A Topological Representation for Taking Cities as a Coherent Whole (1607.07169v3)

Abstract: A city is a whole, as are all cities in a country. Within a whole, individual cities possess different degrees of wholeness, defined by Christopher Alexander as a life-giving order or simply a living structure. To characterize the wholeness and in particular to advocate for wholeness as an effective design principle, this paper develops a geographic representation that views cities as a whole. This geographic representation is topology-oriented, so fundamentally differs from existing geometry-based geographic representations. With the topological representation, all cities are abstracted as individual points and put into different hierarchical levels, according to their sizes and based on head/tail breaks - a classification scheme and visualization tool for data with a heavy tailed distribution. These points of different hierarchical levels are respectively used to create Thiessen polygons. Based on polygon-polygon relationships, we set up a complex network. In this network, small polygons point to adjacent large polygons at the same hierarchical level and contained polygons point to containing polygons across two consecutive hierarchical levels. We computed the degrees of wholeness for individual cities, and subsequently found that the degrees of wholeness possess both properties of differentiation and adaptation. To demonstrate, we developed four case studies of all China and UK natural cities, as well as Beijing and London natural cities, using massive amounts of street nodes and Tweet locations. The topological representation and the kind of topological analysis in general can be applied to any design or pattern, such as carpets, Baroque architecture and artifacts, and fractals in order to assess their beauty, echoing the introductory quote from Christopher Alexander. Keywords: Wholeness, natural cities, head/tail breaks, complex networks, scaling hierarchy, urban design

• The paper proposes a novel topological representation to view cities as coherent wholes, measuring their degree of wholeness using head/tail breaks.
• Case studies across China and the UK, including Beijing and London, demonstrate that city sizes and their wholeness degrees follow a power law distribution.
• This topological approach encourages a shift towards wholeness-oriented urban design, providing a quantitative method to assess urban structure and beauty.

A Topological Representation for Taking Cities as a Coherent Whole

The paper "A Topological Representation for Taking Cities as a Coherent Whole" by Bin Jiang proposes a novel approach to understanding urban spaces by emphasizing the concept of wholeness through a topological framework. This representation moves away from traditional geometry-based methods to present cities as coherent, interconnected entities. The work builds on Christopher Alexander’s notion of wholeness or living structure, offering insights that could influence both theoretical and practical urban design.

The research introduces a topology-oriented geographic representation that characterizes cities by degrees of wholeness, measured through a hierarchical classification based on head/tail breaks. The topological framework allows cities to be abstracted into hierarchical levels, visualizing urban systems as networks of interconnected polygons. This abstraction facilitates the computation of wholeness degrees, providing a novel way of assessing the spatial coherence and adaptability of urban areas.

Four case studies—including analyses of all natural cities in China and the UK as well as detailed examinations of Beijing and London—demonstrate the practical application of this topological representation. Utilizing datasets of street nodes and Twitter locations, the research confirms that both city sizes and their degrees of wholeness follow a power law distribution, illustrating the underlying scaling hierarchy. These findings highlight the properties of differentiation and adaptation, which are fundamental in revealing how different urban elements interact within a larger systemic whole.

The proposed topological representation affords several significant implications. Firstly, it reflects a paradigm shift towards wholeness-oriented design, aligning with Alexander's living structure theory. This perspective emphasizes preserving and extending existing wholeness, promoting urban environments that are more adaptive and harmonious. Practically, it suggests a move away from Euclidean geometric thinking in modern urban planning towards embracing complexity and related geometric principles, thus fostering more vibrant and sustainable urban designs.

Theoretically, the paper contributes to the discourse on urban morphology by offering a method to quantify and differentiate structural beauty quantitatively. It raises foundational questions about the nature of architectural beauty and urban order, proposing that both can be objectively assessed through the degrees of wholeness.

In conclusion, Jiang’s research presents a compelling argument for revisiting urban design principles through a topological lens, advocating for designs that account for wholeness and interdependence. It challenges prevailing models by demonstrating that urban areas, when viewed through this coherent whole framework, exhibit inherent spatial order characterized by intricate scaling hierarchies. This approach provides a template for future research to continue exploring cities as living structures, ultimately extending the conversation into wider domains of modern architecture and environmental planning.