never be built and cannot be built. At best it will be a (virtual) city in which all the aspects of quadralectic philosophy are present. An enormous gap will appear as soon as the (visible) ideas in the mind of the designer are given its actual execution in the practical field. The invisible world of the First Quadrant will disappear and the imaginary world of the Second Quadrant can only be guessed from the given visibility of the Third Quadrant.
The four options of a quadralectic city should be considered on their merits. Its proposals are as good as others, but they have the advantage to be rooted in a system of thought. The quadralectic city is a dynamic system. A multiplicity of different entities (like people, houses and infrastructure) communicates with each other within the boundaries of their constraints given by the type of division thinking. This understanding will result in a measure for the quality of a city.