HIERARCHICAL COOPERATION IN ARCHITECTURE, AND THE MATHEMATICAL NECESSITY FOR ORNAMENT. * * *Nikos A. Salingaros* Division of Mathematics, University of Texas at San Antonio, San Antonio, Texas 78249. E-mail: salingar@sphere.math.utsa.edu /Journal of Architectural and Planning Research/ volume *17* (2000), pages 221-235. © Locke Science Publishing Company, Inc.; posted by permission. ------------------------------------------------------------------------ The case is made that architectural design needs to be organized hierarchically. A method and formula for doing so is derived based on biology and computer science. Fractal simplicity, in which there is self-similar scaling, replaces the outdated notion of rectangular simplicity. Architectural units on different scales are able to cooperate in an intrinsic manner to achieve an emergent property, which is not present in the individual components. The theory of hierarchical systems explains 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 its actual design. This paper gives a scientific proof of why ornament is essential to the overall cooperation of architectural forms, thus revising one of the basic tenets of modernist design. ------------------------------------------------------------------------ Table of Contents: * Attempts to formalize the design process * Forms with and without hierarchical subdivisions * Architectural scales * Simplicity, fractals, and image compression * Different scales in a design * The emotional impact of architectural scales * Hierarchical systems and emergent properties * The mathematical necessity for ornament * Symmetries generate the higher scales * Methods of cooperation among different scales * Practical considerations for design * The ideal scaling factor * Conclusion ATTEMPTS TO FORMALIZE THE DESIGN PROCESS There have been various attempts to formalize the design process in architecture and in other fields. In the 1960s, there was a flurry of activity applying mathematics and systems theory to architectural design, starting with Christopher Alexander (Alexander, 1964). Also 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 today believe that this effort ended in failure, as systems analysis never entered into mainstream architecture. 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. The present paper, which is based on Alexander's later -- as yet unpublished -- work (Alexander, 2000), does not attempt to formalize design. We offer a design /constraint/, and leave the actual design and all its details entirely up to the architect. In offering a universal constraint, we are 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: p. 68). Hierarchical cooperation is necessary, but it does not dictate the form or design details. To the best of the author's knowledge, the explicit notion of hierarchical cooperation was not emphasized by other authors who applied systems theory to architecture. Why did that early body of work have little lasting impact on architecture itself? It could be that much of the work was just too theoretical for architects. Architecture schools stopped teaching mathematics decades ago, and architecture students are no longer required to learn science. Practicing architects tend to be visually oriented, working from images and not from formal rules. It is 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. Modernism offers a remarkably simple universal constraint on built form, which is easy to apply to any type of building. This is the root of its success, more so than any political or philosophical ideology. Its down side is that it is highly specific in the initial design phase, and narrows the solution space considerably (Salingaros, 1997). Even critics of modernism, however, have to admit that contending theories and styles hoping to displace the modernist style have not offered anything so simple. The present theory of hierarchical cooperation is slightly more complicated than pure, rectangular forms. That should be no problem in our computer age, and could indeed provide a challenge to innovative architects wanting to build according to the spirit of the times. FORMS WITH AND WITHOUT HIERARCHICAL SUBDIVISIONS In the early 1920s, a preference for unadorned Platonic solids -- such as simple cubes, triangles, spheres, etc. -- was established as one of the principles of the new architecture (Le Corbusier, 1927). Many people at the time assumed that regular shapes are somehow ingrained within the human consciousness, so that the mind is programmed to prefer them. We now know that this is false (Bonta, 1979). Human beings have to be trained to recognize Platonic solids, which are a purely intellectual concept (Fischler and Firschein, 1987; Zeeman, 1962). What /is/ built into the human consciousness is a recognition mechanism based on hierarchical subdivisions, irrespective of the structure's overall shape. The preference for Platonic solids eventually became part of this century's architectural tradition. In fact, one may argue that modernist architecture owes its success to the fact that Platonic solids are almost never seen in nature as large-scale macroscopic forms. A building with a pure, abstract shape therefore contrasts with the natural environment and stands out. 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 has built unnatural structures such as the pyramids precisely so as to assert man's dominance over nature. Forms that lack hierarchical cooperation can appear exciting by virtue of their being unfamiliar. That feeling, however, is not the emotional elation experienced inside a Medieval Cathedral. We fail to distinguish between an uneasy, distressing excitement, and a deeply satisfying, nourishing visual excitement -- which are totally different in their psychological effect -- and to see how they are produced in buildings. Environmental psychology clearly distinguishes the two cases (Nasar, 1989), yet confusion still reigns in this topic, which provides the basis for architecture's impact on people. Both modernist and classical buildings are rationally understandable, and visually and emotionally evocative, but in almost opposite ways. An architect chooses whether to follow or to violate certain conceptual rules that relate a building to natural forms. This paper proposes a design constraint that comes from systems theory and leads to hierarchical cooperation. It requires only that forms be subdivided in a certain manner, and that the subdivisions be made to relate to each other. All complex systems -- natural as well as artificial -- have distinct scales that 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 logic, starting with three observations: 1. Inanimate natural forms and biological organisms are complex systems that are hierarchically organized. 2. Systems in engineering and computer science have been found to follow the same hierarchical organization rules as natural systems. 3. 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. Architecture creates complex artificial systems. We believe that, in order to achieve emergent properties, architecture must organize matter according to hierarchical system rules. Evidence for this conclusion will be presented throughout this paper, yet the truth of our claim is evident to any observer. If we can detach ourselves from our biases, which are culturally based, then we can check our favorite objects, buildings, and designs. Buildings and urban regions that are hierarchically organized resonate with our own built-in perceptual mechanism. 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. The ratio and degree of cooperation between successive scales determines whether the ensemble is hierarchically organized or not. 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 a coherent whole. This subconscious process might well determine a building's impact independently of such traditional concerns as shape, form, and proportion. Independent support for this theory comes from the science of cognition, though experimental psychology is not used as the primary argument here. There are two reasons for this. 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 (as manifested by the mental completion of missing regions of regular figures) with a non-existent predilection for Platonic solids. ARCHITECTURAL SCALES 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. Material stresses create fractures that show as regular patterns, thus preventing a long-range ordering from continuing throughout macroscopic forms (Smith, 1969). Smoothness and uniformity are alien to natural materials. In nature, structural features exist on different levels of scale, from the macroscopic to the microscopic, through all intermediate scales. We see physical forms that have hierarchical scaling as a result of internal and external forces. Biological forms also exhibit a definite scaling hierarchy. 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 units define a scale. The scales are distinct, yet they are also nested in a complex structure. The same is true for built forms. Architectural scales are defined by similar units of the same size that repeat. Independent scales arise from the materials, structure, and functions, and express an architect's ideas. Methods used throughout the history of architecture to define scales include symmetry, as manifested through shape, fenestration, and columns. For example, 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. Subdividing a window into panes creates a smaller scale. Massing and monumentality define the largest exterior 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). 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 units cooperate 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 (urban spaces, buildings, artworks, artifacts, tools, and machines) are hierarchically integrated. They achieve a balance between different design units according to their size. Our objective is to cast this process into scientific terms so that it can be applied consciously and deliberately. */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. 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 the author's 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 /x_i / at {400, 180, 75, 7, 0.3cm}. It is instructive to compute the ratios /x_i /_+1 //x_i / between consecutive decreasing 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, the magnitude of the last two numbers reveals 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, a hierarchical explanation is proposed. Each surrounding building has subdivisions at roughly 1/3 its overall size, and further subdivisions at roughly 1/7, 1/20, etc. Richly-articulated detail is evident from all directions. The plaza itself is subdivided through the use of contrasting floor paving (Moughtin, /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 relate to any other structure through scaling or similarity, because it has minimal features and subdivisions. The arch itself has very few distinct scales; by far not enough components to connect the structure internally through its design subdivisions, 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 it is imposing, monumental, and under favorable weather conditions exciting; nevertheless, the entire range of human scales is missing. A viewer cannot avoid feeling isolated. One searches in vain for hierarchical subdivisions. This structure is meant to impress and intimidate, but not to relate to human beings. SIMPLICITY, FRACTALS, 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 levels of scale. The more design subdivisions, the more scales there are. Human beings have a basic need to organize complex distributions of units into hierarchies as a means of avoiding information overload. The mind groups similar units of the same size into one scale (Fischler and Firschein, 1987). It then looks for 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 overturned our understanding of simplicity. This is of the utmost relevance to architecture, and is best discussed in the context of mathematical image compression: /an image is simple if it requires the least amount of information to specify/. There are many different ways of encoding an image, and we show how two of them imply radically different notions of simplicity. The first method, called the "Graphics Interchange Format" (GIF), is commonly used for storing pictures electronically 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 newer, more powerful 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 of the same size 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 across 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. 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 proposes 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. 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 units according to size. All such measurements depend on clearly articulated differentiations of the structure on a particular scale. Distinct units arise only by contrast against their adjoining units 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 a unit, so most units will occur in contrasting pairs (Salingaros, 1995). Vague articulations might work close-up, but are insufficient to define a unit when viewed at a distance. Intentionally subtle design transitions work against the hierarchy by concealing or blurring the scales. The size of similar repeating units measures either their width or the length, depending on which is repeating. It frequently happens that two or more different types of units have the same size but different characteristics, and these will define coincident scales. Units 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. (Units 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). Levels of scale 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 from the physical structure, materials, and the need to accommodate stresses. Moreover, the result is crucial to how that 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; Carson, Pirie, Scott store; etc.) succeed in good part because they integrate their different subdivisions into a hierarchy of interconnected scales. THE EMOTIONAL IMPACT OF ARCHITECTURAL SCALES How clearly architectural scales are defined, and how closely they correlate with each other, are now consequences of design decisions based on very different concerns. Contemporary design styles promote hierarchy reversal. 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 the 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 cooperation. Up until now, that has seemed a viable option justified by innovation. In the first case, if one seeks a pure, regular geometric shape, then one does not want any internal structure. An architect is intuitively led to an empty form by the desire to express a Platonic solid in its purest state; or merely to build a cheap building. This creates a source of anxiety and unease. Removing hierarchical cooperation from the environment changes it in a fundamental manner, and affects the emotional and physical state of the people in that environment. There is mounting evidence from experimental psychology supporting this phenomenon, though this work is far from complete (Alexander, /et al/., 1977; Küller, 1980; Mehrabian, 1976; Sommer, 1974). One explanation is that the human mind is programmed to recognize natural and living forms by analyzing their hierarchical cooperation. 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 that view, pattern perception does not begin with information input, then follow with steps that 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 an external structure and the internal structure of our cognitive system. Because this process is instantaneous, it is usually unnoticed by the observer. The organization of a building into distinct scales (or lack thereof) has an immediate emotional impact on the user, which would tend to support the Gibson theory of cognition. In the past, this effect of direct perception usually -- though not consistently -- created a definite positive emotional state. Nowadays, the reaction to most new buildings tends to be negative. The imposition of alien qualities on buildings, however, is quite deliberate. Contemporary architects copy images that define a particular style by virtue of avoiding the integrated hierarchy of natural structures. Three different methods are used, which may be summarized as follows: / / /(a) Too large a gap between scales/. This occurs when substructures intermediate in size between large forms and the smallest natural detail in the materials are suppressed. In buildings that depend on very fine detail, the median and smaller scales are often missing. An exaggerated jump in scale is felt immediately, creating a strongly negative reaction in the user. Boundaries between obvious subsections are usually removed or camouflaged to prevent the subdivision of larger forms into discrete components. / / /(b) Elimination of the smaller scales/. Architects who promote a minimalist style tend to favor amorphous materials such as glass and concrete that have no intrinsic substructure at any scale. These are then employed in such a way that no smaller scales ever arise. A building is allowed to have only very few scales, all of them large. Brutalist use of concrete removes the bottom end of the scaling hierarchy, leaving a user without anything to focus on at arm's length. / / /(c) Scales that are too closely spaced/. Blurring the distinction between scales destroys the scaling hierarchy. This results from too busy a design; one that includes many different non-matching units of not quite the same size. Repetition and rhythm are deliberately prevented through random variations. Hierarchy is prevented when the scales themselves are indistinct, and also if they are defined but are randomly distributed. Some architects plan "randomness" very carefully, but it occurs more often through a lack of cooperation among different components. For a building to be perceived as coherent, it needs a distribution of sharply defined architectural scales that 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 cooperation is possible only if all the subunits cooperate. 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 units fail to correlate to each other, the greatest buildings depend fundamentally on their details. HIERARCHICAL SYSTEMS AND EMERGENT PROPERTIES Having defined the different levels of scale in a structure, we now establish a means of studying their interdependence. This topic has been extensively developed in systems theory and complexity theory, with significant recent applications to computer science and biology (Kauffman, 1995; Mesarovic, /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 according to the thesis of this paper, also to architecture. 1. Units on a particular scale have their own type of interaction, which is independent of those in the other scales. 2. Higher scales result from constraints (expressed in terms of the higher scale) being imposed on lower scales. 3. The interdependence of scales is not symmetric: a higher scale requires all lower scales in order to function, but not vice versa. 4. Interaction across scales leads to correlations among all the different scales, and this process generates 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 units, and the higher scale has the larger units. Life generates hierarchical systems as observable organic structures (Miller, 1978; Passioura, 1979). Computer programs are hierarchical information systems 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, /et al/., 1970). As a consequence of their complexity, these totally artificial entities have evolved a structured hierarchy that has many common features with natural forms, showing how the underlying rules are the same (Booch, 1991). The significance of a unit in a complex structure is clarified as we view it from different scales in the organizational hierarchy. The need for any given unit may not be fully understandable on its own scale: it could be a necessary component for the structure on a higher scale (Mesarovic, /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 complex whole represents something not found in the isolated parts alone. A hierarchy links units together in ways they could not achieve on their own. When units of one scale combine to form the next-highest scale, a new and in some ways unexpected component of the total structure emerges; this is referred to as an "emergent property" (Kauffman, 1995; Miller, 1978). Those units 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. This point is really at the heart of our contribution, and links architecture with the new science of nonlinear phenomena. The assumption of nearly all 19th century mechanistic physics is that a complex system can never be more than the sum of its parts. In the last few decades, however, we have discovered a score of important phenomena that have emergent properties: properties of the system as a whole which cannot be traced back to any of its isolated constituent parts (West, 1997; West and Deering, 1995). That is, many complex systems are irreducible. The present paper argues that the greatest architectural achievements are in fact characterized by emergent properties. Furthermore, our discussion is aimed at understanding what is happening so that it can be successfully applied to new structures. Note that 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 MATHEMATICAL NECESSITY FOR ORNAMENT Natural hierarchies arise from structural and functional reasons, and not to appear beautiful to us. We show a misunderstanding of nature if 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: those structures existed long before human beings did, and actually helped to influence the way our mind developed. 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 scaling downwards from 30cm (the height of a step) right to the smallest limit of visual perception. A modernist might 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/. According to systems theory, the higher scales depend in an essential manner on all the lower scales. If we eliminate any architectural scale for which we can think of no obvious functional argument, then we deny the coherence of the structure as a whole. 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, /et al/., 1977). And yet, these scales -- perceived as visually and emotionally correct in their original creative context -- are necessary in order for the man-made complex system to achieve the emergent properties that give it its coherence. Where does one determine the lower cut-off? Careful thought shows that there really 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 (Salingaros, 1998). At present, architectural theory lacks this argument entirely, and as a consequence leaves design open to major gaps. If an architect feels either justified, or obliged by the prevailing style to eliminate any scales in the hierarchy, then inevitably some scales necessary for the building's coherence will not be included. The twentieth century's neglect of hierarchical cooperation has severely compromised the coherence of buildings and urban regions. SYMMETRIES GENERATE THE HIGHER SCALES Higher scales arise from the geometrical ordering of lower scale units. They could be aligned along a straight line or curve; or to form a pattern having a simple or complex symmetry. A constraint acts on the units 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 different architectural scales, as seen in buildings throughout history. Out of an incredible wealth of mathematical symmetries, contemporary buildings tend to utilize only bilateral symmetry in rectangular forms, and restrict this to the largest scale. There is no functional reason for symmetry on the maximal scale, as most of the functions of a building reside in the intermediate and smaller levels of scale. Nor is a prominent large scale necessary from systems theory, because of the downward dependence of higher scales on all the lower scales, but not vice-versa. Buildings throughout history (with some notable exceptions) have a relaxed, complex overall shape. The Piazza San Marco is not symmetrical. The exceptions are those cases where monumentality is desired, and it is frequently achieved at the cost of internal functions. Such buildings express authority and power, and some intensify the largest symmetric scale by suppressing the intermediate and smaller scales altogether. Examples include the pyramids; Fascist architecture; buildings around the world during the so-called "Heroic Age" of modernism; Canary Wharf; La Grande Arche de la Défense, etc. Some architects disguise architectural units through the use of empty surfaces, thus preventing hierarchical integration; or they arrange units in a way that avoids any symmetry. When units are spaced or aligned randomly, that region becomes incoherent. It can still be incorporated into the hierarchy by a very wide boundary that itself has internal cooperation, so that the two contrasting regions (inner and outer) couple. For this to happen, the boundary has to be as large as the region it bounds (Alexander, 2000). The same effect integrates a large empty surface or space -- as for example a Roman arch or Romanesque doorway -- into the hierarchy. This solution is avoided by contemporary styles, which shrink frames to an ineffective size for what they enclose, thus excluding anything that could act as an integrating boundary for a region. METHODS OF COOPERATION AMONG DIFFERENT SCALES Individual units of design define a scale, and each scale has its own identity. It is still necessary to discuss how the scales are made to cooperate. Specific coupling techniques guarantee the connection between distinct scales. First, coincident scales (two or more scales that have units of the same size) link through contact, and contrasting color or shape. The clustering of coincident scales helps in establishing contrast, which is an essential component of design coherence. Units having complementary shapes and colors can alternate in one direction to provide rhythm, and this is seen in patterns and buildings throughout history (Salingaros, 1995; Washburn and Crowe, 1988). Second, different scales can be linked by similarity: the higher scale units being scaled-up versions of the lower scale units. It is not necessary to duplicate the entire unit; a portion of it will do, as long as the similarity is recognizable. Many fractal patterns are completely self-similar. That is, the units of different scales are similar to each other, so that the whole design is just a combination of an infinite number of scaled-down copies of the same generative unit (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, the relative abundance of units on each scale leads to a mathematical link between two scales. This scaling is apparently perceived as a visual balance between the scales. Elsewhere (Salingaros and West, 1999), we have used entropy arguments to derive an inverse-power law scaling for architectural units. That work shows how the multiplicity of units on each different scale leads to cooperation among the scales themselves. The same process is apparently ubiquitous in both natural and man-made phenomena, including DNA sequences; laws of economics; the evolution of ecosystems; word frequency in linguistics; and the population of cities (West and Deering, 1995). Rather than apply formal rules of cooperation, we should in practice use the built-in capabilities of the human mind, which has evolved precisely so as to recognize hierarchical cooperation in natural forms. It is obvious just by looking at two scales in a structure if they cooperate or not. We might be sidetracked by what we are taught to like; what is exciting; what reminds us of something else, etc. Nevertheless, this direct visual method is still the most powerful and comprehensive method we have to judge cooperation, and to achieve cooperation between design scales. PRACTICAL CONSIDERATIONS FOR DESIGN This section suggests how to establish the architectural 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. The architect's role is to organize inanimate matter consisting of the functional spaces and building materials. It is not enough to put materials together, or to follow some style defined through visual images. A complex coherent whole that has emergent properties can be achieved only via hierarchical organization. The technique alluded to here, and expanded in (Alexander, 2000), 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. At every point, the architect has to visualize what step will enhance the hierarchical cooperation. A building process that is too rigid -- such as those in widespread use today -- does not permit the evolution of coherence that is necessary for emergent properties. This is the only way we can approach the quality inherent in the great buildings of the past, without having to copy their form or details. The building's and the user's functional needs should determine its internal design, as well as portions of the external form. A rigid preconceived design imposed onto a building or a site constrains all the functions to fit into that particular form, with varying and unpredictable degrees of success. Furthermore, a simplistic overall design prevents the emergence of an architectural hierarchy. The process by which cooperation emerges is summarized by Alexander as: every time a scale is created, it must link both to a higher scale and to a lower scale (Alexander, 2000). The conditions for achieving hierarchical cooperation in built structures therefore depends on two design criteria: (1) the ability to define architectural scales; and (2) creating them so that they link via the rules for complex systems. Materials and surfaces are chosen primarily for structural, functional, and climatic considerations, and not exclusively for stylistic effect. The materials and structural subdivisions by themselves create obvious architectural scales, which can be intensified by some (but not excessive) intervention. Remember that contrast is an essential tool for defining units. One adjusts the sizes of subunits so that they correlate with each other, and contribute to the hierarchy. It will be necessary to add subunits that enhance an existing scale or create one that is missing altogether. At every stage, the fundamental rule stated in the previous paragraph guides the whole towards an emerging internal coherence (Alexander, 2000). The design process will itself be facilitated by paying attention to the hierarchical subdivisions. During the process of deciding on the scales, one checks them by computing their successive ratios. To achieve hierarchical cooperation, the ratio should be roughly the same between consecutive 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 continue down to this level of detail; otherwise, one can stop at 1cm or 3mm. Other than this, the architect has complete freedom in the overall 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, 1998). Developing earlier, unpublished findings by Alexander (Alexander, 2000), we 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. Plotting 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 Example B at the beginning of this paper fails because its scales are insufficiently differentiated, and many of the 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} (Salingaros, 1998). Two stylistic features detailed earlier contribute to degrade the cooperation 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 evolution of ecosystems (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 units, the distribution is indeed discrete, and depends on a fixed scaling factor. The analogy with ecosystems can be used to illustrate a point in a dramatic manner: if one level of a functioning hierarchy is removed, then all the higher levels will die out. To avoid this, there will have to be a drastic rearrangement of the hierarchy, with the evolution of existing levels either up or down so that an ordering is reestablished. We have argued that a building's coherence is severely compromised by the removal of a single architectural scale from its hierarchy. The crucial difference between architecture and ecology is that man accepts incoherent structures, whereas nature does not. We can build while violating the scaling hierarchy, but ecosystems are ruthlessly efficient at eliminating incomplete units. CONCLUSION The organization of complex systems inevitably leads to hierarchy and emergent properties that are not present in the individual components. 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 scientific principles that go much further than present-day architectural theories. Such rules do not impose a style; they work on the fundamental level of basic design units, 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. Those practices, originally introduced as innovative, in fact prevent the emergent properties that characterize the most coherent known structures. 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 The author's research into the scientific laws of architecture is supported in part by a grant from the Alfred P. Sloan foundation. 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