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A Complex-Network Perspective on Alexander’s Wholeness

Bin Jiang

Faculty of Engineering and Sustainable Development, Division of GIScience

University of Gävle, SE-801 76 Gävle, Sweden Email: bin.jiang@hig.se

(Draft: February 2016, Revision: April, July 2016)

“Nature, of course, has its own geometry. But this is not Euclid’s or Descartes’ geometry. Rather, this geometry follows the rules, constraints, and contingent conditions that are, inevitably, encountered in the real world.”

Christopher Alexander et al. (2012)

Abstract The wholeness, conceived and developed by Christopher Alexander, is what exists to some degree or other in space and matter, and can be described by precise mathematical language. However, it remains somehow mysterious and elusive, and therefore hard to grasp. This paper develops a complex network perspective on the wholeness to better understand the nature of order or beauty for sustainable design. I bring together a set of complexity-science subjects such as complex networks, fractal geometry, and in particular underlying scaling hierarchy derived by head/tail breaks – a classification scheme and a visualization tool for data with a heavy-tailed distribution, in order to make Alexander’s profound thoughts more accessible to design practitioners and complexity-science researchers. Through several case studies (some of which Alexander studied), I demonstrate that the complex-network perspective helps reduce the mystery of wholeness and brings new insights to Alexander’s thoughts on the concept of wholeness or objective beauty that exists in fine and deep structure. The complex-network perspective enables us to see things in their wholeness, and to better understand how the kind of structural beauty emerges from local actions guided by the 15 fundamental properties, and by differentiation and adaptation processes. The wholeness goes beyond current complex network theory towards design or creation of living structures. Keywords: Theory of centers, living geometry, Christopher Alexander, head/tail breaks, and beauty 1. Introduction Nature, or the real world, is governed by immense orderliness. The order in nature is essentially the same as that in what we build or make, and underlying order-creating processes of building or making of architecture and design are no less important than those of physics and biology. This is probably the single major statement made by Alexander (2002–2005) in his theory of centers, in which he addressed the fundamental phenomenon of order, the processes of creating order, and even a new cosmology – a new conception of how the physical universe is put together. In the theory of centers or living geometry (Alexander et al. 2012), the wholeness captures the meaning of order and is defined as a life-giving or living structure that appears to some degree in every part of space and matter; see Section 3 for an introduction to wholeness and wholeness-related terms. As the building blocks of wholeness, centers are identifiable coherent entities or sets that overlap and nest each other within a larger whole. Unlike the previous conception of wholeness focusing on the gestalt of things (Köhler 1947), the wholeness of Alexander (2002-2005) is not just about cognition and psychology, but something that exists in space and matter. Different from the wholeness in quantum physics mainly for understanding (Bohm 1980), Alexander’s wholeness aims not only to understand the phenomenon

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of order, but also to create order in the built world or art. The wholeness is defined as a recursive structure. Based on this definition, Jiang (2015) developed a mathematical model of wholeness as a hierarchical graph with indices for measuring degrees of life or beauty for both individual centers and the whole. This model helps address not only why a design is beautiful, but also how much beauty it has. However, this previous study had some fundamental issues on the notions of centers and wholeness unaddressed. Specifically, what are the centers, how are they created, and how do they work together to contribute to the life of wholeness? In addition, the wholeness remains somehow mysterious, particularly within our current mechanistic worldview (Alexander 2002–2005). To address these fundamental issues, this paper develops a complex-network perspective on the wholeness. A complex network is a graph consisting of numerous nodes and links, with unique structures that differentiate it from its simple counterparts such as regular and random networks (Newman 2010). Simple networks have a simple structure. In a regular network, all nodes have a uniform degree of connectivity. In a random network, the degrees of connectivity only vary slightly from one node to another. As a consequence, a random network hardly contains any clusters, not to say overlapping or nested clusters. On the contrary, complex networks, such as small-world and scale-free networks (Watts and Strogatz 1998, Barabási and Albert 1999), tend to contain many overlapping and nested clusters that constitute a scaling hierarchy (Jiang and Ma 2015; see a working example in Section 2). The scaling hierarchy is a distinguishing feature of complex networks or complex systems in general. For example, a city is a complex system, and a set of cities is a complex system (Jacobs 1961, Alexander 1965, Salingaros 1998, Jiang 2015c), both having scaling hierarchy seen in many other biological, social, informational, and technological systems. This paper demonstrates that the wholeness bears the same scaling hierarchy as complex networks or complex systems in general. Relying on the complex network perspective, this paper aims to demonstrate that wholeness is not just in cognition and psychology, but something that exists in space and matter. It also aims to show that the concept of wholeness is important not just for understanding the phenomenon of order, but also for creating order with a high degree of wholeness through two major structure-preserving transformations: differentiation and adaptation. I argue that there are major differences between the whole as a vague term and wholeness as a recursive structure. The wholeness comprises recursively defined centers induced by itself, whereas the whole, as we commonly perceive, comprises pre-existing parts. The mantra that the whole is more than the sum of its parts should be more truly rephrased as the wholeness is more than the sum of its centers. This paper examines the notions of wholeness and centers from the perspective of complexity science. It also discusses two types of coherence respectively created by differentiation and adaptation processes, which are consistent with the spatial properties of heterogeneity and dependence for understanding the nature of geographic space. The remainder of this paper is structured as follows. Section 2 briefly introduces complex networks and scaling hierarchy using head/tail breaks – a classification scheme and a visualization tool for data with a heavy-tailed distribution (Jiang 2013a, Jiang 2015a). Section 3 compares related concepts, such as whole and parts versus wholeness and centers, and discusses the theory of centers using two examples of a cow and an IKEA desk. Section 4 presents three case studies to show how wholeness emerges from space and how it can be generated through the two major structure-preserving transformations of differentiation and adaptation. Section 5 further discusses implications of the complex-network perspective and wholeness. Finally, Section 6 draws a conclusion and points to future work. 2. Complex networks and the underlying scaling hierarchy Small-world and scale-free networks are two typical examples of complex networks, which fundamentally differ from their regular and random counterparts. A small-world network is a middle status between the regular and random networks, so it has some nice properties of its regular and

 

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complex networks were understood. More importantly, his thoughts are not just limited to understanding surrounding things, just as what current complex network theory does, but aim for creating things with living structure (i.e., buildings, communities, cities or artifacts). The scaling hierarchy can be further seen from the relevance or importance of individual nodes within a complex network. Through Google’s PageRank (Page and Brin 1998), nodes’ relevance or importance can be computed, as there are far more less-important nodes than more-important ones. PageRank is recursively defined and resembles the wholeness as a recursive structure. This will be further discussed in the following section. 3. The wholeness and the theory of centers The general idea of wholeness, or seeing things holistically, can be traced back to the Chinese philosopher Chuang Tzu (360 BC), who saw the structure of a cow as a complex whole, in which some parts were more connected (or coherent) than others. The butcher who understands the cow’s structure always cuts the meat from the soft spots and the crevices of the meat, so makes the meat fall apart according to its own structure. The butcher therefore can keep his knife sharp for a hundred years. In the 20th century, wholeness was extensively discussed by many writers prominent in Gestalt psychology (Köhler 1947), quantum physics (Bohm 1980), and many other sciences such as biology, neurophysiology, medicine, cosmology and ecology. However, none of these writers prior to Alexander (2002–2005) showed how to represent or formulate wholeness in precise mathematical language. According to the Merriam-Webster dictionary, whole is something complete, without any missing parts. On the other hand, wholeness is the quality of something considered as a whole. However, both whole and wholeness have deeper meanings and implications in the theory of centers (Alexander 2002–2005). In particular, the notion of wholeness is so subtle and profound that Alexander (1979) previously referred to it as ‘the quality without a name’. He struggled with different names, such as alive, comfortable, exact, egoless and eternal, but none of these captured the true meaning of the quality. In this paper, as in Alexander (2002-2005), the three terms wholeness, life, and beauty are interchangeably used when appropriate to indicate order or coherence. Things with a high degree of wholeness are called living structure.

Table 1: Comparison of whole and wholeness using two examples of a cow and an IKEA desk

Whole (noun + adjective)  Wholeness (structure + measure) on the surface  deeper below the surfaceconsists of parts (see below subsection) consists of centers (see below subsection) easy to see, the whole of a cow is the cow itself 

hard to see as a recursive structure, the wholeness of a cow 

hard to sense, one thing is more whole than another 

hard to sense as the degree of coherence somehow like temperature 

easy to sense, a cow is more whole than a desk 

easy to sense, a cow has a higher wholeness than a desk 

a whole assembled from parts like a desk a desk has low wholeness or low coherence a whole unfolded from an embryo like a cow a cow has high wholeness or high coherence Parts  Centerspre‐existing in the whole  induced by the wholenesson the surface  deeper below the surfaceeasy to see  hard to seemechanical  organicnon‐recursive  recursivesimple  complex

Wholeness is defined as a recursive life-giving structure that exists in space and matter, and it can be described by precise mathematical language (Alexander 2002-2005). Although whole and wholeness seem different, they are closely related, and sometimes refer to the same thing. The whole is on the surface and is referred to informally, while wholeness is below the surface and is referred to formally (Table 1). The term whole can be both a noun and an adjective. For example, a cow is a whole, and a

 

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8 

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and robust scientific underpinning (Jacobs 1961, Marshall 2012). The complex network perspective or more truly the wholeness itself will inject scientific elements into urban planning and design. The Sierpinski carpet is an image of the Earth’s surface metaphorically. This is because there are far more small things than large ones across all scales – the spatial property of heterogeneity, and related things are more or less similar in terms of magnitude at each scale – the spatial property of dependence. Both heterogeneity and dependence are commonly referred to as spatial properties about geographic space or the Earth’s surface (Anselin 1989, Goodchild 2004). The carpet is also an ideal metaphoric image toward which our built environment should be made. It implies that any space ought to be continuously differentiated to retain the scaling hierarchy across all scales ranging from the smallest to the largest, and any building or city ought to be adapted to its natural and built surroundings at each scale. The two processes of differentiation and adaptation enable us to create a whole with a high degree of wholeness. In this regard, the wholeness or the theory of centers would have enormous effects on geography, not only for better understanding geographic forms and processes, but also for planning and repairing geographic space or the Earth’s surface (Mehaffy and Salingaros 2015, Mehaffy 2007). Built environments must adapt to nature, and new buildings must adapt to their surroundings. The two spatial properties of heterogeneity and dependence constitute a true image of the Earth’s surface at both global and local scales. Globally, spatial phenomena vary dramatically with far more small things than large ones across all scales, while locally they tend to be dependent or auto-correlated with more or less similar things nearby. These two properties are the source of the two kinds of harmony or coherence respectively across all scales and at every scale. For example, there are far more small cities than large ones across all scales globally (Zipf 1949), whereas nearby cities tend to be more or less similar in terms of the central place theory (Christaller 1933, 1966). However, the geography literature focuses too much on dependence, formulated as the first law of geography (Tobler 1970), but very little on heterogeneity. More critically, spatial heterogeneity is mainly defined for spatial regression with limited variation, governed by Gaussian thinking (Jiang 2015d). This understanding of spatial heterogeneity is flawed, given the fractal nature of geographic space or features. Spatial heterogeneity should be formulated as a scaling law because it is universal and global. The wholeness or the theory of centers in general brings a new perspective to the science of complex networks. For example, the detection of communities of complex networks can benefit from the wholeness as a recursive structure. Instead of a flat hierarchy of community structures, a complex network contains numerous nested communities of different sizes, or far more small communities than large ones (Tatti and Gionis 2013, Jiang and Ma 2015). This insight about nested communities can be extended to classification and clustering. Current classification methods can be applied to mechanical assembly, so mechanical parts can be obtained. For a complex entity such as complex networks, the parts are often overlapping and nested inside each other. This resembles both wholeness and centers as a recursive structure. From a dynamic point of view, a complex network is self-evolved with differentiation and adaptation. A complex network has the power to evolve toward more coherence at both global and local scales, and is self-organized from the bottom up, rather than imposed from the top down. This insight about self-organization reinforces Alexander’s piecemeal design approach through step-by-step unfolding or transformations for a living structure with a high degree of wholeness. 6. Conclusion This paper develops a complex network perspective on the wholeness to make it more accessible to both designers and scientists. I discussed and compared related concepts such as whole and parts versus wholeness and centers to make them explicitly clear. A whole is a relatively coherent spatial set, while wholeness is a life-giving structure – not something about the way they are seen, but something about the way they are (Alexander 2003, p. 14). The wholeness is made of centers rather than

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arbitrarily identified parts. The centers are created or induced by the wholeness and made of other centers, rather than just those pre-existing in the whole. Given these differences, the mantra that the whole is more than the sum of its parts should be more correctly rephrased as the wholeness is more than the sum of its centers. I demonstrated that the complex-network perspective enables us to see things in their wholeness. More importantly, I elaborated on design or making living structure through wholeness-extending transformation or unfolding in step-by-step differentiation and adaptation. Although understanding the nature of wholeness is essential, the ultimate goal of Alexander's concept of wholeness or living geometry is to allow us to make artifacts and built environments with the same order and beauty of nature itself. The complex-network perspective enables us to develop new insights into planning and repairing geographic space. For example, differentiation and adaptation are two major processes for making living structures or geographic space in particular. This study showed that these processes underlie the two unique properties of geographic space: spatial heterogeneity and spatial dependence. These are respectively formulated as the scaling law and the first law of geography. Globally across all scales, there are far more small things than large ones. In contrast, locally at every scale, things tend to depend on each other with more or less similar sizes. These two spatial properties are the source of the two types of coherence respectively at global and local scales. In this connection, the theory of centers or living geometry would significantly contribute to understanding and making geographic space. Our future work points to this direction on how to rely on wholeness-extending transformations to create geographic features with a high degree of wholeness. Acknowledgement XXXXXXXXXXX References: Alexander C. (1964), Notes on the Synthesis of Form, Harvard University Press: Cambridge,

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