Book review: The Mathematics of Urban Morphology

edited by Luca D’Acci and reviewed by Cui Liu

9 Dec 2020, 9:58 a.m.
Cui Liu

The Mathematics of Urban Morphology

Book review: The Mathematics of Urban Morphology

edited by Luca D’Acci and reviewed by Cui Liu

Cham: Birkhäuser, 2019; 564 pp.: 978-3-030-12383-3, €72,79 (pbk), 978-3-030-12380-2, €103,99 (hbk)


Along with the spreading of new technologies and the availability of big data, cities are increasingly viewed not simply as places in space but as systems of networks and flows. Urban studies are undergoing substantial transformation accordingly. Quantitative studies originating from mathematics and computer sciences make urban analysis more precise on the one hand, and on the other hand raise new questions and bring new findings, therefore becoming an important complementary to the classical qualitative methods in urban studies. This book edited by Luca D’Acci provides a comprehensive review of the latest research combining mathematics and urban morphology. Scholars from multiple disciplines such as architecture, urban and regional planning, geography, civil engineering, transportation, environmental sciences, economics, management, computer sciences, mathematics and statistics present diverse mathematical approaches to studying urban forms. The book is organised around the main approaches and is divided into six parts with a foreword by Michael Batty and an introduction by Luca D’Acci.

The first part focuses on the fractal approaches to examining the self-similar structure of the distribution of city size and other urban form elements. Fractal parameters can be used for scaling analysis, spectral analysis and spatial correlation analysis to explain the complexity of urban form and growth, as shown in Chen’s study. All chapters in this part come with a consensus on the applicability of Zipf’s Law, that is, the counter-cumulative relationship between ranks and sizes of cities. That is to say, the frequency of any city size (or other elements) is inversely proportional to its rank in the frequency list. Such a relationship reflects a central place hierarchy which arises from both equilibria and optimal allocations according to the view of Hsu and Zou. The value of fractal parameters and the applicability of Zipf’s Law depend on the level of planning in relation to the organic growth of cities. For example, the study of Nilsson and Gil and that of Jia et al. show that organic cities have higher fractal dimensions and fit Zipf’s Law well while modernist planned cities do not.

Moving from the descriptive tendency of the first part, the second part of the book focuses on simulating and forecasting urban forms through cellular automata. Cellular automata are discrete computer models with a grid of regular cells assigned to one particular state which may change to another state according to certain rules. Both chapters in the second part underline the necessity of integrating cellular automata with geographical conditions in a more realistic way. In particular, to avoid uncertain calibration based on expert knowledge and to achieve reproducible deterministic results, Antoni et al. link the cellular design of geographic space, the Markovian approach to transition processes, the distance weighting in gravity models and the phenomenon of emergence that defines artificial intelligence models in a single formal notation.

The third part talks about urban spatial networks, which are mainly represented as street networks. Volchenkov discusses the graph representations of urban spatial patterns and suggests a quantitative method based on scale-dependent random walks. Boeing underlines the importance of network-based distances rather than straight-line for spatial network analysis due to the circuity difference between walkable and drivable street networks. Amongst the various approaches to studying urban spatial networks, space syntax is the most influential. It originates from graph theory and seeks to explain the effects of spatial configurations on the behavioural patterns of people. The other two chapters in this part, that is, those of Rashid and van Nes, give a very detailed account of space syntax, ranging from its theoretical foundations and mathematical formulas to its practical applications in various fields. Both of these chapters appreciate the contribution of space syntax to integrating human-based spatial analysis into socio-economical processes.

The fourth part discusses the key issues of complexity in urban morphology from different perspectives. Goh et al. attribute the complexity in urban morphology to the inter- and intra-layer coupling of buildings, especially the antiferromagnetic function between residential areas and work places, whereas Jiang views the complexity of cities as a coherent whole, which can be measured through the degrees of wholeness with properties of both differentiation and adaptation. Different from the above two chapters that use mathematical methods to explore the complexity of urban morphology, Bellomo and Terna focus on the most relevant complex features of the overall system, that is, the complex interaction between mathematics and urban morphology. They argue that such complexity is not just limited to exploiting various technical tools offered by mathematics, but also aims to induce new approaches and even new theories to the field of mathematics.

Other quantitative methods in addition to the aforementioned are grouped into the fifth part. These methods are somewhat ad hoc ideas and provide interesting supplements to the aforementioned methods. For example, while most chapters analyse urban forms at one scale only and lack cross-scale linkage, Schirmer and Axhausen propose multi-scalar clustering of urban morphology. Instead of quantifying urban spatial patterns as networks or cells, Huynh views spatial patterns as point datasets and proposes a quantitative method based on continuum percolation. Rather than using different models to examine different aspects of functional spatial structure, Burger et al. develop one gravity model framework to assess urban functional polycentricity and spatial interdependence simultaneously.

While most of the previous chapters deal with technical issues, the last part includes some humanistic and multidisciplinary commentaries. All the chapters in this part, either tackling specific issues or providing comprehensive views, pay high attention to the social implications of urban forms. For example, Lehmann calls for an appropriate ‘quality density’ in the socio-economic context; Sevtsuk and Davis warn about the introduction of automated transportation modes privileging speed, efficiency and function over human-scale interactions and place sustainability. Furthermore, the scholars emphasise the need to integrate urban mathematical models with urban social theories, the quantitative with the qualitative and the new with the classical.

Mathematical exploration in the field of urban morphology, as the book introduces, is still at the beginning stage. In fact, some research outcomes in the book appear to be a quantified reinterpretation of existing qualified studies and urban theories. Moreover, the scholars often focus on their own study areas that are classified from the perspective of mathematics, and have little dialogue with others that use different approaches to study the same aspects of urban forms. This inclination might also be embedded in the structure of the book, that is, more focus on the mathematical approaches and less on their influence on urban morphology.

As Conzen proposes at the end of the book, it really needs cross-disciplinary and cross-conceptual calibration to clarify the effectiveness of various approaches and to enhance our understanding about the mathematics of urban morphology. Such collaboration and integration are a very challenging task because they involve the reframing of the nature of the discipline and of our corresponding understanding of the nature of cities as art and science. Nevertheless, urban studies from the complexity perspective provide a new opportunity to conceive cities between hard and soft sciences, between the scientific and humanistic views of world. As Luca D’Acci explains in the introduction, ‘this new science based on complexity paradigms, is a science that induces art, identified as personal uniqueness’ (p. 16). In that sense, this book is a valuable attempt to promote human-based dialogue between scholars from multiple disciplines and to study urban forms mathematically on a common ground.


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