ARCHITECTURE AND HUMAN PERCEPTIONS

Architectural design can be founded on scientific principles that are analogous to structural laws in theoretical physics and biology. An architectural theory developed by Christopher Alexander applies to create novel forms, while encompassing most traditional architectural styles as well (Alexander, 1996). A set of rules presented in (Salingaros, 1995) summarizes some of Alexander's results. By adopting them, the development of architectural forms can be divorced from specific images that define a particular style.

Human beings possess a basic instinct about forms that is linked to our ability to visually discern potential dangers in our surroundings. Certain mathematical relationships implicit in a building's form, or their absence, will trigger either a positive or negative subconscious response. The presence of essential mathematical harmonies is perceived instinctively, and is emotionally fulfilling - this is the foundation of much of religious architecture. For several millennia, architecture was based on the user's emotional comfort. In our time, however, formal design criteria have taken priority over human feelings, so that many buildings feel unpleasant yet are admired on an intellectual level (Sommer, 1974).

This paper offers a design rule that links human perceptions to geometry and materials (Alexander, 1996; Salingaros, 1995). Using this rule in design gives a result that feels instinctively closer to a natural structure than when the rule is violated. Our objective is to connect buildings to human beings by means of a building's intrinsic design and internal subdivisions. This concerns a large aspect of architectural design: the exterior and interior appearance, and details. The proposed rule says nothing about the plan or overall shape, nor the spaces in a building. It does dictate the subdivisions of the elevation, and it explains our connection to natural materials.

Scaling coherence is achieved by two separate processes. (a) A discrete hierarchy of different scales follows from physics and biology. (b) All the components in the hierarchy are connected - forms are related on each individual scale, and an overall coherence is created by linking forms on the small scale to forms on the large scale. How forms achieve coherence is discussed in detail in (Alexander, 1996; Salingaros, 1995; Salingaros, 1997). This paper is devoted to establishing the scaling hierarchy; the problem of connecting the elements is left to the last section, which includes a helpful checklist intended as a self-contained practical guide for architects.

We believe that contemporary architecture can benefit from the approach presented here. Nevertheless, certain design conventions used today violate the proposed rule, so we need to examine why contemporary design generates the particular forms it does. The use of a scale model in design decisions is singled out for special criticism, because it loses the scaling hierarchy. Suggestions are offered on what changes in the design process should be made in order to implement this program.

THE NATURAL SCALING HIERARCHY

Nature connects to human consciousness through forms and colors, and also via a little-noticed scaling rule. Most natural objects exhibit a hierarchy of scales, starting from their largest dimension, down by approximately factors of 2.7 to the smallest perceivable differentiation. Say a 15cm rock or leaf has smallest details at 1mm. It might have additional visible differentiations roughly of sizes 3mm, 7mm, 2cm, and 5cm. On the other hand, the smallest perceivable scale of a mountain could be 1km, depending on how far away you are standing. The entire mountain could be 3 or 7 or 20 times the smallest visible detail.

An object with scaling coherence has differentiations starting from its largest dimension and decreasing by factors of approximately e = 2.7 down to the smallest perceivable size. Buildings that obey this rule share a coherence with natural forms. (The scaling rule is only one of several requirements for a design to appear "natural"). On the other hand, buildings that violate the scaling rule are perceived as lacking an essential quality. Recently, both architects and scientists have realized that architectural design can be understood in terms of the same laws of complexity as natural systems, although the exact connection has been elusive (Halliwell, 1995).

A building of 20m height should have well-defined structures that are roughly 7m, 3m, 1m, 30cm, 10cm in size, down by factors of 2.7 to the smallest perceivable scale. When using traditional building materials and methods, the limited strength of the materials helps to generate these approximately correct subdivisions, though many architects intentionally suppress the intermediate scales. Stronger modern materials make it easier to avoid the scaling hierarchy entirely. If one does not incorporate the correct subdivisions as part of the design, then the structure is perceived subconsciously as contrasting with natural forms (Alexander, 1996).

Obviously, any man-made structure that does not explicitly copy a natural structure will contrast with natural forms. It is our contention, however, that there is a much deeper connection that follows not from the appearance, but from the underlying mathematical structure. If the latter corresponds to that of natural forms, then a man-made structure will be perceived as natural, even when its appearance is clearly artificial. The great cathedrals do not resemble any natural form, yet we feel perfectly at ease with their structure. We believe that humans have a built-in need for structures with natural scaling.

Previous applications of mathematics to design theory have little in common with the approach of this paper. We apply modern mathematics such as similarity transformations and fractals, and do not use simple ratios or fixed modules. The Golden Ratio Phi = 1.618 was widely used in the past to define the proportions of rectangular forms (von Meiss, 1991). That determines either the overall form or the plan of a building, both of which are of secondary importance in our theory (which focuses on the immediate connections to forms and surfaces). Scaling governs the internal subdivisions of forms. The overall shape and plan are left free by our system, to be determined entirely by a building's practical and aesthetic requirements.

According to Mike Greenberg (Greenberg, 1995a; Greenberg, 1995b), the most fruitful advance in this work may be the choice for the scaling factor as the irrational number e = 2.718, the base for natural logarithms. Modular systems of design -- typical of the Modernist ethic and modern technology, but also of traditional Japanese architecture -- are based on integral multiples of some basic dimension. Fully modular systems tend to have a scaling factor of exactly 2 or 3. The use of e as a scaling factor prevents the rigidity and monotony that are often the consequences of a modular system. One cannot simply repeat a basic unit, but has to define distinct forms at every new scale.

THE ORIGINS OF SCALING IN NATURE

A building's design consists of components having different sizes, some of them repeating to cover a larger area. These are our observables, which define the perceivable ensemble. In quantum mechanics, observable energy states are quantized and not continuous. Architectural observables representing lengths and areas are also quantized. A common feature of all natural forms is the existence of distinct scales. Pure natural forms without subdivisions on a macroscopic scale are extremely rare; most examples around us are man-made. The reasons for this are many and varied, yet the result is universal.

Material fractures due to stresses and strains create a hierarchy of discrete scales in solids. Even in fluids, homogeneity is not possible, as moving fluids generate hierarchical substructures due to turbulence. For entirely different reasons, scaling is necessary for biological forms. Life is the result of complicated chemical and physical connections occurring at many different scales simultaneously. Metabolic and mechanical processes characteristic of living forms require a nested hierarchy of different structures. Biological forms therefore exhibit a discrete hierarchy of interconnected scales.

Suppose that a design has several distinct sizes of objects x_n , with the n-th scale defined by one or more objects of similar size to within a 20% spread. These units are identified by a definite shape, or by a clearly-defined edge. We can rank-order the components of a design from the smallest to the largest. Different components might be totally dissimilar; only their dimension is relevant at this initial stage. The simplest mathematical relation between them is for each size x_n to be related to the next largest size x_n+1 by the same ratio k . Such a relationship gives an explicit formula for the size of the n-th component, with k a constant scaling ratio, and x₀ the minimum size. We have a power law with successive powers kⁿ , n = 0, 1, 2, ... of the same parameter k :

x_n+1/x_n = k implies x_n = kⁿx₀, k = constant

(1)

A design with subdivisions obeying the scaling law (1) will have the following property: some (though not necessarily all) parts of the design will have non-trivial structure when magnified by a factor of k . This process can be repeated n times to look at increasingly smaller details. This scaling property is most clearly shown in fractals, which exhibit some structure on every scale (Mandelbrot, 1983). An additional property of self-similar fractals is that successive magnifications have a similar design. The fractal properties of natural forms are becoming increasingly evident (Mandelbrot, 1983).

DETERMINING THE SCALING FACTOR

The validity of the above power law (1) is independent of the value of k ; further arguments establish the scaling factor k as being approximately equal to e = 2.7. Scaling coherence is a fundamental component of structural morphology in both inanimate and living forms, and is found in all traditional architectures. Alexander originally established the scaling hierarchy phenomenologically by measuring internal subdivisions in buildings, man-made artifacts, natural structures, and biological forms (Alexander, 1996). He gives a figure for the scaling factor k as being somewhere between 2 and 3 (Alexander, 1996). We propose a single working value for the scaling factor k = e = 2.7, based on the law of organic growth.

Exponential growth arises naturally if the change in size is equal to the size at that point. The relation dx/dt = x for the quantity x at any time t has solution x = x₀ exp(t) , where x₀ is the initial value. The values of an exponentially-growing quantity x at integer time intervals t_n = n obey the power scaling law (1) with k equal to e . Exponential growth is a fundamental law of nature. The growth of bacteria and ideal animal populations obeys x = x₀ exp(at), a = constant. The same rule holds true for the shape of sea-shells and horns, whose form is described by an exponential curve in polar coordinates {r, t} as r = exp(at) , a = constant, known as the logarithmic spiral (Thompson, 1952).

Scaling coherence depends on the levels of scale being close enough to relate to each other, yet not so close that the difference in scale is indistinct. A scaling factor k less than 2 fails to sufficiently distinguish between different levels of scale, so there is no discrete hierarchy. For example, a scaling hierarchy based on the Golden Ratio Phi = 1.618 (such as Corbusier's Modulor) has its subunits just too close in size, so they define more of a continuous gradient rather than discrete scales. If, on the other hand, the scaling factor k is much larger -- say, 10 -- structures that far apart in size are so unconnected that consecutive levels of scale fail to relate to one another. This heuristic argument narrows the ideal scaling factor to a number somewhere between 2 and 5, consistent with our choice k = e .

Separate theoretical support comes from fractal patterns in mathematics. There is an infinite number of self-similar fractal patterns, each with similarity ratio r = 1/k . These fractal patterns have the remarkable property that any detail, when magnified by the factor k = 1/r , looks exactly like the whole. It is well known that self-similar fractals can be used to model natural structures such as mountains, coastlines, and snowflakes. Of all the possible Koch, Peano, and Cantor self-similar fractals, those that correspond best to natural forms have similarity ratios r = 1/3 or r = 1/Root[7] = 1/2.65 (Mandelbrot, 1983). If we had to pick a single universal value for the scaling factor, these figures support the choice k = 1/r = e = 2.7.

SOME CONSEQUENCES FOR ARCHITECTURAL DESIGN

The scaling hierarchy explains why some asymmetric structures can appear surprisingly ordered. Examples include buildings by Antoni Gaudí (Zerbst, 1993) and Lucien Kroll (Kroll, 1987). Also, some free-form "organic" buildings may appear strange to academic architects, but they undeniably give pleasure to their inhabitants (Day, 1990). Such buildings need not have classical symmetry, nor follow any recognizable archetypes. According to traditional criteria that also govern Modernist forms, buildings without any obvious overall symmetry are not thought to be ordered, as most people tend to identify "order" with symmetry. Yet those with scaling satisfy one of the several requirements for intrinsic order in a design (Alexander, 1996).

There is another point of considerable practical significance. Weathering patterns are due to fractures and stresses on materials. Morphological features in materials are generated by physical forces, which will tend to follow the scaling hierarchy (Alexander, 1996). This is a remarkable observation, and it is consistent with the fractal development of material surfaces in time (Mandelbrot, 1983).

All structures eventually develop morphologically towards the natural hierarchy of scales, regardless of materials used, or how they were initially put together. If a building's structural subdivisions fail to coincide with the scaling hierarchy, then the weathering patterns might cut across forms. (It certainly doesn't make it easy to maintain the architect's original intentions). This process is clearly evident on large "pure" surfaces. In those cases where an architect pays attention to the weathering, the result definitely has scaling. Weathering tends to reinforce the design of a building with scaling coherence. For this reason, traditional architectures weather well, whereas Modernist forms do not (Blake, 1974).

A SEQUENCE THAT DEFINES DESIGN SUBUNITS

Architects are already familiar with theoretical descriptions of ratios in terms of the Fibonacci sequence. We can present the above results in a more practical manner, by giving a sequence consisting of powers of e rounded off to the nearest integer. Practitioners can then check that the subunits of their designs correspond approximately to the terms in the given sequence. The important point is not strict agreement with each number - these can differ widely - but one should make sure that no terms are missing.

The following sequence of integers approximates the powers of e for n = 0, 1, 2, ... :

{ eⁿ } = {1, 3, 7, 20, 55, 148, 403, 1097, 2981, 8103, ...}

(2)

There are two ways to use this sequence, each of which gives the entire range of relative sizes of subunits. The first way is to choose the smallest base measurement x₀ and then multiply it by the terms of the sequence (2) to obtain the sizes of all the larger units. Alternatively, one may start with X as the largest overall size, and divide by the terms of the sequence (2) to obtain the sizes of all the subunits. Deciding which method to apply depends on which dimension, x₀ or X , is fixed. Either progression - small to large, or large to small - will generate the correct scaling hierarchy.

Readers will recognize that the scaling sequence (2) corresponds approximately to the alternate terms {1, 3, 8, 21, 55, 144, ...} of the full Fibonacci sequence. The correspondence is not exact, and becomes progressively worse for larger terms. Alternate terms of the Fibonacci sequence have scaling ratio k = (Phi)² = 2.618 in the limit as the terms become large, whereas the sequence (2) is an integer approximation of a sequence of numbers with fixed scaling ratio k = e = 2.718. Nevertheless, as the scaling law is approximate, this resemblance is useful and of considerable interest.

THE NUMBER OF DIFFERENT SCALES IN THE HIERARCHY

In principle, scaling coherence defines an infinite number of decreasing levels of scale in any design. For practical purposes, however, we impose a low-end cut off for the minimum detail that we want to show in a building. Taking this lower limit to be (1/4)in = 6mm provides a useful rule for estimating the total number of different levels of scale. If x₀ is the smallest size of a design subunit (corresponding to n = 0 ), and X = e^n-1x₀ is the largest overall size, then we can solve for the number n of distinct scales. Rounding off to the nearest integer value, n is computed as follows (Salingaros, 1995):

n = 1 + lnX - lnx₀

(3)

Here, X and x₀ must be expressed in the same units. For example, if we are going to measure the overall dimension X in meters, then choosing x₀ = (1/4)in = 6.4X10^-3m gives the total number of different scales as approximately:

n = 6 + lnX(m)

(4)

A building of height or width X meters therefore needs to have distinct subunits of n different sizes as given by (4) in order to appear coherent. Even when it has the required number of scales, the relative sizes have to correspond to the scaling hierarchy (1, 2). The degree of coherence will of course depend on the similarities and boundaries of all the different scales (as discussed in the last section). If a building has either significantly fewer levels of scale, or significantly more, it will appear incoherent. Equation (4) tells us that the majority of buildings, which range in size from 5m to 50m, have to have 8 to 10 distinct levels of scale if the smallest detail is going to be (1/4)in.

CONNECTING TO THE HUMAN SCALE

Starting with Vitruvius, writers underline the necessity for architectural forms to have features on a scale to which human beings can relate (Licklider, 1966). There are two independent problems of scale here, the second of which is solved by the scaling hierarchy. The first is use and physical contact: a building's components and dimensions must accommodate people. Room sizes, staircases, placement and dimensions of doors and windows, etc. have to be carefully fixed for maximal ease of use. A sensitive architect fits the geometrical forms and accessible features of a building to activities on the human scale.

The second problem is how a design's subdivisions are perceived. At any given distance, a person will connect to design components that correspond to the entire human structural scale: the whole body; an arm's length; a foot; a hand; a finger's width, etc. (Licklider, 1966). This impression is visual and relative, and depends on the changing distance between the viewer and the structure. It is necessary to define designs that have the complete range of internal subdivisions, regardless of the viewer's distance. Only a building with scaling coherence provides the complete range of human scales to an observer at any distance and from any perspective.

The smallest perceivable size obviously depends on the distance between the observer and the object. If a portion of a building can only be experienced from a large distance, then the smallest perceivable scale is rather large. Some structure should be defined on that scale, but more detail is going to be wasted. (Buildings have unnecessary precision, or expensive materials, where they cannot be experienced). In many situations, the user can move closer or further away from a region, so the smallest perceivable scale is changing continuously. In that case, there must be detail down to the smallest of the "smallest perceivable scales". The detail will be lost at larger distances, but will re-appear when needed at the closest approach.

This principle makes a dramatic difference to a user's experience of larger buildings. If a large building is connected to the range of human scales through the scaling hierarchy, it is perceived in psychologically positive terms. It could then be described as awesome, grandiose, and impressive, in a way that is largely independent of other attributes. On the other hand, if it disconnects from the human scale by eliminating the smaller scales in the hierarchy, it is perceived in psychologically negative terms. Independently of other design factors, it will be felt as aloof, severe, or it might even appear so alien as to be oppressive. Some architects deliberately strive for such an aloofness.

CONNECTING TO BUILDINGS VIA MATERIAL SURFACES

If we don't stop it at 6mm, the scaling hierarchy has no rigidly-defined lower limit. The smallest perceivable size depends on the distance to the viewer, and at the closest approach can go down to less than 1mm. In man and animals, the most expressive forms are highly detailed: e.g., the eyes, nose, and mouth. Here, the details indeed go down to below 1mm. Note that distinguishing details occur locally, and focus attention onto a small region : the same degree of detail does not usually spread over the entire form. It is the existence of detail where it can be experienced that is important, and not its profusion.

Architecture connects to the human consciousness via the smallest details, whether those buildings are in a traditional, or a Modernist style. The psychological need for detail at the smallest perceivable scale is illustrated by the widespread use of natural surfaces such as polished wood and stone, whenever it is economically feasible. Such surfaces provide an emotional connection to details well below 1mm in size. The eye actually perceives the natural structures that characterize real wood or marble, even though they are at the limit of visual perception. One is not easily fooled by Formica even at a distance.

It is possible to highlight the connection between an observer and the microscopic structure of materials, which is obtained via the scaling hierarchy. From the human scale on down, there exists an infinite hierarchy of decreasing scales connecting us to the basic components of matter. This downward scaling links man to the microcosm, and is just as important as the more obvious connection to the larger scales. We establish a strong connection to materials that have a clear hierarchical microstructure, but not to featureless materials that are either amorphous, transparent, or highly reflective.

Materials lacking in natural qualities often result in unresponsive surfaces. That is in part due to the ways they are employed. Modern materials, which as a rule have no ordered microstructure, can establish an emotional connection only through the scaling hierarchy. One needs to differentiate surfaces and to articulate subdivisions much more than with natural materials. That might involve combining matte with shiny materials, and re-introducing detail and color. We do not advocate copying natural materials, but rather finding expressive means to utilize each material's own intrinsic capabilities.

THE USE OF MODELS IN ARCHITECTURAL DESIGN

Whereas a model or plan pays attention to detail at the smallest perceivable scale on the plan, this becomes far too large when built. A building consequently loses its crucial levels of scale. The reason for this is that the different scales in the hierarchy increase exponentially, and therefore lie much closer together at the smaller scales (2). Say a typical model of a 50m tall building shows 3 levels of scale before the detail is lost. The full-size building, however, will have 10 levels of scale if the built detail goes down to 5mm (from (4)). We automatically lose the smallest 7 levels of scale, which are those that we connect to most strongly.

The design process generally begins on an intellectual level, using a set of ideas to plan a new structure. A building's functions should determine the initial design, though in many cases that is subordinated to the building's image. What ultimately defines the success of a building is a set of emotional responses arising from forms at the full scale (Alexander, 1996; Day, 1990). Nevertheless, it is the reduced plans and miniature model that are routinely evaluated by the emotions they produce, and these can be misleading. The user's experience of the final building is totally different.

A model may be judged by its novelty; its overall symmetry; how it fits pieces together; how it resembles familiar forms; if it has "clever" parts. Those decisions rest on factors that are specific to the model's scale. When the building is built, one's experience of the forms changes dramatically, and the original judgements based on the model are no longer relevant. The way that large forms fit together cannot be easily perceived at the full scale. The building's overall shape usually doesn't matter to a user inside. Any similarity between the total plan and familiar forms is apparent only from an aeroplane. Things that look "clever" in a model might turn out to be oppressive and dysfunctional when built.

Critical decisions depend on feedback that cannot be triggered by a miniature model. The only way to determine this accurately is from the interaction between a user and the full-scale structure and its subdivisions. A drawing that shows all the smaller scales has to be at or near actual size. Moreover, the success of architectural forms depends on the view-point of the observer: points close-by to a pedestrian are the most important, and points far away the least important. Relying too narrowly on a scale model reverses these priorities, ignoring the immediate spaces, and emphasizing areas that are invisible or inaccessible to a user. Fortunately, today we can mimic the full-scale experience in a building through virtual reality techniques using computers.

These ideas will be of utility to readers given the interest in finding guidelines in a built environment that is often unresponsive, and is becoming increasingly chaotic and anarchic. There is a comfort in a natural scaling. Computer-aided design offers the potential to assist in three-dimensional design that satisfies the scaling hierarchy. Given a generous tolerance in dimensions, one can determine if any particular scale is missing. This gives a "spell-check" to highlight particularly bad scale relationships before they are built. We will not attempt to describe the vast number of decisions encountered in designing thoughtful buildings. The rule introduced here provides a useful criterion that helps to narrow down the possibilities.

 

A CHECKLIST FOR ACHIEVING SCALING COHERENCE

The scaling hierarchy establishes the proper subdivisions, and the relationship between the different scales in a building. "The small scale is connected to the large scale through a hierarchy of intermediate scales with scaling factor roughly equal to e = 2.7". To achieve coherence, however, it is necessary to go further and link the distinct scales together via similarity techniques as discussed in (Alexander, 1996; Salingaros, 1995; Salingaros, 1997). The following list summarizes how to connect the different levels of scale to each other - these are suggested means and should not be taken as absolute.

1. Define recognizable units through contrast in color and geometry at all scales in the hierarchy.
2. Tie different units together through symmetry, overlapping designs, a common grid, complementary shapes, and matching colors.
3. Every unit needs a boundary that is itself a unit on the next-smallest scale - units should couple sequentially with adjoining units.
4. Units of different size can link with one another by having a similar shape, so the same pattern repeats at different magnifications.
5. Similar patterns of decreasing size can be nested to define a geometrical focus, and this should coincide with a functional focus.

Contrast is necessary because it establishes the different scales that we connect to. Rule 1 opposes the elimination of intermediate scales found in buildings that project a purist image. Such buildings often have subdivisions, but they are deliberately made too subtle, and so fail to differentiate the form internally. The whole idea of coherence is to harmonize components, which requires all the components of a design to be clearly articulated. That is the opposite of trying for uniformity by eliminating components. Many contemporary designs go to great lengths to disguise ordered structure at the small and intermediate scales.

Reflectional and translational symmetries tie together groups of elements. Rule 2 emphasizes the need for multiple symmetries in architecture. The scaling hierarchy guarantees that symmetries can be defined independently on each level of scale. A building with scaling coherence has an enormous number of internal symmetries: there is one overall unit of scale, and an increasing number of units populating each scale of decreasing size; symmetry can act on all of these units. The number of smaller elements could be vast. (With the elimination of the smaller scales, however, the only possible symmetry is an overall bilateral symmetry).

Rule 3 achieves coherence by tying the different scales together through contact. When a form's boundary is the next-smallest term in the hierarchy, and the boundary's boundary is the second-smallest term, etc., then successive forms in the hierarchy are paired geometrically. It is not enough simply to have forms of the right size according to the hierarchy: they must also be touching each other in the proper order. If sequential units are not tied together by proximity (as when an isolated small scale is juxtaposed with a large scale), then there will be a perceptible discontinuity in the scaling connection. It is the smooth progression of scales that leads to coherence.

Proximity of scales means that units must be connected by intermediate borders. This "wide-boundary" concept is due to Alexander (Alexander, 1996), and is found in buildings throughout history. Traditional materials having limited strength usually make a boundary that is of comparable size as the unit unavoidable, and many historical buildings deliberately accentuate this effect. For example, carved Romanesque doorways have a frame that is as large as the door itself. One normally has to work against the materials if one wants to minimize or disguise the connective boundaries in structures.

Sheer walls and large plate-glass windows in today's buildings do not contain the necessary hierarchy of scales. Historically, the thick grout between bricks defined a wide boundary for each brick, but modern bonded brickwork minimizes the mortar. As a consequence, contemporary brick walls embodying a minimalist style lack scaling coherence. In the same way, as long as glass could only be produced in small panes, the supporting framework provided the proper subdivisions; this scaling coherence is absent from plate glass. Glass walls emphasize a technological advance that one cannot relate to.

A building has to fit harmoniously into its environment. The relationship of an object to its surrounding space is determined by the hierarchy of scales. A building's boundary - which is a region of comparable size as the building itself - should couple with the building to define the next-higher level of scale. Therefore, the border requires the same definition and connections as the building's internal subdivisions. A proper boundary region connects the building to its surroundings. A building that draws attention to itself by either ignoring, or clashing with its surroundings, is ultimately unsuccessful (Gehl, 1987).

Sometimes the boundary is the building, as in Medieval Cloisters and Islamic Madrassas. These work by focusing an intensely-detailed surrounding structure onto a central courtyard. Here, a complex boundary intensifies the open space. The coupling of two complementary opposites - the courtyard with its bounding structures - creates a coherent unit. Coupling also occurs between randomness and highly-ordered detail. Some Mosques and Madrassas couple strictly-ordered walls to the courtyard's deliberately irregular paving stones (Blair and Bloom, 1994). Louis Sullivan's banks use bricks with random variations in color and texture to contrast with a regular design in the surrounding border (Weingarden, 1987).

Rule 5 establishes a necessary link between structure and function that is all too often ignored in contemporary buildings (Alexander, 1996). The use of gradients to lead the pedestrian towards a focal point is extensively utilized in Hellenistic and Roman colonnades, later imitated during the Renaissance. Other applications include staircases that become narrower as they ascend, and doorways that focus one's movement through a sequence of concentric arches. Approaches and entrances that connect to the user in a positive and natural way make an enormous difference to a building's use.

CONCLUSION

A scaling rule comprises part of a design theory derived by Christopher Alexander. This is intended to produce buildings that connect subconsciously to users. By controlling internal subdivisions, the design can generate the same positive emotional response as natural forms. The rule derives from physical and biological forms, so buildings designed using this rule have an intrinsic structural similarity to natural forms, even as their shape need not resemble anything specific. The scaling rule is independent of architectural style or particular building shape.

Buildings with scaling coherence look and feel very different from most 20th century buildings. Architects wishing to adopt this paper's proposals will have to overcome a familiar and accepted way of building things. The images that drive contemporary architectural design largely violate the scaling rule. Following our analysis, any thoughtful architect will know what points to emphasize in developing innovative designs that have scaling coherence. Our goal is a certain level of emotional satisfaction from buildings where little or none exists today.