1. INTRODUCTION

Architecture affects mankind in a predictable way, and it has its own set of fundamental laws. We have proposed three laws for architecture ⁽1^, 2⁾ that are based on the work of Christopher Alexander ⁽³⁾. An analysis of architectural forms distinguishes three different aspects: (1) the small scale; (2) the large scale; and (3) linking all the intermediate scales together through hierarchical coherence ^{(1, 2)}. This paper examines how the small and large scales contribute to the success of a building independently of hierarchical coherence.

We can systematize intrinsic qualities that govern architectural forms by setting up a simple model. The first part of the model identifies two distinct qualities and suggests how to measure them. The small-scale structure is described by what is labelled the architectural temperature T : the higher the temperature, the more differentiations, curves, and color (Section 2). The architectural harmony H is identified with the degree of symmetry and coherence of forms, and measures the absence of randomness (Section 3). The harmony H carries the traditional meaning it has in architecture, whereas the temperature T is a new concept.

The second part of the model relates the perceived architectural life and architectural complexity to different combinations of T and H . We define the "architectural life L" as L = TH (Section 4), and the "architectural complexity C" as C = T (10 - H ) (Section 5). The life refers to the degree that one connects with a building in the same way that one connects emotionally to trees, animals, and people ⁽3⁾. The complexity is already understood by architects to mean essentially what we define here. Two of the principal emotional responses to architectural forms are thus formalized in this paper.

This establishes a connection between scientific quantities based on measurements, and intuitive artistic qualities based on feelings. Although it is usually difficult to quantify subjective statements, people agree on the ranking of the emotionally perceived "life" of structures, and what we measure here as the architectural life L (Section 6). We emphasize the model's predictive value in differentiating buildings on a plot of L versus C . There is indeed a pattern independent of personal preferences. One can follow the historical development of architecture in terms of intrinsic qualities rather than styles.

The third part of the model reveals how to endow a building with "life". The method is to judiciously adjust the individual ingredients of forms. By providing a coherent theory on how to do this, the model becomes an extremely valuable tool for both analysis and construction. The process that raises the architectural life is entirely independent of particular styles, or what the forms look like. This model can help someone to understand and control the interplay between life and complexity in any structure.

Section 7 discusses the model's origin from an analogy with thermodynamics. The quantity H represents something analogous to a negative entropy. In defining L and C , the model mimics the thermodynamic potentials; like them, the architectural life and complexity are relative and not absolute values. This analogy suggests that similar physical principles underlie organization and disorder in all structures, thermodynamic as well as architectural.

The final section (Section 8) investigates the link between biological and architectural forms. For example, self-similar fractal patterns have high architectural life, which is why they are widely successful in modeling natural forms ⁽4⁾. The combinations L and C correspond to the organization of matter in living forms, and this resemblance generates the emotional responses to structures with different architectural potentials L and C . Buildings are created by living people, who have a basic need to instill architectural life into inanimate structures. A building is as successful as the degree to which it reflects this.

2. DETAIL AND TEMPERATURE IN ARCHITECTURAL DESIGN

Several factors contribute to perceived qualities in architectural design, and our first task is to distinguish them. The most obvious is the departure from uniformity. A form is either plain, or it is differentiated in terms of the geometry and color. In physics, uniform states in fluids and gases are normally associated with low temperatures. Raising the temperature often breaks the uniformity, leading to gradients and convection cells. Independently of this, heated metals acquire a coloration by radiating.

This suggests that we refer to the degree of detail and small-scale contrast in a design as the "architectural temperature T". (There is a loose analogy between the architectural temperature and nT_th , where T_th is the thermodynamic temperature, and n is the particle number density; see Section 7). The architectural temperature is determined by several intrinsic factors such as the sharpness and density of individual design differentiations; the curvature of lines and edges; and the color hue. Even though people think of architecture as being concerned with the form only, color is an integral part of experiencing a form's surface ⁽³⁾.

We propose a very simple method of measuring the architectural temperature T . This rough guide is by no means the only possible prescription; it is a first step to handling an extremely complex topic. We will distinguish five elements T₁ to T₅ that contribute to T . Each quality is measured on a scale of 0 to 2 according to the scale: very little = 0, some = 1, considerable = 2. The different components are listed as follows:

T₁ = intensity and smallness of perceivable detail

T₂ = density of differentiations

T₃ = curvature of lines

T₄ = intensity of color hue

T₅ = contrast among color hues

T = T₁ + ... + T₅ , 0 < T < 10

(1)

The limit of perceived textural differentiations at arm's length is roughly 1mm, though we propose a more generous cut-off of 5mm. Well-defined detail in those surfaces that a person can touch, regardless of whether it is localized or spread over the entire region, makes T₁ equal 2. On regions farther away, differentiations can be much larger so as to appear the same size as 5mm would be at arm's length. Coarser, or less sharply-defined detail rates T₁ = 1. For detail that is too small or is faintly-defined, T₁ = 0. Smooth or textured monochromatic surfaces rate a 0; to count, detail must be articulated against the ground.

We will treat every geometric differentiation as having the same effect as a greyscale surface design, i.e., T₂ of a colored relief is judged from its flat black-and-white photograph. In this two-dimensional projection, any differentiation or texture is perceived in terms of its contrast in color value, or by the shadows it casts. A high density of sharp differentiations rates T₂ = 2, whereas a plain surface rates T₂ = 0. The color value itself, which represents a particular shade of grey, doesn't contribute to T₂ directly.

A curve can be approximated by a very large number of straight-line segments. An inflected curve (for example, a higher-order polynomial) or zig-zag has a higher structural temperature than a straight line. The temperature is proportional to the curvature of forms. Curves on the intermediate scales rate T₃ = 1; if they have a high degree of curvature, T₃ = 2. Straight lines and rectangular forms rate T₃ = 0.

A richly colored building, even if it is of one hue, has a higher temperature than a grey building (for which T₄ = 0). A design will have T₄ = 1 if it has some color overall; an intense though not necessarily bright color gives T₄ = 2. The actual hue (i.e., yellow, green, or purple) is immaterial.

The architectural temperature is increased further by contrasting color hues, for example, red next to green. If there is any contrast in color hues, give a 1 for T₅ ; if there is a great variety, or the contrast is particularly vivid, give a 2. A uniform color or no color at all rates T₅ = 0.

In different cases, the architectural temperature T , Eq. (1), will depend on each factor T_i to a greater or lesser extent. While there have been periods and cultures that have been more restrained in their detail, curvature, and color than others, the predominance of buildings and artefacts with high architectural temperature throughout history suggests that this satisfies a profound internal need in human beings ⁽³⁾. In Table 1, we have estimated the architectural temperature T of twenty-five buildings. These include famous buildings ⁽5⁾, so there is no need to reproduce photographs here. The numbers given are very approximate, and their derivation is discussed in Section 6, below.

Table 1. The architectural temperature T and architectural harmony H of twenty-five buildings, numbered in chronological order. These estimates provide values for the architectural life L = TH and the architectural complexity C = T(10 - H).

No.	Building	Place	Date	T	H	L	C
1	Parthenon	Athens	-5C	7	8	56	14
2	Hagia Sophia	Istanbul	6C	10	8	80	20
3	Dome of the Rock	Jerusalem	7C	9	9	81	9
4	Palatine Chapel	Aachen	9C	7	9	63	7
5	Phoenix Hall	Kyoto	11C	7	9	63	7
6	Konarak Temple	Orissa	13C	8	8	64	16
7	Cathedral	Salisbury	13C	7	9	63	7
8	Baptistry	Pisa	11/14C	7	8	56	14
9	Alhambra	Granada	14C	10	9	90	10
10	St. Peter's	Rome	16/17C	10	6	60	40
11	Taj Mahal	Delhi	17C	10	9	90	10
12	Grande Place	Brussels	1700	9	7	63	27
13	Maison Horta	Brussels	1898	8	7	56	24
14	Carson, Pirie, Scott	Chicago	1899	7	8	56	14
15	Casa Batlló	Barcelona	1906	8	5	40	40
16	Fallingwater	Bear Run	1936	4	5	20	20
17	Watts Towers	Los Angeles	1954	10	4	40	60
18	Corbusier Chapel	Ronchamp	1955	1	2	2	8
19	Seagram Building	New York	1958	1	8	8	2
20	TWA Terminal	New York	1961	3	2	6	24
21	Salk Institute	San Diego	1965	1	6	6	4
22	Opera House	Sydney	1973	4	5	20	20
23	Medical Faculty	Brussels	1974	7	4	28	42
24	Pompidou Center	Paris	1977	6	4	24	36
25	Foster Bank	Hong Kong	1986	3	7	21	9

3. RANDOMNESS AND HARMONY IN ARCHITECTURAL DESIGN

Randomness is measured by the entropy. Because entropy is not an intuitive concept, we will introduce the "architectural harmony H" to measure the lack of randomness in design. (The relationship between H and a negative architectural entropy is discussed later in Section 7). Where the individual details and shapes relate to each other, the architectural harmony H is high. Symmetrical forms and patterns have high harmony. The harmony H is a property of the whole structure, due to the correlation between the parts on all the distinct levels of scale.

3.1 Estimating the Architectural Harmony

The model depends on direct measurements from perceivable architectural surfaces and forms, i.e., walls, doorways, passages, etc. When thinking about symmetries, most architects immediately look at a building's plan ⁽6⁾. As the plan is not directly perceivable to a user, however, it is not relevant to our model. In a break with both traditional and current practice, we will ignore the aerial view: this model doesn't cover the formal organization of spaces; only the immediate impressions from a human viewpoint.

The architectural harmony H is decomposed into five components, each of which assumes a value from 0 to 2. Again, this is only an expedient that gives very approximate numbers. We will use the scale: very little = 0, some = 1, considerable = 2. The architectural harmony H ranges from 0 to 10, and is the sum of the five components described as follows:

H₁ = vertical reflectional symmetries on all scales

H₂ = translational and rotational symmetries on all scales

H₃ = degree to which distinct forms have similar shapes

H₄ = degree to which forms are connected piecewise

H₅ = degree to which colors harmonize

H = H₁ + ... + H₅ , 0 < H < 10

(2)

An average numerical value has to be assigned for the presence of symmetries on all scales, not just for the largest scale. The quantity H₁ depends on the orientation of the symmetry axis, because gravity defines a preferred direction for both man and materials. Of the possible axes for reflectional symmetry, the vertical one raises the architectural harmony the most. Symmetry about a diagonal axis clashes with natural symmetries created by gravity, and the ensuing imbalance lowers the harmony (i.e., the leaning Campanile of the Cathedral at Pisa). Lack of reflectional symmetry on different scales rates H₁ = 0.

The quantity H₂ measures translational symmetries (and the less common rotational symmetry) on walls, doors, and windows; not on a building's plan. If elements are repeated regularly, then H₂ equals 2. In plain surfaces with no distinguishing elements, H₂ is defined by the edges; if they are parallel, then H₂ = 2. Elements repeated randomly lower H₂ to 0.

Self-similarity raises the architectural harmony: scale up the same figure to several different sizes, then align all the scaled copies. The contribution H₃ measures the similarity of overlapping or spatially-separated figures occurring at different sizes. For example, a group of parallel lines or nested curves is related by a scaling transformation, so H₃ equals 2. Large plain surfaces with no distinct subfigures harmonize by default, so H₃ equals 2. Pieces with different shapes do not harmonize, and H₃ equals 0.

The quantity H₄ estimates the presence of geometrical connections. Internal and external connections can take many different forms: connecting lines or columns; intermediate transition regions; a wide surrounding border, etc. Piecewise connections raise H₄ to 1 or 2. Edges that touch but fail to join, jutting overhangs without obvious supports, and breaks in lines lower H₄ to 0. The main connection of any building is to the ground; if this is not strongly expressed, then H₄ = 0.

A building of a single color or without any color at all has color harmony, so H₅ = 2. If different colors are used, one has to estimate how well the various hues blend to create an overall color harmony. Even with bright colors, a harmonious ensemble has H₅ = 2. The departure from a unified color effect - something unbalanced, clashing, or garish - lowers H₅ to zero.

Together, these quantities give a numerical measure for the architectural harmony H , Eq. (2). In Table 1, estimates of H are provided for twenty-five buildings using the prescription outlined in this section.

3.2 Architectural Harmony and Pattern Recognition

The connection between harmony (as negative entropy) and information in thermodynamics carries over to architecture. Any symmetry in a design reduces the amount of information necessary to specify shapes. A form with bilateral symmetry needs to be specified only on one side, which is then reflected. A design with translational or rotational symmetry is defined by the information contained in a single unit that is repeated. Recognition of an unfamiliar object is greatly simplified if it has as many internal symmetries as possible ⁽7).

Juxtaposing different materials lowers the harmony H by breaking the symmetry across an interface or gap. Disconnected forms near each other create ambiguity, thus lowering the structural harmony. When a form's basic attachments are missing, the brain continues to seek visual information that would establish the necessary connections ⁽⁷⁾. If these are not obvious, the ensemble is perceived as incoherent. Recognition is frustrated whether structural information is missing, or is overwhelming. Since pattern recognition is a low-level brain activity ⁽⁷⁾, we may be intrigued intellectually by a low H form, but our gut reaction is negative.

The harmony of multiple structures unrelated by either symmetry or scaling is raised through piecewise connections. A structural connection relates two separated forms geometrically. Each form connects individually to a third, intermediate region, which itself must be large enough for this to occur. Mathematically, the two original forms relate by establishing a minimal transitive relation via an additional connecting form. The linking is successful only when the two forms and the connecting region together define a coherent larger unit.

3.3 Raising the Harmony by Lowering the Temperature

Adding some of the same hue to areas of different color connects them harmoniously, raising H₅ by lowering T₅ . This is very tricky to do while at the same time preserving any existing contrasts in color hue. In successful examples, areas are brought together by relating their color hues so that the overall result is intense rather than muddy. Further harmonization, however, can eliminate color contrasts that contribute to the architectural temperature. This shows that H and T are related in general.

Two very different techniques raise the architectural harmony in a random design. The first re-arranges existing details to create a maximum number of symmetries, starting from the smallest scales and working up to the larger scales. This maximizes the harmony with the constraint that the architectural temperature remain constant. The second method eliminates all details, curves, and colors, which lowers T . The mostly plain surfaces that are left can be made totally symmetric, which raises the harmony H . This alternative method can raise the architectural harmony the most by eliminating all randomness, but it loses the architectural temperature as well.

3.4 Lowering the Harmony by Raising the Temperature

The architect Lucien Kroll lowers the harmony by randomly arranging small and intermediate-scale components in a building ⁽8⁾ (building No. 23 in Table 1). Design decisions that use random numbers as input escape from the monotony of a strict bilateral or translational symmetry, which in most cases is arbitrarily imposed. Nevertheless, any lack of architectural symmetries in historical buildings is usually the result of accommodating functional or structural needs, and so is not really random. Functionality is inextricably linked to the form, as first stated by Louis Sullivan, and demonstrated by Alexander ⁽³⁾.

Adding small-scale structure that doesn't relate to the ensemble lowers the harmony of a design. This could be either a random pattern, or a regular pattern that fails to connect to existing patterns. Excessive decoration lowers the architectural harmony, and makes an overall coherence difficult or impossible. Many architectural styles evolve historically to reach an overly-decorated Baroque style, which is invariably followed by an anti-decorative reaction ⁽⁵⁾. This cycle re-establishes high architectural harmony.

3.5 Raising the Harmony and Temperature Together

The main message of this paper is that the most responsive architecture is created by raising both H and T together (developed in the following sections). Raising T while not lowering H involves adding detail or color so as to enhance and not clash with existing patterns and colors. One preserves the existing geometry and adds on to it. What is already there acts as a matrix for guiding all additional structure. This is the basis for the theory of architecture proposed by Alexander ⁽³⁾.

There are two approaches to accomplishing this:

1. Add T very selectively, while being careful at each step to raise or at least not to lower H .
2. Add as much T as necessary to bring a design to life, then come back and rearrange everything in order to raise H .

The first method works well on site, and is recommended for the transformation of existing buildings. Most buildings today have acceptably high H , but very low T , so this change is called for. The second method is better applied to the computer-aided design of new buildings, where many changes can be tried and evaluated before the structures are fixed.

4. THE ARCHITECTURAL LIFE OF A BUILDING

Whereas the quantities T and H must be measured in every structure, the combination TH is directly perceivable without any direct measurements. Amazingly, the product TH connects emotionally to an observer. For reasons that will later become clear, we call this the "architectural life L". (Section 8 below discusses the connection between L and biological forms).

L = TH , 0 < L < 100

(3)

Before the 20th century, builders unconsciously tried to achieve the highest architectural harmony consistent with the highest architectural temperature: they didn't measure this, they felt it. The greatest historical buildings maximize their architectural life L , as can be verified by looking at a survey of the world's architecture ⁽⁵⁾ (see Section 6, below). A separate point is that the greatest buildings do not eliminate randomness entirely. The optimal value for the architectural harmony is below its theoretical maximum. Every great building has some degree of randomness, which can manifest itself on different scales.

Achieving architectural life is not trivial, because T is actually a function of H . Raising the harmony H by lessening the degree of small-scale differentiations and straightening out curves can eliminate the architectural temperature T . This lowers the life of a design. The necessity for a high architectural temperature makes it impossible to raise the harmony above a certain value. Visually, the link between T and H is expressed as contrast between regions of different H .

For example, a portion of the building material itself can be arranged randomly so as to contrast with an overall symmetry. Carved human figures break the small-scale symmetry in a portion of a building, which is otherwise highly symmetric (buildings No. 1 and 6 in Table 1). In Islamic buildings, calligraphic script provides the small-scale randomness necessary to contrast with the large-scale symmetry (buildings No. 3, 9, 11). Alternatively, large-scale randomness may contrast with highly-ordered detail (building No. 12).

Many modern buildings have a very low value of architectural life L . We see how to minimize L , Eq. (3), by reducing the architectural temperature or the architectural harmony as much as possible. Lowering T with fixed H is straightforward: eliminate all detail, color, structural differentiations, and curves (see Section 2). We are then left with a plain rectangular grey box (buildings No. 19 and 21). On the other hand, lowering H while T is fixed requires creating randomness on both the small and large scales.

Small-scale randomness lowers H but it also raises T , so it is not entirely effective. With an already low T after eliminating design details, one lowers H by generating randomness on the large scale. This comes from a lack of bilateral and translational symmetries, and a lack of geometrical connections. We obtain a disjoint, unbalanced, and asymmetric large-scale form without details. Large pieces in strange shapes are constructed out of plain materials, with abrupt juxtapositions between surfaces and volumes (buildings No. 18 and 20). If there is any color, it consists of uncorrelated hues. This type of structure is the result of minimizing L by lowering H .

5. THE ARCHITECTURAL COMPLEXITY OF A BUILDING

The "architectural complexity C" of a building or design can be expressed in terms of its architectural temperature and harmony:

C = T (10 - H ), 0 < C < 100

(4)

The quantity T (10 - H ) is perceived directly as a building's complexity, which can range from dull ( C = 0), through exciting (low to medium C ), to incoherent (very high C ). It is the complexity of an object that arouses a viewer's interest; the complexity is the inverse measure of how boring a building is. Using color and contrasting color hues, small-scale differentiations, and curves contributes to the complexity, as does any randomness and asymmetry. The buildings of Antoni Gaudí provide good examples of architectural complexity ⁽9⁾ (building No. 15 in Table 1). The highest C structure in Table 1 is the curious Watts Towers built from pieces of junk by Simon Rodia (building No. 17).

Independently of any particular theory or architectural movement, the commercial sector recognizes that people do not connect to plain walls, but to bright colors and contrasts. Putting this into practice has generated a high T environment defined by a profusion of colored signs and surfaces. Such an environment is uncorrelated, so it has a very low harmony as well as high temperature, and is therefore very high C . All over the world, this process is a major force driving building practice.

The vernacular architecture of the commercial environment is taken seriously by very few architects ^(10, 10⁾. As long as this phenomenon is not understood as arising from certain very basic forces, those forces cannot be controlled and directed, and an incoherent environment continues to proliferate unabated ⁽11⁾. As a result, most of us have to experience built regions with far higher complexity than the Watts Towers during large portions of our daily lives.

At the other extreme, pure 20th century modernist forms have a very low complexity C . Architects reacting against low C forms are today creating buildings with slightly higher values of C . Post-modernist architects define more structure on the small and intermediate scales, and break some of the symmetries of modernist buildings ⁽12⁾. That increases the temperature and lowers the harmony to varying degrees. Some raise C by introducing a host of uncorrelated forms to lower a building's harmony ⁽⁵⁾ (buildings No. 23 and 24). Others use bright overall colors to raise T .

Another movement today, the neo-classical style, is defined by classical symmetry and proportion, which means that the architectural harmony is very high. Since the Renaissance and Palladio, applying and developing the Greco-Roman vocabulary has produced countless successful buildings ^{(5, 10)} (building No. 10). Contemporary neo-classical buildings, however, rarely have the sculptural friezes, nor the degree of detail and color of classical buildings. Consequently, their complexity and life tend to be measurably lower than that of classical buildings, or earlier neo-classical buildings.

6. THE EVOLUTION OF LIFE AND COMPLEXITY IN ARCHITECTURE

Values of the architectural life L and architectural complexity C for prominent buildings are computed in Table 1. The architectural life as defined in this paper corresponds very accurately to what people feel as the "life" of a structure, independently of whether they may like it or not. The numbers given for L and C of a particular building are only approximate, yet most people will agree with the relative ordering of the values. Table 1 is used to produce Figure 1, which permits us to follow the evolution of architectural styles.

6.1 Difficulties of Estimating the Parameters

We have chosen some of the best-known buildings in history. Even so, the readings necessarily reflect the altered state of many buildings. Sculptures, mosaics, and colors are missing; exteriors and interiors have been altered; parts have been extensively re-built; windows are not original. Our estimates are based on their present condition, with a partial correction for their conjectured original forms.

The key to estimating values for T and H is to observe a building's exterior and interior from the human viewpoint. Architecture books favor a view from a great distance that is rarely experienced by a user, but small-scale differentiations are seen only by people close up; in entrances and inside a building, in regions that impact a person immediately. It is necessary to find close-up photos in color showing sufficient detail of the buildings accessible regions - both inside and outside - to determine how far the small-scale patterns are correlated. The author has been to fewer than half the buildings given in Table 1, yet that remains the only accurate way of judging them.

Another problem is to decide on a particular viewpoint - the values of T and H evolve as one approaches and enters a building, sometimes varying drastically. We have selected a single representative value for T and H in each case. The changing experience as one walks through a building will not be analyzed here. It represents the time dimension of architecture, and was carefully controlled by the greatest architects to create the maximum effect on the user ^{(5, 6)}.

6.2 Analysis of the Carson, Pirie, Scott Department Store

Louis Sullivan's Schlesinger & Mayer building, now the Carson, Pirie, Scott & Co. store (building No. 14), is a high point of American architecture ⁽13^, 14^, 15⁾. We illustrate our model by going through the measurements of T and H . First, the glorious cast-iron facade has detail down to 1mm ( T₁ = 2). Looking up from the street shows the detailed pattern outlining the upper windows, which cannot be seen in photos taken from a height ^{(13, 15)}. Altogether, there is a very high density of differentiations ( T₂ = 2). The facade is an organic complex of curves ( T₃ = 2). It is dark green, while the upper stories are faced in white terracotta tiles ( T₄ = 1). There is no contrast in color hue ( T₅ = 0). The original interior did have color contrasts, but it is now completely altered.

The building is almost bilaterally symmetric, and its facades and windows are piecewise symmetric ( H₁ = 2). The windows define rows and columns of translational symmetry ( H₂ = 2). The cylindrical pavillon on the street corner is similar to the curved corner stories on top of it, while the storefront windows maintain the same scaling as the upper windows ( H₃ = 2). The facade is connected internally, and the cylindrical corner portion of the building has columns in relief all the way up; but the facade is not connected to the upper stories ( H₄ = 1). The overall color harmony is pleasing, though the terracotta does not harmonize with the dark metal ( H₅ = 1). The removal of the original roof projection and its replacement with a sheer edge has lowered the architectural harmony ^{(13, 15)}.

6.3 Some Comparisons Between Buildings

To demonstrate the utility of this model, we compare entirely dissimilar buildings in form, that nevertheless have similar values of C or L . Table 1 shows that Charlemagne's Palatine Chapel in Germany (building No. 4) has the same values as the Phoenix Hall of the Byodo-in Temple in Japan (building No. 5). The former follows an octagonal Byzantine plan, whereas the latter is a development of the classic Buddhist temple tradition. No two buildings could be more different in appearance, yet a viewer responds in a comparable way to both of them.

In the same way, the Romanesque Baptistry at Pisa (building No. 8) compares with the Art Nouveau Carson, Pirie, Scott store (building No. 14). Louis Sullivan's great achievement is that he generated the same degree of architectural life without copying anything that had been built before him. By coincidence, the Parthenon (building No. 1) also shares the same values as these two buildings, but this is not a fair comparison, since it has lost most of its sculptures, walls, and coloration.

Two modern buildings that share similar values are Frank Lloyd Wright's Kaufmann house "Fallingwater", in Bear Run, Pennsylvania (building No. 16), and Jørn Utzon's Sydney Opera House (building No. 22). Both are free, innovative, interesting, and generate feelings of similar intensity, despite having entirely different characteristics.

A case of contrast occurs between two religious buildings: Salisbury Cathedral (building No. 7), and the Pilgrimage Chapel of Notre Dame du Haut at Ronchamp, France, by Corbusier (building No. 18). They have about the same value for the architectural complexity - i.e., the same level of interest - but the former has more than thirty times the architectural life of the latter. The comparison belies the statement, common in architecture books, that this particular building by Corbusier is not susceptible to systematic analysis with respect either to his other work, or to other religious buildings.

6.4 The Universal Drive to Raise the Architectural Life

Figure 1 shows that man worked very hard to raise the architectural life of his surroundings, up until the 20th century. People with entirely distinct conceptions of beauty, using very different materials, and driven by similar motivations, managed to build structures that cluster together in the top corner of Figure 1. These buildings do not resemble each other in form. Furthermore, our choice of buildings is only a representative sample: hundreds of buildings from before the 20th century lie in the top corner of Figure 1.

Like animals with the instinct for complicated courtship and nest-building, we have an instinct to build things that embody certain qualities. For thousands of years, structures were built that do not meet any obvious utilitarian need; and yet they occupy a central role in cultures, requiring vast commitments in manpower and time. A simple shelter does not require the incredible sophistication that people have invested in buildings. Throughout history, buildings have reflected mankind's drive to transcend materials and produce something to which we can relate directly on a deep emotional level.

What about houses and ordinary buildings? This model applies to all structures, and not just to important historical buildings. Vernacular architecture has reflected the values of L and C of "official" buildings throughout history. For instance, classical Greek and Roman houses were sufficiently detailed and coherent to give high values for L similar to those of contemporary temples, despite having an entirely different form. Although houses and commercial buildings in our time are strongly influenced by architectural fashion to have low L , their inhabitants instinctively raise L by decorating interior surfaces.

6.5 The Limits of Architectural Life and Complexity

Figure 1 includes all buildings within a large triangle, whereas the older and modernist buildings are each restricted to within much smaller triangles. This is a mathematical consequence of our definition of L and C . From equations (3) and (4), we have the identity:

L + C = 10T

(5)

The maximum possible value for T is 10, so Eq. (5) defines an upper limit for the architectural life in terms of the complexity as L = 100 - C . This relationship is represented by the diagonal in Figure 1. All structures in history, and all structures yet to be built, therefore lie inside the large triangle of Figure 1.

Measurements establish the fact that traditional buildings strive for a high value of architectural life L . They inhabit the top corner of Figure 1, shown as the upper small triangle. It is already explained in Section 4 why the complexity C of high L buildings does not vanish, and for this reason the triangle enclosing them is displaced from the L axis.

A similar but distinct relationship holds for modernist buildings. Their architectural temperature is very low, and this provides an upper bound for both the architectural life and the architectural complexity. The purest modernist buildings included here are numbers 18, 19, and 21, which occupy the triangle L < 10, C < 10 with the diagonal bound L = 10 - C . The pure modernist idiom, whether defined by aesthetic principles, or by pioneering buildings that are used as models by succeeding architects, is restricted to a very narrow range of parameters represented by the lower small triangle in Figure 1.

7. THERMODYNAMICS AND THE ARCHITECTURAL MODEL

The model we have described is inspired by thermodynamics, which provides insight into the fundamental processes of architecture. We have taken concepts such as symmetries and coherence that are well known in architecture ⁽⁶⁾, and combined them into a sort of thermodynamic potential. Whereas in the past they have always been considered separately and qualitatively, we made the relevant qualities measurable, and then synthesized the values obtained into a consistent and robust model that has predictive value.

The model depends on the degree of randomness, which is measured by some sort of architectural entropy S . The word entropy is used in a very particular sense, and its meaning is analogous to but not the same as the thermodynamic entropy in physics. The entropy of a design is defined as the degree of randomness in the patterns. We used the architectural harmony H in order to measure S indirectly as S = 10 - H on a scale of 0 to 10. Since the architectural entropy S represents the absence of symmetries, connections, and harmony, it is more difficult to measure than the presence of those qualities.

The architectural entropy is, like the thermodynamic entropy, an extensive or bulk function. What we define here as the architectural entropy is the average over the entire form. This is not the entropy directly, but it permits us to compare the architectural entropy of two buildings of different sizes. Without normalization, the architectural entropy of the sum of two buildings would be the sum of the entropies. Since the architectural entropy is normalized, the entropy of the sum is the entropy of the ensemble, which is more useful for architectural purposes.

The thermodynamic temperature is an intensive or point function that measures quantities locally. The temperatures of two separated points do not add. The architectural temperature T as defined here, however, takes into account both local and average qualities. Each component T_i of T measures the maximum point value anywhere in the design, as well as the average of that quantity over the entire form. This combined method is the best way to measure local differentiations as departures from equilibrium, and at the same time, as a value on an absolute scale.

In physics, T and S have different units. Here, T and S are dimensionless numbers, and are combined to get other dimensionless numbers such as H , L , and C . It is the thermodynamic potentials that characterize a system, so we define the architectural complexity C as the product TS . This makes C look like the internal energy or the enthalpy. The architectural life L = 10T - TS would then correspond to something like the Gibbs potential or the Helmholtz free energy. Combinations such as these also characterize the state of an architectural system, which is the key to our model. The idea is that similar laws should govern organization in thermodynamics as well as in architecture.

8. THE LINK TO BIOLOGICAL LIFE

The notion of "life" in architecture is due to Alexander ⁽³⁾, who has worked very hard to achieve it in his own buildings ⁽16^, 17 ^, 18). Our formulation attempts to codify some of Alexander's results. More than just creating a utilitarian structure, mankind strives to approach the intrinsic qualities of biological forms in its traditional and vernacular architectures. This result is not obvious, because very few buildings actually copy living forms: the resemblance is obtained by raising L via the structural temperature and harmony.

Starting initially from a traditionalist point of view, Charles, the Prince of Wales has also discovered style-independent rules that raise the architectural life. He calls these his ten principles ⁽¹⁰⁾. Although the approach and details are different, these developments are supported both by Alexander's results, and by the model of this paper. The links between biological and architectural life are now being recognized formally. We are witnessing a convergence of ideas coming from several different directions, and forming an entirely new approach to architecture.

One class of examples of artificial objects that mimic living forms is beautiful self-similar fractal curves. The design temperature T of fractal curves is very high; the harmony H is also very high because they are self-similar (any portion, when magnified by a fixed factor, looks exactly like the original form) ⁽⁴⁾. Therefore, they have a high degree of architectural life L . As is well-known, fractal pictures resembling natural objects provide excellent representations ⁽⁴⁾, and this property serves to support our model.

The connection between biological life and architecture arises from the thermodynamics of living forms. Life is the result of an enormous amount of purposeful complication. Biological organisms are marvelously connected on all different levels, and they are characterized by very high design temperature and harmony. The connective thought processes underlying cognition themselves mimic the thermodynamic and connective structures that are characteristic of living forms. This helps to explain our instinct to relate to forms having a high degree of architectural life.

The architectural temperature mimics the activity of life processes, which is highly organized and structured. It should not be surprising that living beings instinctively copy the intrinsic qualities of living systems in their own creations. How can humans put an image of life into a building? Apart from figurative icons and statues, we work with emotions: structures are carefully tailored to generate positive psychological and physiological responses. Far from merely being a plausible hypothesis, this model suggests that humans have a basic need to raise the architectural life of their environment.

9. CONCLUSION

A model for architectural forms was inspired by thermodynamics. By measuring the architectural temperature T and the architectural harmony H , we estimate the architectural life L of a building by analogy to a thermodynamic potential. Incredibly, the value computed for L in this way corresponds directly to the emotional perception of a building's "life". A different potential, the architectural complexity C , is a distinct combination of T and H . Again, the computed value for C in any building corresponds directly with what is emotionally perceived as its "complexity". This establishes a link between intrinsic qualities of architectural forms, and the subconscious connection they establish with people.

The model represents a first attempt at analyzing architecture with a quantitative method. While the results depend on the detailed definition of the variables, the basic principles are fairly robust. One of the results is to critically distinguish older historical buildings from those of the 20th century. This was illustrated dramatically in a plot of the architectural complexity C versus the architectural life L of twenty-five famous buildings. We interpreted this in terms of the attempt to mimic fundamental processes in nature. Traditional buildings derive their structure from physical and biological processes, whereas modernist forms seek innovation through features that do not occur in nature.

This quantitative description of architecture was based on the buildings themselves. We looked at buildings in isolation and gave them a quantitative place on a scale. The model, which follows ideas of Christopher Alexander, also applies to the impact of a building on its environment. Alexander's approach is holistic, and considers a building and its environment to be a whole unit. The measure of architectural harmony applies especially to the juxtaposition of a building with adjoining buildings, natural scenery, the sky, and the ground. Even if a building is internally harmonious, it creates a low-harmony impression when its edges clash with the environment. A built environment with high architectural life therefore connects all structures to their surroundings.

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