HIERARCHICAL COOPERATION OF ARCHITECTURAL SCALES, AND THE MATHEMATICAL NECESSITY FOR ORNAMENT

(preliminary version)

© Nikos A. Salingaros, 1997

Division of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249

 

Abstract

This paper makes a case for complex systems such as architecture to be organized hierarchically. It derives a method and formula for doing so based on biology and computer science. The notion of fractal simplicity, in which there is self-similar scaling, replaces the outdated rectangular simplicity. Architectural elements on different scales are able to cooperate in an intrinsic manner to achieve coherence. Repeating structural and design elements of the same size define an architectural scale. The properties of hierarchical systems explain how to relate different scales to each other. In buildings, the correlation between architectural scales determines whether a structure is perceived as coherent or incoherent, independently of the design. This paper reviews existing results, and derives new ones which help in achieving integrated structures.

INTRODUCTION

In the early 1920's, Ozenfant and Jeanneret (Le Corbusier) published a series of articles in their own journal L'Esprit Nouveau, which were later incorporated into Vers Une Architecture (Le Corbusier, 1927). In these articles, they stated - without proof - that the Platonic solids are somehow ingrained within the human consciousness, so that the mind is programmed to recognize (and prefer) simple cubes, triangles, spheres, etc. We now know that this is false (Bonta, 1979). Human beings have to be trained to recognize Platonic solids, which are purely an intellectual concept (Fischler and Firschein, 1987). What is built into the human consciousness is a recognition mechanism based on natural structures being hierarchically subdivided, irrespective of their overall shape.

The validity of the above assumption about Platonic solids was never questioned, and eventually became part of this century's architectural tradition. In fact, one may argue that modernist architecture owes its success to the opposite effect. Platonic solids are almost never seen in nature as large-scale macroscopic forms, so that any building with a pure, regular form contrasts with the environment and appears "novel". The sun and full moon - both exceptional cases of perfect disks - were worshipped by ancient peoples. The same was true for monoliths. Mankind throughout history built unnatural structures such as the pyramids precisely so as to assert man's dominance over nature.

Architecture is an artificial process. An architect has a choice of whether to follow or to violate certain conceptual rules that relate a building to natural forms. I propose a design constraint that comes from systems theory and leads to hierarchical coherence. It requires only that forms be subdivided in a certain manner; every other aspect of the design is left up to the architect. All complex systems - natural as well as artificial - follow the rule in which distinct scales cooperate hierarchically to define a coherent whole. How to achieve this will be discussed in detail. Arguments are presented as to why this universal rule applies to architecture; and also why it is built into the human consciousness. The underlying argument is based on the following four-step logic:

  1. Natural forms and biological organisms are hierarchically organized.
  2. Systems in engineering and computer science have been found to follow the same hierarchical rules of organization as natural systems.
  3. Architecture is a complex artificial system: to achieve coherence, it must also satisfy hierarchical system rules.
  4. The human mind evolved partly in order to recognize and analyze hierarchical structures in nature, so artificial structures that are not hierarchically organized are perceived as alien.

Independent support comes from the science of perception, though experimental psychology is not used as the primary argument here. First, the nature of human perception has never been completely described, and is still a subject of much investigation. Second, the critical experiments that would demonstrate this phenomenon directly have not been performed, and the evidence, while supportive, is scattered and circumstantial. Readers confuse the need for symmetry (manifested by the mental completion of missing regions of regular figures) with a non-existent predilection for Platonic solids.

Confronted with a man-made object or structure, we grasp all the different scales at once, automatically establishing a scaling hierarchy. In cases where the scales are ambiguous, our perception of the structure is frustrated. If the scales are spaced the same way as in natural structures, and if they also correlate with each other, we perceive the structure as coherent. The ratio between successive scales determines whether the ensemble is perceived as hierarchically organized or not, i.e., if it defines a coherent whole. This subconscious process might well determine a building's impact independently of such traditional concerns as shape, form, and proportion.

Forms that lack hierarchical coherence can appear exciting by virtue of their being unfamiliar. That feeling, however, is very different from the emotional elation experienced inside a Medieval Cathedral. Architects ought to be sensitive enough to distinguish between an uneasy excitement and a deeply satisfying visual comfort - which are really opposite in their psychological effect - and to see how they are produced in buildings. Up until now, confusion has reigned over this topic, which is the basis for architecture's impact on persons. Both modernist and classical buildings are rationally understandable, and visually and emotionally evocative, but in almost opposite ways.

In this paper, I introduce the idea of a more sophisticated simplicity, in which forms have a hierarchy of subdivisions. This concept comes from the mathematical theory of fractals. I then define an architectural scale as a collection of similar elements of the same size. The different scales in design define a hierarchical system. If the scales cooperate, the hierarchy achieves coherence, otherwise known as an emergent property, and the result is greater than the sum of its parts. After an overview of systems theory, I apply it to architectural design. I then analyze what consequences violations of the proposed rule in architectural forms may have on the building's user. The importance of the scaling ratio and the linking between scales has previously been overlooked. A later section is devoted to helping a practitioner apply the method in designing a building. The last section discusses theoretical design, and speculates as to why systems theory apparently failed when first introduced into architecture in the 1960's.

GLOSSARY

Architectural scale: One or more repeating or similar structural and design elements having the same size (width or length).

Coincident scales: Two architectural scales having the same size but dissimilar, usually contrasting elements.

Hierarchical system: Any structure with clearly-defined substructures that relate to each other, where the higher scales depend on all the lower scales.

Quantization: A discrete (as opposed to a continuous) distribution of measurable quantities.

Scaling hierarchy: When a form's subdivisions get smaller and smaller, with a fixed ratio between successive subdivisions.

 

ARCHITECTURAL SCALES

All complex natural systems have a hierarchical structure, regardless of whether they are biological or inanimate (Simon, 1962; Smith, 1969). Most inorganic materials are crystalline, and a few are amorphous. Even complex compounds have crystalline structure with long-range ordering. Material stresses create fractures that show as regular patterns in crystals, preventing an overall ordering from continuing throughout the piece (Smith, 1969). Smoothness and uniformity are alien to natural materials. We see physical forms that have hierarchical scaling as a result of internal and external forces. Structural features exist on different levels of scale, from the microscopic to the macroscopic, through all intermediate scales.

Biological forms exhibit a definite hierarchy of scales. In decreasing order of size there are communities of organisms, organisms, organs, tissues, cells, organelles, membranes, molecules, atoms, and elementary particles, with many possible intermediate scales to these (Miller, 1978; Passioura, 1979). At different sizes, structurally coherent elements define a scale. The scales are distinct, yet they are also nested in any complex structure. The same is true for architectural forms. Architectural scales are defined by similar elements of the same size that repeat. Independent scales arise from the materials, structure, and functions, and also express an architect's ideas.

Methods used throughout the history of architecture to define scales include symmetry, as manifested through fenestration and columns. Windows - if they are of the same size - create a distinct scale. They can be repeated in a symmetrical pattern to define a larger scale. The subdivision of a window into panes creates a smaller scale. Colonnades define several scales: the column's width; the inter-column spacing; and the column's base and capital (with fluting generating yet one more smaller scale). Massing and monumentality define the largest exterior scale. Interior scales are created by window and door frames, baseboards, and trim in various sizes, aided by contrast in materials, surface texture, and color.

Design elements link when a distinguishing characteristic connects them visually; if they have some portion of design in common, and if they have a similar texture or color. Although existing design methods can organize materials on a particular scale, there is no general theory of how to space the scales themselves, nor of how to correlate the different scales. And yet, most human creations prior to the 20th century (buildings, artworks, artifacts, tools, and machines) are hierarchically integrated. They achieve a balance between different design elements according to their size. Our objective is to cast this process into scientific terms so that it can be applied consciously and deliberately.

The effect that we seek is one of connectedness: a viewer should relate to all parts of a building, and the building ought to relate to its surroundings. This is best seen in an urban setting as opposed to talking about a single building. A building can mislead by being exciting but failing to connect, either internally, or to anything else. This confusion is routinely exploited by architectural magazines that illustrate only a distant, symmetric view of an isolated building. There is usually no attempt to show how one would experience such a building close-up (at touching distance), nor within its surrounding architectural and urban space (Moughtin, Oc et al., 1995). As a result, it is impossible to judge the building's actual impact.

Example A. Spanish Oak Tree. Despite the apparently amorphous character of the tree, it is in fact subdivided into distinct scales. The trunk and main branches have a narrow distribution of lengths and roughly the same width. The secondary branches are about 1/3 the width of the main branches, and so on. Leaves are grouped into clusters that are about the same size as the width of the trunk; they are distributed neither uniformly, nor randomly. The inter-leaf spacing on twigs tends to be regular. This hierarchical structure is evident once one looks for it. Organized detail goes all the way down to the microscopic scale. The size of leaves, twigs, acorns, and bark articulations define one or two scales, while their fine structure defines many scales smaller than these. There exists a hierarchy of scales from the height of the tree at several meters, down to below 1mm.

Example B. Small Bedroom. The largest scale of a small room in my apartment, painted white throughout, is 4m. Two windows built together define a scale at 180cm. The width of each window equals the door's width, which defines another scale at 75cm. The window trim and door frame are both 7cm wide. There is no other scale until we get down to 3mm surface detail in the faint wall texture. Altogether, we have four obvious scales xi at {400, 180, 75, 7, 0.3cm}. It is instructive to compute the ratios xi+1/xi between consecutive scales, which are approximately 2.2, 2.4, 11, and 23. (We don't count the microscopic scales in the materials). As explained in a later section, these numbers reveal the room's poor hierarchy.

Example C. Piazza San Marco, Venice. This is one of the world's great outdoor urban spaces. It has been diagrammed, replicated, but its success is still incompletely understood. Here, I propose a hierarchical explanation. Each surrounding building has subdivisions at roughly 1/3 its overall size, and further subdivisions at roughly 1/9, etc. Richly-articulated detail is evident from all directions. The plaza itself is subdivided through the use of contrasting floor paving (Moughtin, Oc et al., 1995). Each building is hierarchically coherent, and the strongly established individual hierarchies link across space to create a coherent whole. It is the subdivisions, or architectural scales, of the disparate and visually dissimilar buildings around the plaza, that cooperate with each other and with the pavement to make us experience this space as a magnificent ensemble.

Example D. Grande Arche de la Défense, Paris. The pavement in front of and inside the arch does not connect to anything through scaling or similarity, because it has minimal features and subdivisions. The arch itself has very few distinct scales; by far not enough to connect the structure internally, or to a pedestrian, or to the plaza. Overall, the deliberate lack of hierarchy leads to the failure of the parts, despite their studied simplicity, to unify into a whole. The structure is of such enormous size that, under favorable weather conditions, it is imposing, monumental, and exciting; nevertheless, the entire range of human scales is missing. A viewer cannot avoid feeling isolated. There is no material point to connect to. This structure is meant to impress; even intimidate, but not to relate to human beings.

 

SIMPLICITY AND IMAGE COMPRESSION

Hierarchical scales in design influence the viewer because they facilitate the process of human cognition. We are able to perceive a complex structure easily by reducing it to a number of distinct scales. The more design subdivisions, the more scales there are. Human beings have a basic need to organize elements into hierarchies as a means of avoiding information overload. The mind groups similar elements of the same size into one scale (Fischler and Firschein, 1987). It then looks for mathematical similarities or links between all the different scales. Since the mind has evolved in response to patterns and scaling hierarchies found in nature, a certain set of rules is "hard-wired" within our perceptive mechanism (Fischler and Firschein, 1987).

The idea behind proposing the Platonic solids as fundamental in the early days of modernism was to find simplicity in design, and to apply it to new built forms. Inevitably, that notion of simplicity is outdated, and belongs to ancient science; modern science has totally revised our understanding of simplicity. This is best discussed in the context of mathematical image compression: an image is simple if it requires the least amount of information to specify. I will now discuss two different ways of encoding an image, and show how they imply radically different notions of simplicity.

The first method is called the "Graphics Interchange Format" (GIF), and is commonly used for storing pictures on the World-Wide Web. The algorithm divides an image into pixels on a rectangular grid, and looks for both horizontal and vertical repetitions. A repeating sequence on one row is coded compactly by entering the repeating group, along with its multiplicity. Horizontal lines that are the same do not need to be coded more than once; the same line is multiplied. In this way, any horizontal or vertical regularity is compressed. More sophisticated connections, ones that are seen automatically by the eye, pose serious problems for this particular algorithm of pattern recognition in wishing to emulate human perception (Fischler and Firschein, 1987).

A second, newer method is called "Fractal Image Compression" (FIC), and works much more closely to the way the mind itself works (Barnsley and Hurd, 1993; Fisher, 1995). Roughly speaking, fractal image compression identifies self-similarity at different distances and at different scales. It deals with pieces of the picture rather than individual pixels. Repeating units along any direction or directions are encoded as one design scale. Similar units that differ only by scaling are also grouped (corresponding to a cooperation between scales). This method works very well on faces, trees, and natural scenes. Neither GIF nor FIC can simplify random textures, because those have no spatial regularities.

All this is relevant to architecture because the two methods represent two very different concepts of simplicity. In the first method (GIF) plain rectangular forms are the simplest, requiring the smallest amount of information to encode. In the second method (FIC), plain rectangles are simple, but so are fern leaves, snowflakes, and rocks. Any structure that is hierarchical and self-similar (resembling the well-known pictures of fractals (Mandelbrot, 1983)) requires very little encoding. By contrast, rectangular GIF compression cannot handle such complex scenes well. This paper is proposing an architecture that reflects the much more sophisticated simplicity associated with fractals, and which moreover has a far deeper connection to both natural and artificial complex structures, and to our own perceptual mechanism.

 

HOW SELF-ORGANIZATION LEADS TO HIERARCHY

Complex biological and physical systems are usually organized into a hierarchy of scales (Miller, 1978; Simon, 1962; Smith, 1969). A hierarchical structure is the response to complexity for many, if not all, systems. By developing an appropriate hierarchical organization, a system organizes itself without losing its internal complexity. Its elements can interact in a more coherent manner through the creation of subsystems. Initially chaotic processes and components are stabilized through the creation of discrete hierarchical scales. All substructures are then linked internally, with both higher and lower scales interacting.

Systems differ in the amount of information required for their description, which is reflected in their complexity. This is an intrinsic property of the system being analyzed. Nevertheless, we observe many diverse and apparently unrelated systems - natural as well as artificial - evolving towards new degrees of order and stability, while obeying the same organizational rules. Disordered systems can spontaneously crystallize and acquire a high degree of order. The transition from chaos to order occurs through self-organization.

Order refers to any internal regularity and connectivity exhibited by a system. The stronger the correlations among the different components of a system, the more ordered a system is. Chaos on the other hand refers to a system lacking any kind of correlation among its components. Chaotic systems may be homogeneous or inhomogeneous. The criterion is connectivity, not appearance: in a highly ordered system, every component is related to all the other parts. Highly symmetric forms without internal connections are very weakly ordered because they are devoid of structural hierarchy.

Contemporary science confronts the problem of conceptualizing organized complexity directly. This task is very different from trying to understand complex phenomena by simplifying them. Macroscopic patterns either self-organize, as in crystals and biological systems (Kauffman, 1995), or they are man-made, as in computer programs and buildings (Booch, 1991). Architecture is not natural; man has to impose organizational rules on his own designs. The key is to imagine the process by which a form emerges. Entities that evolve must exhibit spatiotemporal continuity and at the same time establish internal boundaries and differentiations. In short, they must form individual subunits. Any irreversible process of organization operating on individualized entities produces a hierarchy (Simon, 1962).

Although this paper does not address the mechanism of the design process (see the last Section), we need to mention that design depends on the generation of complexity. Students imagine incorrectly that complexity is given, and all they have to do is to organize it. That is not the case: complexity in spaces and materials first has to be there in order to be organized. Entropy has to be allowed into the situation at many stages of the design process, and at each architectural scale (Salingaros, 1997a). This will broaden the spectrum of possibilities that the designer can choose from. The results depend a great deal on which way the complex field is generated initially.

 

DIFFERENT SCALES IN A DESIGN

What determines the visual and emotional impact of a building? Some factors, such as form and color, are obvious; just as important, though perceived only subconsciously, are its architectural scales. In a building (already built, or in the process of being designed), we can measure the size, x , of all clearly-defined substructures. Different situations might require different measures such as area, width, or length. Any unit of measurement (cm, inch, foot, or meter) may be used as long as features of all sizes are measured in the same unit. We have to estimate the size of curved sections, the idea being to group similar elements according to size.

All such measurements depend on clearly articulated differentiations of the structure on a particular scale. Distinct elements arise only by contrast against their adjoining elements and background. There are several means of achieving this: sharp differences in grayscale value or color hue; an outline; a change in texture and materials; relief; etc. For this reason, the background or boundary will also define an element, so most elements will occur in contrasting pairs. Vague articulations might work close-up, but are insufficient to define an element when viewed at a distance. Intentionally subtle design transitions work against the hierarchy by concealing or blurring the scales.

The size of similar repeating elements measures either their width or the length, depending on which is repeating. It frequently happens that two or more different types of elements have the same size but different characteristics, and these will define coincident scales. Elements could be aligned with translational, rotational, or reflectional symmetry, but that is not necessary for defining their own particular scale. An overall symmetry defines a new higher scale. (Elements related by a similarity transformation - scaled up or down in size - do not form a scale, because they don't have a measurement in common; this is instead a method for linking different scales together).

Scales have been defined with great care in the architecture and human artifacts of the last several thousand years. The concept of "modularity" is anchored in the fundamental need to perceive sharp architectural scales. A molding used throughout a building ties the entire space together through repetition. The possible scales in a building's hierarchy arise naturally from the physical structure, stresses, and materials. An architect chooses to include some scales in a building and exclude others, and the result is crucial to how a building is perceived. The greatest buildings (Parthenon; Hagia Sophia; Dome of the Rock; Palatine Chapel; Phoenix Hall, Kyoto; Konarak Temple, Orissa; Salisbury Cathedral; Baptistry, Pisa; Alhambra; Maison Horta, Brussels; Carson, Pirie, Scott department store; etc.) succeed in good part because they integrate their different subdivisions into a hierarchy of interconnected scales.

 

HIERARCHICAL SYSTEMS AND EMERGENT PROPERTIES

Having defined the different scaling levels of a structure, we establish a means of studying their interdependence. This topic has been developed in systems theory and complexity theory, with significant recent applications to computer science and biology (Kauffman, 1995; Mesarovic, Macko et al., 1970; Passioura, 1979). The general properties of hierarchical systems (also called layered systems) can be summarized as follows. These rules apply to any discipline that deals with complex structures, and to architecture in particular.

  1. Elements on a particular scale have their own type of interaction, independent of the other scales.
  2. Higher scales result from constraints (in the language of the higher scale) being imposed on lower scales.
  3. Interaction between scales is not symmetric: a higher scale requires all lower scales, but not vice versa, in order to function.
  4. Interaction across scales leads to correlations among all the different scales, and this creates a coherent whole.
  5. Emergent properties add new components of structure to a complex, organized system, making it "greater than the sum of its parts".

A lower scale has the smaller elements, and the higher scale has the larger elements. Life generates hierarchical systems that have observable organic structures (Miller, 1978; Passioura, 1979). Computer programs are hierarchical systems of information with distinct, interconnected scales that have to cooperate (Booch, 1991). The increasing complexity of man-made systems has made it necessary to organize them internally in some practical manner, simply in order to understand them (Mesarovic, Macko et al., 1970). As a consequence of their complexity, these totally artificial entities have evolved a structured hierarchy in common with natural forms, showing how the underlying rules are the same (Booch, 1991).

The significance of an element in a complex structure is clarified as we view it from different scales in the organizational hierarchy. The need for any given element will not be fully understandable on its own scale: it may be a necessary component for the structure on a higher scale (Mesarovic, Macko et al., 1970; Passioura, 1979). In an organized structure, every scale in the hierarchy contributes - with the downward dependence of larger on lower scales - yet the total effect is an effect of the system. A complex system does not depend solely on any single scale; neither can any scale of organization be neglected or eliminated. Each scale has its own particular goal, which is indirectly supporting the whole.

The whole represents something not found in the isolated parts alone. A hierarchy links elements together in ways they could not achieve on their own. When elements of one scale combine to form the next-highest scale, a new and in some ways unexpected structure emerges; this is referred to as an "emergent property" (Kauffman, 1995; Miller, 1978). Those elements combine into something novel, which is not explainable in terms of the lower scale. A more encompassing whole includes the contributions of all the lower scales, while adding its own organizational principle.

The higher scales of a hierarchical system depend on the proper definition of all lower scales. Every scale must work together and in the proper sequence if the whole is to function properly. A fundamental rule that governs all complex structures, organic as well as mechanical, is that all lower scales are necessary for the higher scales to work. In plant physiology, this explains the effect of a herbicide (Passioura, 1979). A chemical blocks the working of a lower scale, and that is sufficient to sabotage (and kill) the organism. Similarly, a lower-scale bug crashes a big computer program. The same might be said of viruses and germs for the animals.

 

THE EMOTIONAL IMPACT OF ARCHITECTURAL SCALES

How clearly architectural scales are defined, and how closely they correlate with each other, are unnoticed consequences of design decisions based on very different concerns. Contemporary design styles promote hierarchy reversal. (This has serious implications which will be discussed in a later section). Such buildings intentionally ignore the integrated, complex, scale hierarchies of natural structures, defeating the organizational process that generates complex coherent forms. In one approach, whole scales of articulation are removed. This minimizes complexity by trivializing forms. At the opposite extreme, both hierarchy and integration are prevented through randomization of substructures; this disorganizes the complexity so that it eludes human comprehension (Simon, 1962).

Either of the above extremes will result in a structure that lacks hierarchical coherence. Up until now, that has seemed a viable option justified by innovation; if one seeks a regular geometric shape then one might not want any internal structure. Indeed, an architect can be intuitively led to such an empty form by the desire to express a Platonic solid in its purest state; or merely to build a cheap building. We claim that that creates a source of anxiety and unease. Removing coherence from the environment changes it in a fundamental manner, and disturbs the psyche and well-being of the people in that environment.

There is mounting evidence from experimental psychology supporting this phenomenon, though this work is far from complete (Alexander, Ishikawa et al., 1977; Küller, 1980; Mehrabian, 1976; Sommer, 1974). One explanation is that the human mind is "hard-wired" to recognize natural and living forms by analyzing their hierarchical coherence. Any form that lacks those qualities creates alarm and so raises the adrenaline level. An unnatural, alien form attracts attention and uses up the brain's energy as it tries to figure out the form's internal organization. There is no steady-state coexistence with such a form: it goes against the ordering processes inherent in the mind, so it can never be experienced as visually (and psychologically) comfortable.

The model presented here is consistent with Gibson's theory of "direct perception" (Gibson, 1979; Michaels and Carello, 1981). According to this view, pattern perception does not result from a discrete sequence of steps which begins with information input, then processes the information according to different sets of criteria by turns; instead, patterns are perceived at the same time as they are seen. That is because the brain is closer to a massively parallel computer than to a sequential computer (Fischler and Firschein, 1987). The mechanism involves a kind of resonance established between the external structure and an internal structure of the cognitive system. Because this process is instantaneous, it is usually unnoticed by the observer. The criterion for cognition in this model is that our mind's internal structure resonates with the structure of the perceived object.

In practice, the organization of a building into distinct scales has an immediate emotional impact on the user. In the past, this effect was usually (though not consistently) positive. Nowadays, it tends to be negative. Often, the imposition of alien qualities on buildings is deliberate. An architect copies images that define a particular style by virtue of avoiding the integrated hierarchy of natural structures. Two different methods are used:

 

(a) Too large a gap between scales. This occurs when the natural substructures - intermediate in size between large forms and the smallest natural detail in the materials - are suppressed. The median and smaller scales are often missing (as from Le Corbusier's chapel at Ronchamp, and from most buildings by Mies van der Rohe). An exaggerated jump in scale is felt immediately, creating a strongly negative impression in the user. Architects who promote a minimalist style tend to favor materials such as glass and concrete that have no intrinsic substructure at any scale. They are then employed in such a way that no scales ever arise. Boundaries between obvious subsections are removed or camouflaged to prevent the subdivision of forms into discrete components.

 

(b) Scales that are too closely spaced. The opposite technique destroys the scaling hierarchy by blurring the distinction between scales. This is the inevitable result of too busy a design; one that includes many different non-matching elements of not quite the same size. Repetition and rhythm are prevented through random variations. (Examples include the Monastery of Sainte Marie de la Tourette by Le Corbusier, the Lloyd's building and Centre Pompidou by Rogers, and the unplanned commercial strip with billboards). Hierarchy is prevented when the scales themselves are indistinct, and also if they are defined but are randomly distributed. Some architects (such as Libeskind) plan "randomness" very carefully, but it occurs more often because of a lack of cooperation between different components.

 

For a building to be perceived as coherent, it needs a distribution of sharply defined architectural scales, and for those scales to cooperate hierarchically. Whereas shape is purely a matter of choice in design, its subdivision into a hierarchy of scales is not. Just as in biological systems, the smaller scales are relevant because they support and anchor the forms on the larger scales. Internal coherence is possible only if all the subunits cooperate. In architecture, a form or detail is irrelevant only if it doesn't integrate into the whole. While we find many examples of incoherent structures (in both traditional and modern styles) where large and small elements fail to correlate to each other, the greatest buildings depend fundamentally on their details.

 

VISUAL VERSUS FUNCTIONAL NEEDS

Today, it is very difficult to talk about the various architectural scales in design. The existence of intermediate and smaller scales is not even part of contemporary architectural thinking, so there is no vocabulary to treat them. We have only the words "form" for the highest scales, and "ornament" for the lowest scales. The word "ornament" is negatively loaded, since it has come to represent irrelevant structure added on for no obvious reason, and not an integral part of the whole. This prejudice hinders our analysis of architectural scales, in which every subunit in a building has its own independent validity.

Natural hierarchies arise from structural and functional reasons, and not to appear beautiful to us. We show our total misunderstanding of nature when we interpret hierarchical structure from our own, narrow viewpoint and declare it a visual style or effect. The marvelously complex hierarchy of a leaf or spider web has nothing to do with its perception by humans. Hierarchy is a fundamental structural rule that precedes mankind's emergence as a dominant species, because in nature hierarchy follows function.

This is obvious in much of architecture itself. For example, a Greek theatre is correctly subdivided into a hierarchy of cooperating scales, from its diameter down to the height of a step. Every one of those architectural scales has a functional basis. The ensemble happens to be beautiful, but that is after the fact. One can also find details that continue the scales downwards from 30cm (the height of a step) to 3mm. A modernist would argue that these smaller scales are irrelevant. Nevertheless, all the larger and intermediate scales arise out of functional requirements, and are therefore necessary, yet they are part of the same hierarchy as the smaller scales.

Architecture is a man-made complex system, and according to systems theory, the higher scales depend in an essential manner on all the lower scales. There is a range of scales, however, that is hard to justify from functional needs. These are the scales between 30cm and 3mm, and which exist in all traditional architectures as ornament (Alexander, Ishikawa et al., 1977). And yet, these scales - perceived as visually and emotionally necessary in their original creative context - are necessary in order for the system to achieve the emergent properties that give it its coherence. If we eliminate any architectural scales for which we can think of no obvious functional argument, then we deny the coherence of the structure as a whole.

Where does one determine the lower cut-off? I propose that there is no smallest scale; all scales must decrease by steps until they meet the natural texture of the materials, which is at the limit of visual perception. At present, architectural theory lacks this argument entirely, and as a consequence leaves design open to major errors. If an architect feels either justified, or obliged by style, to eliminate scales in the hierarchy, then it is inevitable that some scales necessary for the building's functions will not be included. I suggest that this century's abhorrence of hierarchical coherence has severely compromised the functions in buildings and urban regions.

 

SYMMETRIES GENERATE THE HIGHER SCALES

Higher scales can arise from the geometrical ordering of elements from a lower scale. Elements are aligned along a straight line or curve; or to form a pattern having a simple or complex symmetry. A constraint acts on the elements of the lower scale, and is expressed as symmetry in the language of the higher scale. Translations and rotations repeat a unit in a straight line or circle; reflections double a unit and tie it together with its reflection; and glide reflections move a unit, then reflect it (Washburn and Crowe, 1988). These act on each architectural scale at different sizes. Regularity in design is seen in buildings throughout history until the modern age. Out of an incredible wealth of possible symmetries, contemporary buildings may utilize only bilateral symmetry of rectangular forms, and restrict this to the largest scale.

Some architects disguise architectural elements, preventing structural integration through the use of empty surfaces; or they arrange elements in a way that avoids any symmetry. When elements are spaced or aligned randomly, the region becomes incoherent. It can still be incorporated into the hierarchy by a very wide boundary that itself has internal coherence, so that the two contrasting regions (inner and outer) couple. For this, the boundary has to be substantial enough to "tie" things together. The same effect integrates a large empty surface or space, as for example a Roman arch or Romanesque doorway. This solution is avoided by contemporary styles, which exclude anything that could act as an integrating boundary for any region.

Modernism narrowed the vocabulary of forms and symmetries down to minimal rectangles and reflections. Twentieth-century architectural education conditions students to see and think only in terms of bilateral symmetry of pure minimal shapes. There is no room for complex, hierarchical, internal symmetries in such a conception of the world. The reader should not interpret "symmetry" in this trivial and highly restricted context. That idea is founded on a misconception about the role of Platonic solids, as was explained in the Introduction. An effective antidote is to study the complex symmetries in Islamic and Romanesque architectures, and closer to our times, the buildings by Gaudí, Sullivan, Horta, and Guimard.

 

MONUMENTALITY AS DECORATION

Modernism intensified the largest scale through the use of reflectional symmetry, while at the same time suppressing the intermediate and smaller scales. This amounts to hierarchy reversal, and according to the modernists themselves, it was intentional (Bonta, 1979; Le Corbusier, 1927). There is no functional reason for a monumental maximal scale, as most of the functions of a building reside in the intermediate and smaller levels of scale. Nor is a large scale necessary from systems theory, because of the downward dependence of higher scales on all the lower scales, but not vice-versa. The functions on the intermediate and lower scales do not require a massive higher scale. Most useful buildings throughout history (with some notable exceptions) have a relaxed and complex overall shape. The Piazza San Marco is not symmetrical.

The exceptions are those cases where monumentality is desired, and is frequently achieved at the cost of internal functions. Such buildings express authority and power. While local government may decide to sacrifice some space to provide the people with a symbol of their civic pride, State Capitols in the United States do not often resort to hierarchy reversal. Both Wren and Lutyens used overall symmetry while avoiding gigantism and retaining hierarchy. Totally symmetric buildings are most often associated with megalomania in the client or architect. Examples include the pyramids; Fascist architecture; buildings around the world during the so-called "Heroic Age" of modernism; Canary Warf; La Grande Arche de la Défense, etc. Hierarchical systems theory shows us that symmetry on the largest scale is functionally and hierarchically superfluous.

 

METHODS OF COOPERATION AMONG DIFFERENT SCALES

I have described how individual elements of design define a scale, and also how each scale has its own identity. It is still necessary to discuss how the scales are made to cooperate. Several techniques specific to design guarantee the connection between distinct scales. First, coincident scales (two or more scales that have elements of the same size) link through contact and contrast, either in color or shape. The clustering of coincident scales helps in establishing contrast, which is an essential component of design coherence. Elements having complementary shape and color can alternate in one direction to provide rhythm, and this is seen in patterns and buildings throughout history.

Second, scales of different size can be linked by a scaling transformation or similarity: the higher scale elements are scaled-up versions of the lower scale elements. It is not necessary to duplicate the entire element; a portion of it will do, as long as the similarity is recognizable. Many fractal patterns are completely self-similar. That is, each scale is composed of similar elements so that the whole design is just a combination of an infinite number of scaled-down copies of the same generative element (Mandelbrot, 1983). It is for this reason that designs with fractal properties - such as natural scenery - are easily compressed by a fractal image compression program (see discussion in an earlier Section).

Third, explicit visual connections that provide a path for the eye to travel between two scales help. These can be any type of strong boundary that connects rather than separates two regions. A boundary can itself be subdivided to define a lower scale. Rather than apply formal rules of cooperation, however, we should in practice use the built-in capabilities of the human mind, which has evolved so as to recognize hierarchical coherence in natural forms. It should be obvious just by looking at two scales if they cooperate or not. This might not be all that easy to do if we are sidetracked by what we are taught to like; what is exciting; what reminds us of something else, etc. Nevertheless, this is still the most powerful and comprehensive method we have to judge coherence, and to achieve cooperation between scales in design.

 

THE IDEAL SCALING FACTOR

The scaling hierarchy is supported independently by the laws of organic growth, and by the mathematical theory of fractals (Salingaros, 1995; Salingaros, 1997b). Developing earlier, unpublished findings by Alexander (Alexander, 1998), I propose that consecutive scales in the hierarchy satisfy the same ratio given approximately by the constant e ~ 2.7, the base of natural logarithms. This corresponds very closely to the spacing of scales found in psychologically comfortable structures. Measurements of the most successful buildings - including the great buildings of the past, and vernacular architectures - reveal a discrete distribution of scales. If we plot the different scales on a logarithmic graph, we find an evenly-spaced distribution, with roughly one scale (or group of coincident scales) for each integer 1, 2, 3, etc.

On the basis of this rule, the room discussed as an example at the beginning of this paper fails because its scales are insufficiently differentiated, and many smaller scales are missing. Recall that the actual room's scales were at {400, 180, 75, 7, 0.3cm}. Ideally, a 4m room should have eight clearly-defined scales measuring approximately {400, 150, 50, 20, 7, 3, 1, 0.3cm}. Two stylistic features detailed earlier contribute to degrade the coherence of this room: (a) the elimination of the smaller scales at 20, 3, and 1cm; and (b) the almost complete lack of contrast (all the room is painted white).

The theory sketched out in this paper has a remarkable parallel in animal populations. Hierarchical concepts have found an especially fruitful application in the study of ecosystems and their evolution (Allen and Starr, 1982; Salthe, 1985). Ecosystems exhibit a quantization of sizes. Animals comprising a one-dimensional ecosystem define a discrete sequence, in which the mass of each different animal type increases geometrically. An ecosystem cannot support animals with body masses that are too close; on the other hand, a large gap in the distribution of body masses will be filled by some animal evolving either from above or from below. Plotting all the animals' weights on a logarithmic plot reveals a discrete evenly-spaced distribution (May, 1973). The actual scaling factor for animal body mass has been determined in one case to be roughly equal to 2 (Hutchinson, 1959). While body mass cannot be compared directly to the size of architectural elements, the distribution is indeed discrete, and depends on a fixed scaling factor.

 

PRACTICAL CONSIDERATIONS FOR DESIGN

This section addresses the needs of a practicing architect who wishes to apply the theoretical postulates set out in this paper. I am using unpublished results of Christopher Alexander, who is developing an approach to architecture that arises directly from scientific principles (Alexander, 1998). Here I will focus on defining the architectural scales, and suggest how to use them in establishing the scaling hierarchy. While this approach is straightforward, and at first sight can be seen as helping existing design practice, in fact it challenges much of what we have accepted unquestionably in the 20th century about good design.

If the architect is honest, a building of whatever size will have to arise out of the building's and the user's functional needs. These should determine much of the building's internal design, as well as portions of its external form. It is preferable not to start with a rigid pre-conceived "form" or "concept", but rather to allow the design to evolve from the inside out. We wish to avoid the problems that arise when a particular form is imposed onto a building or a site, because then all the functions are constrained to fit into this form, with varying and unpredictable degrees of success.

Materials have to be chosen for structural, functional, and climatic considerations, and not primarily for stylistic effect. Once those decisions have been made, the architect should put some effort into seeing how the materials and structural subdivisions by themselves create obvious architectural scales. The scales should be decided based on those possibilities, and then intensified by some (but not excessive) intervention. That is, one can adjust the sizes of elements so that they correlate, and add subdivisions that enhance an existing scale or create one that is missing altogether. At every stage, contrast is an essential tool for defining elements.

During the process of deciding on the scales, one ought to check them by computing their successive ratios. To achieve hierarchical coherence, the ratio should be roughly the same between scales, and not too different from 2.7 (any number from 2 to 4 is adequate). It is essential, moreover, that subdivisions be continued downwards so as to create the smallest scales near the limit of perception. If there is any texture in the chosen material surfaces, then the man-made scales ought to scale down to this scale of detail; otherwise, one can stop at 1cm or 3mm. Other than this, the architect has complete freedom in the overall design.

The technique alluded to here, and expanded in (Alexander, 1998), uses feedback from the structure at each stage of its construction in deciding what to do next. This presupposes a certain freedom in design that is not currently part of the building process, although it was previously so for several millennia. While making sure that the hierarchy is satisfied, the architect can visualize or "feel" what innovative design can bring the building to life. There is an infinite number of possibilities. Most often, the practical design will itself be aided by the hierarchical subdivisions. The architect takes on the role of an agent who organizes inanimate matter consisting of the functional spaces and building materials into a coherent whole.

 

ATTEMPTS TO FORMALIZE THE DESIGN PROCESS

There have been various attempts to formalize the design process in architecture and in other fields. In the 1960's, there was a flurry of activity applying mathematics and systems theory to architectural design, starting with Christopher Alexander (Alexander, 1964). Just as relevant is the work of Bruce Archer (Archer, 1970), Bill Hillier (Hillier, 1996; Hillier and Hanson, 1984), Christopher Jones (Jones, 1970), and Horst Rittel (Rittel, 1992). Sophisticated mathematical tools were utilized in setting up general models to handle the design process. The early work is summarized in (Broadbent, 1973). Many architects believe that this effort ended in failure, as systems analysis never entered into mainstream architecture. I will discuss this impression, and relate my own work to the pioneering work of the above authors.

It is true that the initial promise of a formal theory of architectural design based on mathematics never materialized. Nevertheless, such an approach found more fertile ground in engineering and programming, where it is now established as a major topic of research (Booch, 1991; Cross, 1989). Some of the above authors eventually expressed reservations that a comprehensive design theory was at all possible. My own work, which is based on Alexander's later - as yet unpublished - work (Alexander, 1998), does not attempt to formalize design. I offer a design constraint, and leave the actual design entirely up to the architect. This paper argued that failure to satisfy this constraint will result in a building that is perceived as alien, irrespective of its form.

In offering a universal constraint, I am in keeping with Hillier's prescription for a theory of design: "What is needed are theories ... that are as nonspecific as possible to particular solutions in the generative phases of design in order to leave the solution field as large and as dense as possible, and as specific and rigorous as possible in the predictive phases in order to be able to deal predictively with unknown forms where the need for effective prediction is greatest" (Hillier, 1996). Hierarchical coherence is necessary, but it does not dictate the form or design details. To the best of my knowledge, the explicit notion of hierarchical cooperation was not emphasized by other authors who applied systems theory to architecture.

Why did that body of work have little lasting impact on architecture itself? Though only a personal opinion, I believe that much of the work was just too theoretical and mathematical for architects. Architecture schools stopped teaching mathematics several decades ago, and architecture students are no longer required to learn science. Practicing architects tend to be visually oriented, working from intuition and not from formal rules. It is extremely difficult to communicate a method unless it can be done so on intuitive terms. Also, the application of those early theories did not lead to any radical improvement of buildings, even in test cases. The formalism was useful in comprehending the complexity of the problem, but was incomplete as a practical design tool.

 

CONCLUSION

The self-organization of complex systems inevitably leads to hierarchy. We are able to understand the world precisely because it is hierarchical; those aspects of it which are not elude our understanding and observation. This paper applied the rules of hierarchical systems, developed to describe any complex structure, to aspects of architectural design. Systems theory relates the organizational mechanisms underlying design to analogous processes taking place in biology, physics, and computer science. Cast in this scientific setting, architecture can profit from results already established in other disciplines. This framework provides a way in which to derive practical design rules based on universal principles.

Such rules do not impose a style; they work on the fundamental scale of the basic design elements, and are satisfied by the greatest historical buildings and vernacular architectures. Style is a matter of choice, but architectural order is of profound importance to the human experience. Buildings with a quantized distribution of scales predominate until we come to the 20th century, when the architectural scales are either suppressed, or are distributed randomly. This result has a remarkable parallel in population biology: the laws governing the distribution of architectural scales in a building are analogous to the laws governing the size distribution of animals in an ecosystem.

 

ACKNOWLEDGMENTS: This work utilizes and expands on the ideas of Christopher Alexander. I have profited from working with the unpublished text of "The Nature of Order". I am also very grateful to John Miller for bringing some important references on hierarchical systems to my attention, and to Michael Benedikt for many useful suggestions.

 

 

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