A multifractal method based on Spacematrix type units for analysing cross-scale characteristics of urban morphology

Abstract

The urban morphological system, developed over multiple phases, exhibits complex cross-scale characteristics, with significant scale discrepancies among morphological elements at the same hierarchical level, which suggests that the cross-scale sliding model holds the potential to reveal additional characteristics of urban morphology. This paper introduces a multifractal method that integrates Spacematrix morphological classification for the analysis of detailed urban building data within defined boundaries. Using the Nanjing Old City in China as a case study, the results reveal a dense yet balanced urban form, showing annular differentiation characterized by fragmented fringe belts at the macro level and a uniform mixture of diverse land use types and building types at the micro level. The typical scale invariance and multifractality are not consistently observed across single-type analyses. The study identifies height uniformization and spaciousness differentiation in the scaling of urban morphology, attributing the multifractal mechanism to the interweaving and transformation of multiple types across scales. This enhanced multifractal approach improves spatial mapping capabilities, aiding in the elucidation of the formation mechanisms of urban morphology.

Keywords

Urban morphology / Cross-scale / Urban morphology type / Multifractal / Generalized dimension / Spacematrix

References

[1] Batty, M. , Longley, P. , 1994. Fractal Cities-A Geometry of Form and Function. Academic Press Professional, Inc.

[2] Berghauser Pont, M. , Haupt, P. , 2023. Spacematrix. Space, Density and Urban Form. Republished by TU Delft OPEN Publishing 2023.

[3] Boeing, G. , 2018. Measuring the complexity of urban form and design. Urban Des. Int. 23 (4), 281- 292.

[4] Bruyns, G.J. , Higgins, C.D. , Nel, D.H. , 2021. Urban volumetrics: from vertical to volumetric urbanisation and its extensions to empirical morphological analysis. Urban Stud. 58 (5), 922- 940.

[5] Cai, Z. , Demuzere, M. , Tang, Y. , Wan, Y. , 2022. The characteristic and transformation of 3D urban morphology in three Chinese mega-cities. Cities 131, 103988.

[6] Cao, J. , Zhu, J. , Zhang, Q. , Wang, K. , Yang, J. , Wang, Q. , 2021. Modeling urban intersection form: measurements, patterns, and distributions. Front. Archit. Res. 10 (1), 33- 49.

[7] Chen, Y. , Feng, J. , 2017. Spatial analysis of cities using Renyi entropy and fractal parameters. Chaos, Solit. Fractals 105, 279- 287.

[8] Chen, Y. , Huang, L. , 2019. Modeling growth curve of fractal dimension of urban form of Beijing. Physica A 523, 1038- 1056.

[9] Chen, Y. , Wang, J. , 2013. Multifractal characterization of urban form and growth: the case of Beijing. Environ. Plann. Plann. Des. 40 (5), 884- 904.

[10] Chen, Y. , Wu, M. , 2022. Analyzing recent morphological evolution in central Beijing during relocating non-capital functions based on POI data and digital maps. Cities as Assemblages 3, 101- 113.

[11] Conzen, M.P. , 2009. How cities internalize their former urban fringes: a cross-cultural comparison. Urban Morphol. 13 (1).Article 1.

[12] Conzen, M.P. , Gu, K. , Whitehand, J.W.R. , 2012. Comparing traditional urban form in China and europe: a fringe-belt approach. Urban Geogr. 33, 22- 45.

[13] Conzen, M. , 1960. Alnwick, northumberland a study in town-plan analysis. Trans. Inst. Br. Geogr. 27, 1- 122.

[14] D'Acci, L. , 2019. The Mathematics of Urban Morphology. Springer International Publishing.

[15] Fort, B.L. , Visscher, J.-P.D. , 2020. Typo-morphological diversity and urban resilience: a comparative study of three heterogeneous blocks in Brussels. Urban Morphol. 24 (1). Article 1.

[16] Frankhauser, P. , Tannier, C. , Vuidel, G. , Houot, H. , 2018. An integrated multifractal modelling to urban and regional planning. Comput. Environ. Urban Syst. 67, 132- 146.

[17] Gallagher, R. , Sigler, T. , Liu, Y. , 2020. Targeted urban consolidation or ad hoc redevelopment? The influence of cadastral structure and change on the urban form of Brisbane, Australia. Urban Geogr. 41 (2), 183- 204.

[18] Gao, C. , Lu, H. , Ding, W. , Larkham, P.J. , 2024. The relationship between urban form and land-use regulation in China: the case of Nanjing. Urban Morphol. 28 (1). Article 1.

[19] Ge, X. , Han, D. , 2020. Sustainability-oriented configurational analysis of the street network of China's superblocks: beyond Marshall's model. Front. Archit. Res. 9 (4), 858- 871.

[20] Hameed Basee, D. , Riadh Abdulla, Z. , 2022. Transformation of urban morphology, vulnerability and resilience: haifa Street Area, as a case study. Ain Shams Eng. J. 13 (4), 101718.

[21] Hentschel, H.G.E. , Procaccia, I. , 1983. The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8 (3), 435- 444.

[22] Lagarias, A. , Prastacos, P. , 2020. Comparing the urban form of South European cities using fractal dimensions. Env. Plan. BUrban Anal. City Sci. 47 (7), 1149- 1166.

[23] Li, X. , Cheng, S. , Lv, Z. , Song, H. , Jia, T. , Lu, N. , 2020. Data analytics of urban fabric metrics for smart cities. Future Generat. Comput. Syst. 107, 871- 882.

[24] Li, Y. , Rybski, D. , Kropp, J.P. , 2021. Singularity cities. Env. Plan. Burban anal. City Sci. 48 (1), 43- 59.

[25] Liu, H. , 2017. Research of the morphological types of blocks in the old city of nanjing. Proceedings 24th ISUF 2017-city and territory in the globalization age. 24th ISUF 2017-City and Territory in the Globalization Age.

[26] Liu, J. , Deguchi, A. , 2024. The mechanism of block form diversification in urban morphological transformation: case study of grid blocks in Beijing, China. Front. Archit. Res. 13 (4), 928- 943.

[27] Liu, Z. , Liu, H. , Lang, W. , Fang, S. , Chu, C. , He, F. , 2022. Scaling law reveals unbalanced urban development in China. Sustain. Cities Soc. 87, 104157.

[28] Long, Y. , Chen, Y. , 2021. Multifractal scaling analyses of urban street network structure: the cases of twelve megacities in China. PLoS One 16 (2), e0246925.

[29] Mandelbrot, B.B. , Wheeler, J.A. , 1983. The fractal geometry of nature. Am. J. Phys. 51 (3), 286- 287.

[30] Murcio, R. , Masucci, A.P. , Arcaute, E. , Batty, M. , 2015. Multifractal to monofractal evolution of the London street network. Phys. Rev. E 92 (6), 062130.

[31] Potapenko, A. , Moor, V. , 2020. Vladivostok city morphology: space matrix as a tool for the urban form analysis. IOP Conf. Ser. Earth Environ. Sci. 459 (5), 052039.

[32] Saeedimoghaddam, M. , Stepinski, T.F. , 2021. Multiplicative random cascade models of multifractal urban structures. Physica A 569, 125767.

[33] Schirmer, P.M. , Axhausen, K.W. , 2016. A multiscale classification of urban morphology. J. Transp. Land Use 9 (1). Article 1.

[34] Sevtsuk, A. , Kalvo, R. , Ekmekci, O. , 2016. Pedestrian accessibility in grid layouts: the role of block, plot and street dimensions. Urban Morphol. 20 (2), 89- 106.

[35] Song, Y. , Pang, Y. , 2023. Measuring the superblock based on a hierarchy matrix of geometry, configuration, network, and area: the case of Nanjing. Env. Plan. B-Urban Anal. City Sci. 50 (4), 1057- 1071.

[36] Song, Y. , Zhang, Y. , Han, D. , 2022. Deciphering built form complexity of Chinese cities through plot recognition: a case study of Nanjing, China. Front. Archit. Res. 11 (5), 795- 805.

[37] Song, Z. , Chen, Y. , Li, Y. , Zhao, X. , 2020. An analysis of the allometric and multifractal features of a development in the urbanrural area in the lower reaches of the yangtze river: 2012 crosssectional data of four provinces and one city. Complexity 2020, 6963827.

[38] Song, Z. , Jin, W. , Jiang, G. , Li, S. , Ma, W. , 2021. Typical and atypical multifractal systems of urban spaces?using construction land in Zhengzhou from 1988 to 2015 as an example. Chaos, Solit. Fractals 145, 110732.

[39] Steadman, P. , 2014. Density and built form: integrating 'spacemate' with the work of martin and march. Environ. Plann. Plann. Des. 41 (2), 341- 358.

[40] Tan, X. , Huang, B. , Batty, M. , Li, J. , 2021. Urban spatial organization, multifractals, and evolutionary patterns in large cities. Ann. Assoc. Am. Geogr. 111 (5), 1539- 1558.

[41] Tannier, C. , Vuidel, G. , Houot, H. , Frankhauser, P. , 2012. Spatial accessibility to amenities in fractal and nonfractal urban patterns. March. Environ. Plan. B-Plan. Des. 39 (5), 801- 819.

[42] Thomas, I. , Frankhauser, P. , Badariotti, D. , 2012. Comparing the fractality of European urban neighbourhoods: do national contexts matter? J. Geogr. Syst. 14 (2), 189- 208.

[43] Wang, J. , Lu, F. , Liu, S. , 2023. A classification-based multifractal analysis method for identifying urban multifractal structures considering geographic mapping. Comput. Environ. Urban Syst. 101, 101952.

[44] Wang, P. , Gu, C. , Yang, H. , Wang, H. , 2022. The multi-scale structural complexity of urban morphology in China. Chaos, Solit. Fractals 164, 112721.

[45] Wang, S. , Gu, K. , 2020. Pingyao: the historic urban landscape and planning for heritage-led urban changes. Cities 97, 102489.

[46] Whitehand, J.W.R. , Gu, K. , Whitehand, S.M. , Zhang, J. , 2011. Urban morphology and conservation in China. Cities 28 (2), 171- 185.

[47] Whitehand, J.W.R. , Morton, N.J. , 2004. Urban morphology and planning: the case of fringe belts. Cities 21 (4), 275- 289.

[48] Whitehand, J. , Gu, K. , 2017. Urban fringe belts: evidence from China. Env. Plan. B-urban anal. City Sci. 44 (1), 80- 99.

[49] Wu, C. , Wang, J. , Wang, M. , Kraak, M.-J. , 2024. A Spacematrix and clustering approach to understanding the morphology of Singapore's housing development board (HDB) estates. AGILE: GIScience Series 5, 1- 7.

[50] Ye, Y. , Li, D. , Liu, X. , 2018. How block density and typology affect urban vitality: an exploratory analysis in Shenzhen, China. Urban Geogr. 39 (4), 631- 652.

[51] Zhang, H. , Nijhuis, S. , Newton, C. , 2023a. Advanced digital methods for analysing and optimising accessibility and visibility of water for designing sustainable healthy urban environments. Sustain. Cities Soc. 98, 104804.

[52] Zhang, L. , Ding, W. , 2018. Changing urban form in a planned economy: the case of Nanjing. Urban Morphol. 22 (1). Article 1.

[53] Zhang, P. , Ghosh, D. , Park, S. , 2023b. Spatial measures and methods in sustainable urban morphology: a systematic review. Landsc. Urban Plann. 237, 104776.

[54] Zhao, F. , Liu, J. , Zhou, Y. , 2023. Sandbox edge-based algorithm for multifractal analysis of complex networks. Chaos, Solit. Fractals 173, 113719.

[55] Zheng, W. , Du, N. , Yang, Y. , Wang, X. , Xiong, Z. , 2022. Multi-fractal characteristics of spatial structure of urban agglomeration in the middle reaches of the Yangtze River. Acta Geograph. Sin. 77 (4), 947- 959.

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