We also compare the accuracy of approximating a given design such as a curve. This serves as an easily-grasped metaphor for design in general, where the ability to represent a complex curve depends critically on having many different scales of structure. Architects as a rule wish to have available a large number of possible solutions, so as to enhance their ability to generate novel designs. Working within a modular system of design (with or without supplementary conditions such as internal structure in a module), however, restricts the number of possible results in a drastic manner. The restriction imposed by empty modularity may eliminate all of the possible designs that relate visually and functionally to human beings. We will argue that using empty modules reverses ancient practices that lie at the basis of humanity's connection with nature. The alternative to modular design is creating a form through subdivision or differentiation. A structure and its components can in this way have any dimension or shape, and at the same time, the structure can utilize materials in a variety of sizes and shapes. Design can be freed up by subdividing the building materials to achieve a spatial coherence as determined by human functions, movement, the psychological perception of space, connectivity, etc. This is the opposite from rigidly fitting human functions into a geometrical frame that is determined primarily by the size of pre-fabricated construction panels. Today we try to fit ourselves into some arbitrary dimensions fixed by an architect without any regard for the complexity of our spatial and emotional needs. Modular arrangements often define the aesthetics of a "style". Reasons for adopting a modular design system in architecture include those of economy of thought and action: it is easy to repeat a design unit that has worked before. It is true in almost all architectures, including various vernacular traditions, that a style is defined after a successful modular system has been developed, which is then formalized into a design canon such as the Classical orders. For example, the transition from wooden to marble construction in early Greek architecture occurred after the former material was developed into a successful system for building temples. The resulting style re-makes sensible wooden modules out of marble, which is not all that practical, even though the results are wonderful.
Qualitatively, the result of the study by Salingaros and West [1999] may be stated as follows: "Substructure exists in a hierarchy that follows an inverse proportionality: many smaller subelements; fewer intermediate ones; and very few larger ones". This multiplicity rule, derived for general complex structures, explains why contemporary neoclassical buildings don't always achieve the appropriate visual impact. Their forms clearly refer to classical forms, but the distribution of subdivisions is closer to the early modernists such as Mies van der Rohe and Le Corbusier. That is because substructure is cut off at a relatively large size; or if smaller elements are present at all, they are not numerous enough for visual balance. For this reason, despite an obvious attempt to mimic Classical prototypes, recent neoclassical buildings tend to resemble in spirit the modernist buildings they are trying to contrast with. Looking at the architectural and urban failures of the twentieth century, modularity is one candidate for critical scrutiny since ugly, boring buildings look the same, and many of them consist of large empty rectangular panels or exposed untreated concrete, and lack ornament and color. The distribution of sizes is cut off at the size of the empty modules and so the scaling does not continue downwards. Thus, the mathematical connection of the structural scales to the scales of human perception and movement, all the way down to the minute scales present in natural materials, is eliminated. The worst urban mistakes are again characterized by rectangular blocks arranged in precise modular alignment. The more such buildings repeat, the more they create an inhuman habitat whose negative effect is proportional to the area covered by the modules. The deficiencies of uniformity become dramatic on the urban scale. The requirements of totally different functions would normally preclude mixed-use building groups from sharing the same module, yet the industrial production of the same modules encourages architects to create similar buildings and monotonous urban zones. The alternative is to abandon a pointless strict modularity on such a large scale. Instead of doing that, however, we have eliminated mixed use and variety, partly in order to preserve a visual geometrical modularity on the urban plan. Modularity applied in this manner thus concentrates functions. Our cities now assume their form based on visual templates that are totally alien to the complexity of human perception, functions, and movement. A free design process that allows for numerous subdivisions permits mathematical substructure on many different scales. By abandoning an empty modularity, one has access to solutions based on a far richer approach to design that creates visually successful buildings. Art Nouveau architects like Antoni Gaudí used small modular elements (such as standard bricks) to create curved large-scale structures. This freedom of form contrasts with those instances where a building reproduces the shape of an empty rectangular module. The small scale can link to the large scale mathematically, because scaling similarity in design is a connective mechanism of our perception. Therefore, the materials can and do influence the conception of the large-scale form, and the larger a module, the stronger the influence. When one chooses to use large, empty rectangular panels, these will necessarily influence the overall building; often implying a monotonous, empty rectangular façade.
In describing the user-layout design process, Alexander says:
Realizing that this result is fundamental for all design (and
is not just restricted to office layout and furniture), Alexander
has developed it at length in his new book,
The derivation presented in the following sections answers Alexander's question, and proves his intuition correct.
While the eye and mind cannot actually perceive the difference between a very large number and an infinite number of different objects, a person can indeed experience a responsive design when compared to other designs that are deficient in some way. It may be difficult to put one's finger on what is lacking in a particular building, yet the fact that one experiences a visceral, positive emotional experience in a great historical building is incontrovertible. The point we are trying to make is that placing restrictions on design, as a modular system certainly does, might well eliminate the few optimal design solutions for that project. In particular, the vast majority of the world's favorite buildings -- representative of all periods of architecture, and coming from different stylistic traditions and geographic regions -- are excluded by a modular system of empty units. Counting the possible arrangements of modular units in a design
is a straightforward exercise. Without any loss of generality,
we construct a one-dimensional model where the proof is much
easier to follow. Consider a line of length equal to one on some
arbitrary scale. The length of the line is going to correspond
to the size of a building or urban ensemble. We are going to
imagine
Now we ask: how many possible re-arrangements of these We will use the analogy that each re-arrangement of the units
making up the line corresponds to a particular design solution.
It follows that the total number of possible re-arrangements
equals the number of design possibilities inherent in this design
system. In practice, however, architects working with modular
design usually define only Figure 2.
Design based on a single empty module labeled 1 restricts
the possibilities to one unique case, because all rearrangements
lead to the same result.
Figure 3.
Free, non-modular design corresponds to subdividing a line
into an n-element partition. The numbers show the order in which
the partitions are made, and it takes n - 1 cuts to separate
the line into n elements.We now count the number of possible ways to affect such a
partition. There exist an infinite number of possibilities of
where to place the first mark, since it can be made anywhere
on the line. Once we start, the positions of the second and all
subsequent marks also have an infinite number of possibilities.
Altogether, therefore, each partition of the line into
A practitioner is drawn
to use as large a module as possible in order to obtain the presumed
economical advantages of a modular design system. When justifying
such a system, however, proponents of modularity turn instead
to the opposite case of theoretically small modules, where the
number of modules n increases. This is misleading for
two reasons.Shrinking the size of the basic units increases the number
of different units, hence the number of design possibilities.
The re-arrangements in our one-dimensional model become more
numerous as
Figure 4.
A simple self-similar triangular fractal composed of triangles
on five different scales.A fractal's substructure is evident at every magnification, so that at no scale is the structure empty. This represents the opposite of a simplistic geometry of rectangular blocks, where structure is defined on only two scales, usually those of the module and the overall size of the building. An important feature of all fractals is that they satisfy the multiplicity rule for the distribution of subelements [Salingaros and West 1999]. Natural processes generate subelements in inverse proportion to their size. Material stresses create fractures that show as regular or irregular patterns, which prevent a continuous ordering throughout the form. Initially smooth materials will generate a fractal structure over time [Ball 1999]. Pure, non-fractal objects are unstable under weathering, which is precisely the reason they are not often found in nature. Modularity in design is related to the use of non-weathering materials in the following way. As explained above, natural forces and weathering generate substructure, thus destroying scale dominance [Ball 1999]. If we create an artificial non-fractal structure by emphasizing one particular scale -- say, the scale of a plain module -- then we are violating natural processes. To be in any way effective, such structures must resist weathering. It is for this reason that the topics of modularity, geometrical purity, and non-weathering materials are inextricably bound together. A preoccupation with shiny, smooth, or transparent materials follows the desire to avoid fractal substructure. It is well known that early modernist architects founded their new style of building on a search for "new" materials that they hoped would not weather at all. Going upwards in scale, same-size elements in nature tend to join in higher-order groupings. Units of similar size connect into superstructures or communities. Forces are distributed among many different scales, thus ensuring a more efficient linking. The emergence of higher scales corresponds to upward fractal growth, a process that guides morphogenesis and creates complex order while minimizing randomness. The separateness of repeated units existing on one scale is sacrificed to the coherent structure of a higher scale. This generative process is reversed by empty modularity, which tries to maintain the emptiness of its original module with no internal substructure. Architectural modules of this type are often used to build an empty scaled-up version of the module. An inanimate object can have different degrees of organized complexity. There is a process by which this complexity is developed and maintained. We repair buildings and the urban fabric from wear and tear and physical decay, which is analogous to the feeding and maintenance of organisms or machines. The difference is that in artificial systems as opposed to biological systems, repair is done by human beings rather than by the entities themselves. A structure that doesn't weather at all appears alien to a human being. For buildings to weather well, they have to allow for the development of a fractal structure: they must be designed from the beginning with a fractal hierarchy of scales, otherwise the inevitable patterns that develop as a result of weathering will cut across the original design intentions. The only way to prevent this inconsistency is to anticipate a fractal form in the original design.
We propose to link ordered complexity as characterized by levels of substructure, on the one hand, to design freedom, on the other. This can be seen in the simplest design problem: creating a curved surface to accommodate human movement, psychological space, or other factor optimizing people's physical and emotional interactions with a building. We will show that a non-trivial curve cannot be constructed without abandoning the simplistic modularity discussed earlier. Any curve can be obtained as an approximation of straight segments of connected lines. Nevertheless, a good approximation depends on the length of the basic segments that are utilized. In a modular system of several elements, one of them with a fixed length is going to be the smallest. The size of the smallest element limits the curvature of curves that can be approximated. Consider the case of a single module. Comparing the three approximations to a semi-circle shown here, we observe for example that the possibility of creating an accurate representation of a smooth circle of given radius gets better as the length of the modular segment becomes smaller. Figure 5.
A curve can best be approximated by having as small a straight
module as possible.One might think that approximating curves can be improved by introducing modules with curvature. We could then have arcs of different length and curvature. In contrast to the case of the straight segments, however, the curves with greater curvature will not necessarily be better approximated by the smallest modules of arc. The approximation is still limited by the length and shape of the modular segment. Neither approximation shown below is particularly satisfying, which reveals the inadequacies of any modular design system. This can be remedied by combining small and large units of different curvature together. Different units will fit different sections of the curve. This results in by far the most accurate representation of a continuous figure by means of an encoding scheme. We are thus led away from strict modularity, and into fractal approximation that works on many different scales simultaneously. Figure 6.
Approximating a curve (shown in solid black) with a curved
module is inadequate regardless of the size of the module. Here,
two quarter-circle modules with different radii are used.After this simple analysis, we can see why modular architecture abandons curved designs on the smaller scales: the reason is that realizing a curve using relatively large modules is mathematically impossible. Complex designs require components on different scales. Empty modularity restricts the curvature of design, and so empty modularity is incompatible with mathematically-rich designs such as curves. Thus, the architecture of rectangular modules eliminates one of the principal components of natural forms. We identify this exclusion as a major reason for the dull uniformity of rectangular boxes. Recent buildings such as the Denver airport and the Bilbao Art museum take advantage of new materials and computer-aided design and fabrication to break out of the rectangularity inherent in the old early modern mechanical reproduction of many identical components. The metaphor of approximating a complex curve shows the need to allow for different structural units at different scales. A building need not be curved necessarily, yet it should ideally follow a similar design freedom: the geometry must arise from subdivision rather than modular juxtaposition. We thus propose a metaphor for design that in general responds better to human physical and psychological needs. The freedom necessary to achieve those aims requires the ability to adjust the built form according to complex inputs. If we follow a rigid visual image too faithfully, we cannot at the same time create a design plan that serves unanticipated but important needs. The reason for this contradiction is the impossibility of predicting in advance all the uncountable effects and interactions between human beings and the built form.
The Modular construction was accepted by the post-war architectural community as a necessary component of building for the future. We have inherited the idea of modular design in a watered-down version that leads to the most boring and minimally satisfying results. Today, both apartments and tract houses tend to be built throughout a region according to standardized plans; not an optimized plan for patterns of family life and climate as in vernacular architecture, but simply a dimly-remembered floor plan from the heady days of early modernism. Housing "developments" usually repeat a single poorly-adapted house plan with only minor variations, which moreover offers few of the advantages of modular construction. Such houses are still built by hand, using the cheapest possible materials. A rectangular grid exists in countless examples of post-war
buildings and construction. We mention here the 3 ft. 4 in x
8 ft. 3 in planning grid used in England by the Hertfordshire
school system, and the 3 m x 3 m grid used by Constantine Doxiades
in building several new villages in the Arab world. Ludwig Mies
van der Rohe employed an empty module of 24 in x 33 in for the
1950 Farnsworth house. People continue to think that there is
something fundamentally advantageous to such a restriction, which
is questionable. It might look more efficient on paper, but it
does not lead to life in the built environment. Le Corbusier
wrote at length on the presumed advantages of a Cartesian grid
in his own journal, Among the few modernist architects who refused to adopt modular
design in buildings as a matter of principle was Alvar Aalto,
who is reported to have said: "my module is the millimeter".
Even the organic post-modernist architecture of Lucien Kroll
was praised for using a 10 cm module previously developed by
the Dutch group
Journal of the
Royal Institute of British Architects 66.Alexander, Christopher, A. Anninou, G. Black
and J. Rheinfrank. 1987. "Towards a Personal Workplace."
Pp. 131-141 in Alexander, Christopher. 2000. Ball, Philip. 1999. Kroll, Lucien. 1987. Mandelbrot, Benoit B. 1983. Salingaros, Nikos A. and Bruce J. West. 1999.
"A
Universal Rule for the Distribution of Sizes." Pp. 909-923
in
(Doctorate in Algebra
1981, University of Montpellier, France; Ph. D. in Topology 1993,
University of North Texas, Denton) is Professor of Mathematics
at the Universidad Nacional de Colombia, in Medellin. During
the academic year 1999-2000, she was a visitor in the Mathematics
Department at the University of Texas at San Antonio. Dr. Tejada's
principal publications are in the area of Knot Theory. Recently
she has become interested in the mathematical foundations of
Architectural form and Urbanism.Débora M. Tejada
Copyright ©2001 Kim Williams top of
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