If the life’s work of Alexander [1
]—The Nature of Order
—had to be summarized in one word, “beauty”, “life” and “wholeness” would be the three top candidates. If allowed two words, it would be “living structure”. What do these terms really refer to? Instead of getting into their detailed meanings (see Section 2
), let us use an analogue to clarify them first. If wholeness were compared to temperature, then beauty or life would be like the feeling of warmness or coldness. The higher the temperature, the warmer one feels, and the lower the temperature, the colder one feels. The higher the wholeness, the more beautiful or the more life one feels; the lower the wholeness, the less beautiful or the less life one feels. Therefore, a thing or structure that exhibits a high degree of wholeness is called a living structure. Opposite to living structure is non-living (or dead) structure. There is a wide range between the living and the dead, so living is always to some degree or other, just as the feeling of warmness relates to a range of temperatures. Living structure is what Alexander [1
] discovered and further pursued, and it is independent of any style or culture from for example Indonesia, Japan, Russia, Africa, Turkey, Iran, India, or China. Having said that, Alexander has no particular style of buildings, contrary to what his rivals or critics tend to think. To put it different, living structure is a living style, just as nature itself, being able to trigger a sense of belonging or well-being or healing in people who are exposed to it. To know whether a thing or space exhibits living structure, one can simply examine whether it possesses “far more smalls than larges” across all scales ranging from the smallest to the largest. For example, at the multiple levels of scale or in a recursive manner—an entire tree, its branches, and its leaves (in terms of the detailed texture)—there are always “far more smalls than larges”. Therefore, a tree is beautiful or alive structurally, regardless of whether it is alive biologically.
Based on the notion of “far more smalls than larges”, a simple shape that lacks detailed smaller structures is neither beautiful nor alive. This is for the same reason why sans-serif fonts are less beautiful or less alive than serif ones. For example, the font “I” (when shown as a sans-serif) is not a living structure (one vertical line only) without “far more smalls than larges”, whereas the font “I” (when shown as a sans-serif) is a relatively living structure (one vertical line and two little bars) with “far more smalls than larges”. The difference between the non-living and living fonts may be hardly sensed when the two fonts are too small, in particular when the letter’s meaning is focused on. As a matter of fact, serif fonts in general are objectively more beautiful than sans-serif ones. It is based on this kind of structural fact—actually the phenomenon of living structure—that Alexander [1
] established a scientific foundation of architecture. Unfortunately, the phenomenon of living structure has not yet been well accepted by the scientific community as a fact but been sidelined as a human taste or personal preferences. This situation constitutes a major motivation of this paper. Human history is full of many great builders or architects who made great buildings, but few of them really made it clear—or even intended to think about—how to make great buildings and why the great buildings are great.
As an architect who was initially trained in science, Alexander wanted to make beautiful buildings, and he wanted to know in particular why beautiful buildings are beautiful. His classic work on the pattern language [2
] is widely read by ordinary people looking to make beautiful rooms, houses and gardens, and to facilitate their daily lives, for example, where to put a lamp or a flower vase, and how to lay a tablecloth. His design thoughts are therefore very practical—down-to-earth—and his research is essentially empirical. On the other hand, his research is deeply philosophical and visionary, up to heaven, touching the fundamental issues of what the universe constitutes, and where our consciousness comes from. He conceived and developed—from the phenomenon of living structure—a third view of space, and a new cosmology in which we human beings—not only the body but also the mind—are part of the universe [1
] (Volume 4, c.f. Section 5
for a more detailed discussion). The third view of space states that space is neither lifeless nor neutral, but a living structure capable of being more living or less living. This new view of space sets a clear difference from two traditional views of space: absolute space by Isaac Newton (1642–1727) and relational space by Gottfried Wilhelm Leibniz (1646–1716), both of which are framed under the mechanistic world view of Descartes [3
]. This new view of space constitutes part of the new cosmology that unifies the physical world and our inner world as a coherent whole.
Despite a large body of literature on or inspired by Alexander’s work [4
], living structure has not yet been well recognized as a physical phenomenon or mathematical concept, for people to understand the objective or structural nature of beauty. This paper is an attempt to fill this gap, by setting up a dialogue with those who are skeptical about Alexander’s design thoughts. It is intended to clarify some doubts in order for skeptics to understand three main points. First, the essence of beauty is structural or objective, lying in the notion of “far more smalls than larges”, which accounts for a majority of our feeling on beauty. There is a clear sign that beauty is beginning to be accepted as an objective concept in the literature of philosophy [13
]. Second, the phenomenon of living structure is universal and pervasive, not only in nature but also in what we made and built across all cultures, ethics, and religions, involving ancient buildings and cities, as well as ancient carpets and other artifacts. Thus, there is no so-called Alexander’s style of architecture; if there is, it is the living structure (just as nature itself), which is able to trigger a sense of beauty or life in the human mind. Third, there is no mystery at all regarding the “quality without a name
], which is actually living structure, yet the mystery of a non-material world view remains.
The remainder of this paper is structured as follows. Section 2
introduces and illustrates the living structure as a physical phenomenon, using a sketch by Alexander, in terms of its governing laws (scaling law and Tobler’s law), its design principles (differentiation and adaptation), and its 15 structural properties (Table 1
). Section 3
argues why living structure is scientific or empirical by drawing evidence from Alexander’s own works such as the pattern language. Section 4
further presents case studies to demonstrate that living structure is objective or structural rather than just a matter of opinion. Section 5
discusses the metaphysical aspects in order to make better sense of the living structure in terms of why living structure evokes a sense of beauty or life in our minds. The paper concludes with a few remarks and suggestions for future work.
2. Living Structure: Its Governing Laws and Design Principles, and 15 Structural Properties
The four terms mentioned at the outset of this paper can be placed into two categories—wholeness and living structure in the first group, and beauty and life in the second group—representing the outer and inner worlds, respectively. The central concept among these four is wholeness, which can be defined mathematically [1
]. It exists pervasively in our surroundings; in an ornament, in a room, in a building, in a garden, and in a city. It was previously referred to by Alexander as the “quality without a name”: “a central quality which is the root criterion of life and spirit in man, a town, a building, or a wilderness. This quality is objective and precise, but it cannot be named
]. The term wholeness is also a key concept in Gestalt psychology [16
], in quantum physics [17
], and in many other religious and philosophical contexts. Semantically, there may be some overlap across these different fields, but Alexander’s wholeness is unique with its distinguishing features. It is not only a static structure, but also a dynamic process, through which living structure emerges. In the next part of this section, we will use a sketch by Alexander [1
] (6a in Figure 1
) to introduce and illustrate living structure or wholeness.
The sketch shows the evolution of a living structure, demonstrating many of the 15 properties (Table 1
). It consists of at least 19 different sized mutually overlapping, nested shapes or centers in Alexander’s terms, namely the four outmost black dots, the square, the big circle, the eight tiny triangles, the four small circles, and the tiny dot in the middle. Among many of the other properties, the first property of the levels of scale is the most distinguishing one. The sketch is with six levels of scale, indicated by the six colors with red being the highest, blues being the lowest, and other colors in between (6b in Figure 1
). It should be noted the six levels of scale is not in terms of their sizes, but the supports they receive. Overall, there are “far more blues than red”, and some in between the blues and the red in the colored sketch, the spectral coloring in terms of the degree of wholeness of individual centers. For example, the tiny center in the middle has the highest degree because it receives many supports from other centers. The notion of “far more smalls than larges” reflects the very first property of living structure, namely levels of scale (Table 1
), or “scaling hierarchy” or scaling law [18
]. As a reminder, the scaling hierarchy of “far more smalls than larges” should be—more correctly—understood in a recursive manner, implying that “far more smalls than larges” for a whole, for a sub-whole, and a sub-whole of sub-wholes and so on (c.f. the above tree example). The scaling hierarchy of “far more smalls than larges” recurs multiple times rather than just once, except for some simple cases like font “I”. In the living structure, the recurring happens five times (steps 2–6 in Figure 1
), leading to six hierarchical levels. The living structure can be said, more precisely, to be evolved, which implies that centers are well adapted to each other as a coherent whole. The living structure is not simply an assembly of pre-existing components. In this regard, the less-living structure (6c in Figure 1
) is indeed an assembly of pre-existing units.
The less-living structure (6c and 6d in Figure 1
) looks smooth, glassy, and uniform, but it exhibits a lower degree of wholeness. There are several reasons for this, a few of which are highlighted here. First, the less-living structure is created (at once) by assembling rather than generated (step by step) by adaptive design. The lack of adaptation can be clearly seen from figure and ground relationship [19
], or the fact that the space between these geometric shapes is not well shaped, or not convex like ripening corn, “each kernel swelling until it meets the others, each one having its own positive shape caused by its growth as a cell from the inside
]. Second, the four dots outside the square of the less-living structure are less integrated into the whole inside the square. This is because the space outside of square is fully open and therefore lacks a sense of belonging for the four dots, as strikingly shown in the living structure. Third, the less-living structure misses the property of roughness. Interested readers can compare against the 15 structural properties (Table 1
) to learn why one structure is more living than the other: the more structural properties, the more living a structure is.
There are two fundamental laws of living structure, scaling law and Tobler’s law (Table 2
), which also underlie the 15 structural properties (Table 1
). The first structural property (levels of scale) reflects scaling law, as elaborated above, while the remaining properties are largely a reflection of Tobler’s law [20
], probably except for the property of “not separateness”. Tobler’s law, which is commonly called the first law of geography, states that “everything is related to everything else, but near things are more related than distant things
”. Essentially, Tobler’s law is complementary to—rather than contradictory to—scaling law (Table 2
), indicating that on each scale, centers are “more or less similar”. It is important to stress that centers that are “more or less similar” are more beautiful or more alive than centers that are precisely the same. For example, in the living structure, the four small circles are “more or less similar”, and eight triangles are “more or less similar”, so they are much more living than if they were precisely the same in size, as shown in the less-living structure. With the perfectly drawn shapes, some of the 15 structural properties are no longer available, such as positive space and roughness, which can be phrased as “the perfection of imperfection
]. These two properties are the most important for naturally evolved things, such as cell structures and maize grains. On the surface, naturally evolving things may look rather rough or irregular, yet they tend to exhibit the essence of natural beauty. As for Tobler’s law or the notion of “more or less similar” on each scale, we can add another example: A coastline with the same degree of complexity as the Koch curve [22
], with which things (or segments) are precisely the same at each of scales such as 1/3, 1/9, 1/27 and so on. The coastline at each of its scales exhibits the property of “more or less similar” segments rather than precisely the same ones, so the coastline is more natural, more beautiful, or more living than the Koch curve.
From a design or dynamic point of view, a space or structure is continuously differentiated toward scaling hierarchy of “far more smalls than larges”. Actually, these two laws are largely statistical, which does not guarantee living structure. To make the structure really living or beautiful, we must consider geometric aspects. This is the idea of adaptation: on each level, things should be “more or less similar”, or nearby things should be “more or less similar”. Note that “nearby” is usually referred to in a geometrical distance, but a topological distance is better in many instances. For example, my neighbor is defined within a certain geometric distance, but an airport’s neighbor is better defined in terms of topological distance of flight connections. My neighbor’s house should look “more or less similar” (in size and shape) to my house, whereas Heathrow Airport should look “more or less similar” (in size or capacity) to the Paris Charles De Gaulles Airport rather than the Gatwick Airport, because there is no flight between the Heathrow and the Gatwick. Along these two laws, there are two design principles: Differentiation and adaptation. The living structure in Figure 1
is continuously differentiated to reach the status of living structure.
It is important to realize—as Alexander noted repeatedly—that the evolution process is not simply about adding new centers; more correctly, centers are induced by the wholeness. In other words, it is incorrect to say a whole comes from parts, or a whole consists of parts; it is the wholeness that induces centers to generate a coherent whole. It is incorrect to say a flower consists of petals; it is the flower as a whole that induces petals. Another design principle is adaptation. On each level of scale, saying that things are “more or less similar” implies things are adapted to each other. Again, this notion of “more or less similar” things should really be understood literally. If things are exactly the same, it tends to generate a structure that is less living or less beautiful; see also examples mentioned above about the coastline versus the Koch curve. It should be noted that adaptation could imply things adapted across scales. This is again a good example of Alexander’s observation (see more in Section 3
). Alexander found that, across levels of scale, the scaling ratio should be between 2 and 3; otherwise structure would look less living (see Figure 4 in Section 3
). It is in this sense that Alexander’s living geometry generally surpasses fractal geometry [23
]. Fractal geometry hardly cares about whether the generated pattern is beautiful or not, and it only cares about automation of some structure, good or bad, beautiful or ugly.
3. A commonsense and Humane Approach to Architecture
The phenomenon of living structure exists not only in human-made or -built things, but also in nature. Alexander’s approach to architecture is very much commonsense and humane. More importantly, he wanted to be inspired by nature and to make sure that what he observed from human-made or -built things also applied to nature. For example, the 15 properties of living structure are pervasively seen not only in the built environment, but also in nature. In this section, we draw evidence from Alexander’s earlier works to learn why living structure is scientific and empirical, and why this is a correct way of conducting science and art.
Alexander first described the idea of living structures in a corner of an English country garden, where a peach tree grew against a wall:
“The wall runs east to west; the peach tree grows flat against the southern side. The sun shines on the tree and, as it warms the bricks behind the tree, the warm bricks themselves warm the peaches on the tree. It has a slightly dozy quality. The tree, carefully tied to grow flat against the wall; warming the bricks; the peaches growing in the sun; the wild grass growing around the roots of the tree, in the angle where the earth and roots and wall all meet.”
In this living structure of the garden corner, there are many interconnected living centers, such as the wall, the peach tree, the sun, the bricks, the wild grass, the roots of the tree, and even the garden. This is a very good example of Alexander’s miniscule observations on nature and on our surroundings.
Considering another example of embryogenesis, a growing mouse foot is a living structure that comes from continuous differentiation and adaptation (Figure 2
]. In the course of the step-by-step development of the five days, many of the 15 structural properties can be observed, such as strong centers, thick boundaries, gradients, levels of scale, contrast, local symmetries, and finally, good shape of the whole. Alexander started his research on architecture—nearly from scratch—not only from traditional buildings and cities, but also from ancient artifacts such as carpets. Two of his books have accurately documented his miniscule observations: one on built environment [2
]—part of the trilogy with Alexander et al. [14
]—and the other on carpets [26
]. His dream was to build beautiful buildings and cities, sharing the same order or beauty—or spirit—of nature, and his understanding of the kind of natural beauty that exists in deep structure rather than on the surface, such as thermal comfort, energy saving or illumination of surfaces. Nowadays, so-called “green” buildings are not really green according to Alexander, and “a world built according to the present sustainable paradigm, the technical sustainability paradigm, would be quite a horrible place
”. Instead, living structures “represent true sustainability, they sustain the heart, and sustain the soul. They sustain the humanness of the person, and they sustain the Earth
]. When nature and the built environment are treated as one, when the physical world and our human’s inner world are treated as one, and when the precious Earth’s surface and our surroundings are treated as our garden, we will be able to reach the true sense of sustainability.
Back in the 1970s, when he was granted a project by the National Institute of Mental Health in the United States to investigate on relationship between the built environment and human well-being, Alexander and his colleagues started learning what works and what does not. To a great extent, traditional buildings and artifacts were his great teacher. His research is not limited to a certain type, but all types of building—across all cultures and countries—that people know, deep in their heart, are universally beautiful or alive. For example, he noted that a waist-high shelf can be very convenient for people to leave things while searching for keys (Figure 3
a), whereas a room with a sunbeam is more uplifting (Figure 3
b). Thus, both the shelf and sunbeam are supportive to human well-being. It is these kinds of materials that constitute the major content of pattern language [2
Alexander describes his early work on pattern language as follows:
“To get my feet on the ground, and to have something solid that I could be sure of, I started by examining the smallest particles of functional effect, that I could discern in buildings, with small and sometimes barely significant aspects of the ways that buildings affect people. My purpose in doing this, was to focus on the smallest particles of fact that I could be certain of something that was extraordinarily difficult when faced with the porridge of mush that then passed for architectural theory. In the early years my studies were based on the most ordinary, miniscule observations about usefulness and the effect of buildings on the people who lived in them, always keeping the observations modest, reliable—small enough and solid enough so that I could be sure that they were true.
At first, I included very small particulars of functional effect in any matter that actually made a practical difference to daily life… a shelf besides the door where one could put a packet down while searching for ones keys, for instance, or the possibility of a sunbeam coming into a room and falling on the floor.
But I quickly realized that some of these details were very much more significant than others. Those like the first (the shelf) tended to be pedestrian, even though useful; while those like the second (the sunbeam) were more uplifting, and clearly mattered more in some obvious but profound sense. I began to focus on those miniscule points which mattered more, in the sense of the second example. Gradually, then, I was able to pave the way to the possibility of seeing how buildings support human well-being—not so much mechanical or material well-being, but rather the emotional well-being that makes a person feel deeply comfortable in himself. And as I studied these small effects carefully, gradually I was led to a conception of wholeness, wellness, and spiritual support that might, under ideal circumstances, be present between buildings and human beings.”
Built on this earlier work of pattern language, Alexander realized that there are some structural aspects—the 15 properties—that are the most fundamental to human well-being. For example, the beauty of the blue sky and clouds comes from the property of positive space (Figure 3
c); not only the white clouds but also the blue sky are well shaped. The same positive space appears between a tree’s branches (as Alexander sketched himself in Figure 3
d). This property of positive space is particularly important for urban environments. It implies that not only buildings, but also the space between buildings, should be well shaped like convex spaces, as we discussed with the living versus less-living structures presented in Figure 1
Goodness of space is a matter of fact rather than opinion or personal preferences, and good space has a healing effect, sustaining and promoting health. For example, two spaces that are either too close or too open are transformed into positive spaces to which people can develop a sense of belonging (Figure 4
a,b). This sense of belonging further triggers human well-being, security, or safety. Well-being or comfort provided by environments or space is an important factor for human healing. In this regard, Ulrich [29
], an architect, found that the view from a window may influence a patient’s recovery from surgery; that is, natural scenes are better than urban scenes for post-surgery recovery. Taylor [30
], a physicist, found that fractal patterns, if they are living structures, in nature and art have stress-reducing effect on people. Since both natural scenes and living structures are living rather than non-living, I conjecture that, essentially, it is living structures that have healing effects on people.
Alexander made another important discovery related to the property of levels of scale or scaling hierarchy. He found that the scaling ratio between two consecutive scales should be between 2 and 3; having it too close or too far apart would reduce the goodness or coherence of space or structure (Figure 4
c–e). The major difference between fractal geometry [23
] and Alexander’s work—or living geometry [31
]—is that the former is largely for understanding complexity of fractals. Although fractal geometry is able to create artificial fractals, it often ends with “pretty pictures, pretty useless
]. By contrast, living geometry aims not only to understand complexity, but also to create organized complexity, or beautiful or living structures, which are able to trigger a sense of beauty or life in our minds or hearts. The creation or the making of living structures is what makes Alexander [32
] differ from other pioneers in complexity science.
The following quote shows how Alexander explained the property of the levels of scale or scaling hierarchy pervasively seen in nature and in what we make and build, and how it triggers the feeling of life in our hearts:
“I would like to summarize the content of this new kind of empirical complex in the following way. In any part of what we call nature, or any part of a building, we see, at many levels of scale, coherent entities or centers, nested in each other, and overlapping each other. These coherent entities all have, in varying degree, some quality of “life.”
For any given center, this quality of life comes about as a result of cooperation between the other living centers at several scales, which surround it, which contain it, and which appear within it. The degree of life any one center has, depends directly on the degree of life that is in its associated centers at these different scales. In short, I had identified a kind of wholeness: in which the life of any given entity depended on the extent to which that entity had unfolded from the wholeness.”
In addition to what is summarized above, there are many other empirical findings [33
] that support living structure as a physical phenomenon, as well as a well-defined mathematical concept. In following section, we will carry out some case studies to support that living structure is not only a phenomenon and concept, but also can be used to objectively judge quality of things.
This paper is intended to defend living structure as a physical phenomenon and mathematical concept for people to understand objective or structural nature of beauty, or to setup a dialogue with those who are skeptical about Alexander’s design thoughts. Living structure is a mathematical structure of physical space, which is able to reflect in our minds psychologically: the more living the structure is, the more beautiful one feels. By drawing evidence from Alexander’s work and through our own case studies, this paper has shown that beauty is essentially objective or structural. In other words, beauty exists in the deep structure of details, or in the scaling hierarchy of “far more smalls than larges”. Beauty and ugliness can be clearly defined by scaling law; that is, a structure with a flat scaling hierarchy—with maximum two levels of scale only—is objectively considered to be ugly, whereas a structure with a steep scaling hierarchy—with at least three levels of scale—is objectively considered to be beautiful or alive.
Armed with the kind of simple analysis on scaling hierarchy, people can understand why beautiful buildings beautiful, and why ugly buildings are are ugly. Simply put, a building is beautiful because of its steep scaling hierarchy, or ugly because of its flat scaling hierarchy. By claiming objective or structural beauty, our intention is not to deny idiosyncratic aspects of beauty, which account for only a small proportion of our feeling. This dominance of the objective over the subjective can be compared to any statistical regularity with a majority of agreement, such as an r square value of 0.75 instead of 1.0. In addition to the scaling hierarchy or scaling law, Tobler’s law plays an important role in the objective or structural beauty as well. As one of the two laws of living structure, Tobler’s law—or the notion of “more or less similar”—recurs on each level of scale. The true meaning of “more or less similar” is neither “completely same” nor “completely unique”, but something between the same and the unique. These two complementary laws work together, governing living structures, with the scaling law being primary, and Tobler’s law being secondary.
The I-hypothesis is a powerful concept that makes better sense of the inner meaning of living structure than purely psychological or cognitive explanations. The third view of space provides a fresh look at our surroundings, whereby everything is a living structure, and should become more living or more beautiful through our daily makings. The new cosmology solves the problem of the bifurcation of nature [50
], for we human beings are not separate from, but are part of the universe, thus making our daily lives more meaningful. Human beings can be uplifted by good space, reflected by good architecture, and eventually united and re-united to the hypothesized “I”. To end this paper, we would like to claim that living structure may actually be the “bead game conjecture
], a mechanism that unites all structures or forms in mathematics, science, art, philosophy, and religion. This claim requires further research on living structure from these multiple disciplines or contexts, and implies that living structure is not a dogma, and can instead be further discussed, argued and even challenged.