**Pierre Frankhauser**, in answering the contribution from Cécile Tannier of 12 March point by point, supplements her discussion and fuels the debate:

Are urban fabrics fractal? Is there a general process that gives rise to fractal urban fabrics? Are fractal models useful for planning? These recently asked questions call for a reminder that, through its intrinsic properties, this approach can be used to introduce a different reading of the spatial organization of cities and to contribute otherwise to the debate on whether or not there are “laws” or whether “norms” are useful in this context.

Fractal geometry is used in research into the structuring of urban spaces, as in other domains, to explore the arrangement of the constituent parts of a structure across scales, which Euclidian geometry does not do. By focusing on the principle of nested scales, fractal geometry can directly compare largely irregular empirical structures with theoretical structures constructed via the associated law of distribution, the hyperbolic law. It is thus possible both to develop graphic models and to explore the spatial structure of urban fabrics and of networks in terms of the relative positions of their component parts.

Graphic models have been used over the centuries and by many a discipline working on cities, often by reference to figures such as the circle or the hexagon. Examples include the models of Burgess and Hoyt of the Chicago School, various urban economics models, and planning models by Howard and Eberstadt-Möhring-Petersen or Christaller. However, cities are neither circular nor square, unless they are the outcome of some deliberate decision to make them so, as with Roman cities or certain bastide towns such as Aigues Mortes. A graphic model, like any model, is used to illustrate certain aspects of the object under study in a simplified way. It is in this spirit that constructed fractals have been used to illustrate certain characteristics of the structures studied (e.g. Batty and Longley 1994, Frankhauser 1994, Thomas et al. 2008, Chen and Wang 2013) by focusing on a cross-scale reading and it comes as no surprise that urban fabrics are not fractal in the strict sense of the term.

**Scaling behaviour as a dominant ordering principle**

As Mandelbrot made clear in speaking of “prefractals”, any real-world structure can only display behaviour of this type with a limited range and that behaviour may change at certain scales. Neither trees, nor alveoli, nor ferromagnetic structures are fractal… and yet this approach has proved useful across a wide array of disciplines. It would certainly be more useful to speak of scaling behaviour. If all cities or subdivisions of cities were to display a scaling spatial organization within a limited range, we would then be in a position to assert that this was a “law-like” general phenomenon for a restricted range of scales. But, even in this case, the presence of a “dominant” type of spatial organization would not mean that the values of the dimensions or other associated characteristics were identical for all cities. This is the same in other disciplines. For trees or their root systems, their fractal character will depend on various factors even for one and the same species, although the scaling character is invariably present (Zeide and Gresham 1991, Oppelt et al. 2000).

The position is much the same for urban fabrics. A fine analysis of scaling behaviour can identify different types of urbanization according to the historical context (Thomas et al. 2008, 2010, 2011). This does not detract in any way from the fact that the analyses of urban fabrics conducted by a good many researchers show that the fractal approach is a relatively stable model for characterizing urban fabrics within a realistic range (Batty and Longley 1994, Frankhauser 1994, Shen 2002, Qin and Liu 2004, Tannier and Pumain 2005). In a large number of cases, the goodness of fit of fractal curves remains very high (Thomas et al. 2008, 2011). Tests show that this goodness of fit remains high for that matter even if it is estimated from increments and not from cumulative data, which is an issue that is sometimes raised. It should be recalled, too, that a structure made up of different fractals is itself a fractal. If the districts of a city can be described by different fractal dimensions, then the whole would still be consistent with fractal logic for the range of scales in question.

However, it would be overstating things to claim that this “scaling order” is to be found in all urban fabrics at all times. From the standpoint of complexity theory, it would be better to speak instead of a particular type of organization, of a state of order such as a “phase” that develops quite often, but that may or may not be stable depending on circumstances. In this way, urban growth may also modify the scaling character of an urban fabric as shown by the multifractal analysis of Beijing (Chen and Wang 2013).

**Taking an interest in the processes that generate scaling behaviour of urban fabrics **

It seems unlikely that urban growth should obey a simple iterative logic. Different multi-scale densification processes or differential growth of certain parts of an agglomeration, or the projection of further urbanization may change spatial organization over the course of time, which would not be contrary to a final scaling organization (cf. Benguigui et al. 2000). For some agglomerations, it has been shown that spatial organization corresponds increasingly to a scaling logic as urbanization continues and this has been shown by an ordering parameter (Frankhauser 2000). This means that buildings are added in keeping with a certain mode of growth. The example of Montbéliard, set out in Frankhauser (2000), shows, though, that as urbanization advances the structuring of space may come to be dominated by one particular activity, such as industry, which fits into space modifying the scaling behaviour of certain sectors of the agglomeration.

Without seeking to claim this is the case, it would besides be possible to model “fractal” growth by introducing the spread of an agglomeration conforming to an iterative logic, while admitting that the outlying areas of the existing “urban flecks” become more compact. This would seem then to be more a model of allometric growth. It is in this sense that the “general” model introduced by Schweitzer and Steinbrink (2002) should also be seen.

Chen and Wang’s (2013) conclusion that less controlled growth of peripheral spaces generates more complex fabrics seems quite plausible. A wider range of multifractal dimensions is mathematically a clue that the spatial organization is more complex. In an unpublished multifractal analysis of Besançon, it turned out that the dimension ranges were quite narrow for the city centre and the Corbusian La Planoise district, but wider for the inner ring. These subdivisions are a mixture of various types of buildings constructed at different periods of urban growth.

As the fractal approach is morpho-descriptive, it does not provide any direct interpretation of the emergence of the forms observed. In biology it has been possible to find an explanation through genetic codes and it has been possible in physics to explain the emergence of “fractal” structures by laws of physics. The emergence of urban fabrics has been modelled first through descriptive morpho-dynamic models (Batty 1991) and then by looking for explanations for the emergence of this type of shape. Batty and Kim (1992) and then Batty and Longley (1994) recall that many constituent components of the city such as the road network or the organization of services are part of a hierarchical logic and conclude that “cities show all the properties of fractals”. White, Engelen and Ulje (2015) make similar arguments and conclude “a city is fractal in several respects”.

Going further, it will be attempted to find reasons for growth, whether “fractal” or otherwise, through socio-economic processes that are liable to contribute to a certain mode of growth. It is a question, therefore, of finding reasons for the emergence of a “macroscopic” structure as a result of interactions among various public and private actors. This is how the approaches by Salingaros and West (1999), Frankhauser (1994) or Schweitzer and Steinbrink (2002) should be interpreted. In their cellular automata model, White and Engelen (1993) also use rules that can be explained by the repellent effects of certain spaces.

**Fractals, optimization and normativity**

The question of whether a fractal organization corresponds to an optimization principle is not a metaphysical one. Thing are optimized with respect to an objective or a constraint; for example in economics with respect to a budget constraint, in physics with respect to a given condition. In this way, a deductive approach based on urban economics applied to a fractal model of an agglomeration has made it possible to explore the connection between household preferences for various types of amenities, travelling expenses and land rent (Cavailhès et al. 2004). A wholly compact and dense city would certainly satisfy households looking for no green amenities and who are not bothered by urban heat islands. In biological systems, scaling behaviour appears to be advantageous when two spatial systems must be articulated efficiently maximizing their area of contact as is the case for trees or lung alveoli. Mathematical properties inspired the use of fractal concepts in urban planning (Frankhauser et al. 2011, 2018). The close articulation of built-up areas and green areas makes it possible to preserve an interconnected system of green zones that are readily accessible and that reach into the city. This system avoids the fragmentation of green areas and maintains ventilation corridors for urban centres. Added to this is the objective of making urban and leisure amenities accessible by taking account of their frequentation and residents’ needs in a hierarchical logic that is inspired by central place theory. Accordingly, certain fractal dimension values are liable to respond better than others to certain planning objectives.

It would be entirely possible to use fractal measures to ensure compliance with certain urban development objectives in the same way as other standards used for regulating urbanization, which are often based on the criterion of density. Such measures set limits while leaving a degree of freedom as to the details, which is also true of fractal dimensions. There is an infinity of spatial patterns that correspond to one and the same fractal dimension. It could therefore supplement existing measures by adding cross-scalar information that is currently not taken into account in transcribing hierarchical aspects of cities. In order to ensure maximum contiguity of green areas and prevent their break-up, supplementary indicators, based on a topological approach could prove useful.

**The aesthetic virtue of fractals**

Geography takes little interest in aesthetics – things are different, though, in city planning and a fortiori in architecture. Aesthetics has always played a part in urban planning and also finds in it references to various objects, such as the fan-shaped plan of Karlsruhe (18th century) and many others. Sometimes natural objects serve as references, as with Schumacher’s palm-leaf plan for Hamburg. Although Jiang and Sui (2014) try to forge a link between fractal geometry and aesthetics, it should be emphasized that this question was raised in the 1990s for architecture (Bechhoefer and Bovill 1994, Bovill 1996), for example, concerning the blueprints of Frank Lloyd Wright or Le Corbusier (Ostwald et al. 2008, Ostwald 2013). It can be imagined that the draughters of these architectural plans were pursuing some aesthetic objective.

To what extent fractal structures correspond to aesthetic criteria is a question raised with the earliest works on fractals and often somewhat intuitively (Peitgen and Richter 1986). Some authors claim that simulated graphics are seen as particularly aesthetically satisfying if their fractal dimension corresponds to certain values (Draves et al. 2008). Lorenz et al. (2017) are more doubtful as to these findings but do not rule out that scaling behaviour might play a non-negligible part in aesthetic theory and underscore the connection between the fractal dimension and the golden ratio used, for example, by Le Corbusier. Other works have shown that fractal images reduce stress (Taylor 2006).

**Conclusion**

Through its intrinsic properties, fractal geometry opens up the gate to a cross-scalar reading of spatial structures. Furthermore, it forges a direct link between the construction of geometrical objects and a statistical law, which distinguishes it from Euclidean geometry. Application of this concept based on a principle of simple nesting of scales has made it possible to better understand the morphology of urban fabrics which are often perceived as having no precise shape. This makes it possible to think differently about the spatial organization of cities both in their comparability and their diversity, which is not contradictory. The mathematical properties of fractals are also valuable in terms of structuring metropolitan areas for the purpose of reconciling urbanization, quality of life and ecology. By way of conclusion, let us leave the last word to Mike Batty when he says that “form follows function” (Batty and Kim 1992, Batty 1994).

**References **

Batty M. (1991). Cities as Fractals: Simulating Growth and Form. In: Crilly A. J., Earnshaw R. A., Jones H. (eds), Fractals and Chaos. Springer, pp 43-69.

Batty M. and Kim S. K. (1992). Form Follows Function: Reformulating Urban Population Density Functions. Urban Studies, 29, 7, pp. 1043-1070.

Batty M. and Longley P. (1994). Fractal Cities. A Geometry of Form and Function. London: Academic Press, 394 p.

Bechhoefer W. and Bovill C. (1994). Fractal analysis of traditional housing in Amasya, Turkey. Tunis: IASTE. 362 p.

Benguigui L., Blumenfeld-Lieberthal E. and Czamanski D. (2006). The Dynamics of the Tel Aviv Morphology. Environment and Planning B, 33 pp. 269-284.

Benguigui L., Czamanski D., Marinov M. and Portugali Y. (2000). When and where is a city fractal?, Environment and Planning B: Planning and Design, 27, pp. 507-519.

Bovill, C. (1996). Fractal Geometry in Architecture and Design. Boston, Birkhäuser. 195 p.

Cavailhès J., Frankhauser P., Peeters D. and Thomas I. (2004). Where Alonso meets Sierpinski: An urban economic model of fractal metropolitan area. Environment and Planning A, no. 36, pp. 550-578.

Chen Y. and Wang J. (2013). Multifractal characterization of urban form and growth: The case of Beijing. Environment and Planning B: Planning and Design, 40(5), pp. 884-904.

Draves S., Abraham R., Viotti P., Abraham F. D. and Sprott J. C. (2008). The Aesthetics and Fractal Dimension of Electric Sheep. International Journal of Bifurcation and Chaos, 18 (4), pp. 1243-1248.

Frankhauser P., Tannier C., Vuidel G. and Houot H. (2018). An integrated multifractal modelling to urban and regional planning. Computers, Environment and Urban Systems, (67)1, pp.132 146.

Frankhauser P., Tannier C., Vuidel G. and Houot H. (2011). Une approche multi-échelle pour le développement résidentiel des nouveaux espaces urbains. In: Antoni J.-P. (Ed) Modéliser la ville. Forme urbaine et politiques de transport. Economica, Coll. Méthodes et approches, pp. 306-332.

Frankhauser P. (2000). La fragmentation des espaces urbains et périurbains : une approche fractale. In: Structure des villes, entreprises et marchés urbains. Harmattan, pp. 25-50.

Frankhauser P. (1994). La fractalité des structures urbaines. Anthropos (Ed.), 291 p.

Jiang B. and Sui D. (2014). A new kind of beauty out of the underlying scaling of geographic space. The Professional Geographer, 66(4), pp. 676-686.

Lorenz W. E., Andres J. and Franck G. (2017). Fractal aesthetics in architecture. Appl. Math. Inf. Sci. 11, No. 4, pp. 971-981.

Oppelt A., Kurth W., Dzierzon H., Jentschke G. and Godbold D. (2000). Structure and fractal dimensions of root systems of four co-occurring fruit tree species from Botswana. Annals of Forest Science, 57 (5), pp. 463-475.

Ostwald M. J., Vaughan J. and Tucker C. (2008). Characteristic visual complexity: Fractal dimensions in the architecture of Frank Lloyd Wright and Le Corbusier. Nexus VII: Architecture and Mathematics, 7, pp. 217–232.

Ostwald, M. J. (2013). The fractal analysis of architecture: Calibrating the box-counting method using scaling coefficient and grid disposition variables. Environment and Planning B, 40, 644–663.

Peitgen H.-O. and Richter P. H. (1986). The Beauty of Fractals. Springer 203 p.

Salingaros N. and West B. (1999). A universal rule for the distribution of sizes. Environment and Planning B, 26(6), pp. 909-923.

Schweitzer F. and Steinbrink J. (2002). Analysis and Computer Simulation of Urban Cluster Distributions. In: Humpert K., Brenner K. and Becker S. (eds) Fundamental Principles of Urban Growth. Müller + Busmann, pp. 142-157.

Shen G. (2002). Fractal dimension and fractal growth of urbanized areas. International Journal of Geographical Information Science, 16, pp. 419–437.

Taylor R. P. (2006). Reduction of physiological stress using fractal art and architecture. Leonardo, 39(3), pp. 245-251.

Thomas I., Frankhauser P. and Badariotti D. (2011). Comparing the fractality of urban districts: Do national processes matter in Europe? Journal of Geographical Systems, 14(2), pp. 189-208.

Thomas I., Frankhauser P., Frenay B. and Verleysen M. (2010). Clustering patterns of urban built-up areas with curves of fractal scaling behavior. Environment and Planning B, 37-5, pp. 942-954.

Thomas I., Frankhauser P. and Biernacki C. (2008). The morphology of built-up landscapes in Wallonia (Belgium): A classification using fractal indices. Landscape and Urban Planning, no. 84, pp. 99-115.

Qin Y. and Liu K. (2004). Advances in applied studies of fractal theory in geography in China. Journal of Geographical Sciences, 14, pp. 62–68

Tannier C. and Pumain D. (2005). Fractals in urban geography: A theoretical outline and an empirical example. Cybergeo: European Journal in Geography article 307.

White R., Engelen G. and Uljee I. (2015). Modeling Cities and Regions As Complex Systems: From Theory to Planning Applications. MIT Press, 330 p.

White R. and Engelen G. (1993). Cellular automata and fractal urban form: A cellular modelling approach to the evolution of urban land-use patterns. Environment and Planning A, Vol. 25, 8, pp. 1175-1199.

Zeide B. and Gresham C.A. (1991). Fractal dimensions of tree crowns in 3 loblolly-pine plantations of coastal South-Carolina. Canadian Journal of Forest Research 21(8), pp. 1208-1212.

Zhang D., Samal A. and Brandle J. R. (2007). A method for estimating fractal dimension of tree crowns from digital images. International Journal of Pattern Recognition and Artificial Intelligence, 21(3) pp. 561-57.