From form to information: Analysing built environments in different spatial cultures
FF ROM FORM TO INFORMATION :A NALYSING BUILT ENVIRONMENTS IN DIFFERENT SPATIALCULTURES
Vinicius M. Netto
Associate ProfessorDepartment of UrbanismUniversidade Federal Fluminense (UFF)Visiting ScholarCenter for Urban Science and Progress (CUSP NYU) [email protected]
Edgardo Brigatti
Associate ProfessorInstitute of PhysicsUniversidade Federal do Rio de Janeiro (UFRJ) [email protected]
Caio Cacholas
ResearcherGraduate Programme of Architectural and Urban Studies (PPGAU UFF)June 29, 2020 A BSTRACT
Cities are different around the world, but does this fact have any relation to culture? The idea thaturban form embodies idiosyncrasies related to cultural identities captures the imagination of many inurban studies, but it is an assumption yet to be carefully examined. Approaching spatial configurationsin the built environment as a proxy of urban culture, this paper searches for differences potentiallyconsistent with specific regional cultures or cultures of planning in urban development. It does sofocusing on the elementary components shaping cities: buildings and how they are aggregated incellular complexes of built form. Exploring Shannon’s work, we introduce an entropy measure toanalyse the probability distribution of cellular arrangements in built form systems. We apply it todowntown areas of 45 cities from different regions of the world as a similarity measure to compareand cluster cities potentially consistent with specific spatial cultures. Findings suggest a classificationscheme that sheds further light on what we call the ‘cultural hypothesis’: the possibility that differentcultures and regions find different ways of ordering space. K eywords Spatial information, built form, entropy, order, spatial cultures.
Cities of different cultural types and different scales embody different spatial identities... human societies order theirspatial milieu in order to construct a spatial culture, that is, a distinctive way of ordering space.Hillier [38, p.5-6]The idea that urban form embodies idiosyncrasies that express cultural identities seems to be a frequent assumption inurban studies. It has to do with the contextual role of custom and institutional settings, from regional idiosyncrasiesassimilated to traditional ways of building to the dichotomies of planned and unplanned cities, shaped through top-down agencies or as chance-grown arrangements [46]. However, can local cultures actually leave traces in urbanspace? Despite its persistence in the urban imagination, the problem of how built environments might embody specificcultural identities seems yet to be fully addressed in urban morphology. To begin with, there is an “evident lack of aquantitatively rigorous, comprehensive and systematic framework for the analysis of urban form” [68]. In this sense, a r X i v : . [ phy s i c s . s o c - ph ] J un PREPRINT - J
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29, 2020historically- and culturally-informed quantitative methods are essential for uncovering forms and patterns resultingfrom city organization processes [17].In this paper, we look closely into that assumption, and address the question of whether cities find distinct regionalcharacteristics as material forms and cultural milieu, or take on physically specific forms under certain culturalconditions [38]. This implies examining the existence of contextualised ways of shaping cities – and features that mighttranscend context. We shall do so approaching the spatial configurations of the built environment as a proxy of urbanculture, looking into the very constituents of urban form. Differently from emphases on street networks [42, 58], ourapproach focuses on the elementary components shaping the tangible spaces of cities: buildings and how they areaggregated in complexes of built form. It also means taking into account a feature that seems to differentiate cities fromnon-urban settlements: the systems of built forms arranged in urban blocks. Closely related to systems of streets andopen spaces, the urban block has become emblematic, uniquely defining the form of cities in urban societies emerged inregions and cultures seemingly with no contact with one other [55].We will look into 45 cities around the world and measure their spatial configurations to assess differences and similaritiesbetween them. In order to do so, we shall lay down an approach based on Shannon’s [65] measure of information andentropy. We will argue that Shannon’s measure is particularly suited for the task of capturing amounts of informationrelated to randomness and order in configurations of built form. Our approach takes the following steps: • Inquire into built form as ‘spatial culture’. • Propose a measure of configuration of built form based on Shannon’s entropy. • Apply this measure to examine cities of different regions of the world. • Finally, use the results as a similarity measure to compare and cluster the studied cities, as ‘informationsignatures’ potentially associated with specific regions or spatial cultures. ‘Culture’ was famously described by Raymond Williams [71] as one of the most complex words in the English language,an elusive phenomenon notoriously difficult to conceptualise and frequently challenged as an explanatory category[51, 32]. We use the term not to refer to an ‘independent entity’ with explanatory force [25] but as an ongoing processinvolving the practices, works and products of human activity situated in time and place. Therefore, ‘culture’ isembedded in material contexts and social frameworks, and relates to institutions and institutionalised behaviours, values,meanings and orientations, and capacities for self-regulation [15]. Such processual notion of culture also as a field ofaction [37] takes into account forms of self-organisation and coordination between agencies in material production. Inthis sense, we wish to explore urban form as an expression of cultural systems as inherently material processes, awareof potential contingencies that must be considered in empirical analysis.One of the under-examined assumptions about the connections between society and urban form is that the latter maysomehow express cultural identities that constitute the former. Conzen [21] was a pioneer in studying patterns ofchange in urban form in relation to changes in the economic, social, and technological milieu, proposing a cyclicalnature of the development of urban form [59]. Going a step further, Aldo Rossi [61] argued that the material formof the city is intrinsic to its sociological and cultural reality. Later on, Hillier [38] addressed the possibility of citiesof different cultural types embodying different spatial identities. His analytical approach allowed him to claim thathuman societies order their built environment to construct a ‘spatial culture’, a ‘distinctive way of ordering space’.Cities take on different forms in different cultural conditions in non-contingent ways, as spatial arrangements shapethe field of encounters that animate different social cultures. Physical space is systematically ordered to reproduceculturally-specific patterns of social behaviour based on co-presence as a principle for ordering social relations.Recent discussions have enriched this construct by approaching a spatial culture as “a fundamentally performativeand temporal process” [33, p.xxiv] Focusing on “questions of cultural specificity in the formation of space”, theseworks assess how culture affects spatial formation, and the possibility of “encoding and transmitting social and culturalinformation” in urban space [45]. Contingencies are added by the possibility that “different cultures invest differentlyin space, be it in regards to what is manifested, or to what extent society is manifested through built form” [45] (p.i).Difficulties are also of an epistemological nature, since the space-culture relation may be indivisible analytically [57],and therefore hard to be scrutinised.In a careful opposition to some of these views, the late urban historian Spiro Kostof [46] was suspicious of the beliefthat buildings and city-forms fully embody recognisable idiosyncrasies enough to be medium of cultural expression.Even though his works relate processes like ‘reading’, ‘encoding’ and ‘information’ to culture, he claimed that a sameurban form does not express an invariable human content. Despite this position, the quest for underlying explanations2
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29, 2020for systematic differences in urban form has led to the idea that physical patterns encapsulate an extra-physical reality,as different social and cultural agencies are seen to shape physical space. These agencies can range from traditionand custom, material requirements of interaction, associations with socially shared symbols or principles of societalorganisation. For instance, cities with irregular physical patterns are thought to be the result of development left entirelyto individuals, as bottom-up processes leading to the random ways of the unplanned city. In turn, top-down processestriggered by governing bodies would be able to guide the organisation of urban land and built form, leading to uniformlyordered cities [Castagnoli in [46], p.43)].Some studies looked into spatial features, logics or organising principles in comparative studies of cities consistent withdistinct regions. To be sure, most of these works deal with street networks rather than built form systems. Medeiro’s[50] topological analysis of betweenness centrality and depth in the street networks of 164 cities in different parts ofthe world identified regional differences. For instance, American and Canadian cities appear prominently with thehighest levels of accessibility, as opposed to Brazilian cities in South America, the most spatially segregated. Loufand Barthelemy [48] searched for the ‘fingerprints’ of cities analysing the distribution of blocks extracted from streetnetworks of 131 city centres. Their classification scheme is based on information about the area and a simplified proxyfor the shape of blocks. The method identifies that nearly two thirds of American cities in their sample are structurallydifferent from European cities. Rashid [59] carried on a comparative study of urban morphology in 104 cities in sixcontinents. Using uni-variate statistics of data for 44 spatial measures of street configurations and basic geometricmeasures of built form, like block perimeters and areas, he found limited differences between downtown areas indeveloped and developing countries. Furthermore, disaggregated measures of geometry of urban layouts have littlepower to describe actual form (say, of blocks) and do not grasp information encoded in relations between componentsof built form.In turn, our configurational approach focuses on buildings as the elementary components shaping cities, and how theyare aggregated in combinations and complexes. By looking into frequencies of cellular arrangements representingbuildings in selected cities, we wish to understand if and to what extent their configurations can be seen as particularcultural features, regardless of whether these features are intentionally embodied in urban space. Recognising thaturban structures are different around the word, and approaching spatial configuration as a proxy of urban culture, weattempt to measure such configurations to assess their differences and similarities. For that, we shall explore Shannon’sview of ‘information’ and ‘entropy’ to investigate whether spatial cultures entail ‘distinctive ways of ordering space’, asHillier suggests. A number of works have explored information and entropy measures in relation to urban systems, beginning withWilson’s [72] pioneering study of utility-maximising systems in 1970. The entropy-maximising paradigm was frequentlyused to derive model formulations for spatial interactions and urban distributions, microeconomic behaviour and input-output analysis [7]. Batty developed a number of studies on entropy in spatial aggregations and interaction since theearly 1970s [8, 9, 10]. More recently, Batty et al. [13] proposed a measure of complexity based on Shannon informationable to grasp the complexity of cities as they vary in scale, size and spatial distribution of population, dealing with spatialentropy related to the distribution of information, and with information density related to city size. Other approachesused modifications of Shannon entropy and information-theoretical metrics as methods to capture, quantify and groupsimilar two-dimensional spatial patterns in landscape ecology, including efforts towards a universal classification ofconfiguration types in a linear sequence according to increasing values [20, 4, 56].Entropy measures have also been applied to purely urban morphological problems, namely in street network analysis.Gudmundsson and Mohajeri [34] developed a method based on Shannon’s entropy to measure angular variation betweenstreets, applied to 41 British cities. Boeing [17] applied Gudmundsson and Mohajeri’s method to analyse 100 citiesaround the world focusing on street networks downloaded from Open Street Maps (OSM). However interesting asmorphological approaches, these applications do not seek to uncover spatial information patterns, focusing instead onentropy as a measure of variation in street angles and lengths. The entropy measure of the distribution of crossing anglesdoes not necessary capture the global degree of order/disorder of street networks. Even if they do, street orientation doesnot describe spatial configurations in a relational sense. These considerations are reflected by the fact that the resultingvalues of the measure applied are concentrated in the extremes, suggesting that it is not so sensitive. Furthermore,entropy measures applied to street orientation are not a comprehensive morphological approach since they not takeinto account entropy in built form. In other words, they ignore discrepancies between levels of order in street networksand built form systems. Cities can be physically disordered even if their street networks are perfectly ordered. AsKostof [46] put it, “[s]treets that read as straight and uniform on the city plan may be compromised by the capriciousbehavior of the bordering masses” (p.44). In short, we can have low entropy in street orientation, yet highly disorderedmorphological structures (figure 1). 3
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29, 2020Figure 1:
A same street network can support very different built form systems.
More comprehensively, Haken and Portugali [35, 36] focused on how the built environment actually embodiesinformation. They explore Shannon information quantitatively in connection with Haken’s synergetic qualitativeapproach to semantic information, in order to empirically assess how basic cellular arrangements and categorisationsof building facades convey different amounts of information. Finally, other approaches to spatial information haveadopted different measures of entropy, distribution of spatial co-occurrences, or information density to assess theamount of redundancy and grouping related to cognitive efforts to extract task relevant information from the builtenvironment [73, 60]. In turn, our approach will explore Shannon entropy to measure levels of randomness and disorderin physical space, namely in cellular arrangements of built form. We shall look into the possibility that consistentdifferences between cities can be perceived at this scale, and that cultures and regions find different ways of orderingsuch configurations, which may be captured by this measure.
Our first procedure involves a reduction of urban form to two-dimensional arrangements based on building footprints.Since Giambattista Nolli’s 1748 Map of Rome, the figure/ground diagrams have become a classic methodologicalresource in urban studies, showing built/unbuilt distinctions. For instance, Nineteenth century scholar Camillo Sitte[66] represented public buildings and the ordinary fabric of the city exploring such diagrams. More recently, Roweand Koetter [62] have described the theoretical significance of the figure-ground map or ‘Nolli map’ [59, 44, 69]. Thefigure-ground diagram provides a spatial data-driven method to analyse and study the urban form and circulationnetworks that structure human activities and social relations [17].Our second procedure looks into different cellular arrangements of built form and attempts to characterize theirconfigurations. We do so analysing the probability distribution of built form configurations, by estimating the Shannonentropy [65] of Nolli maps of different cities of the world. Of course, this has to do with the level of randomness in thecellular arrangements of built form in cities. By analysing cellular arrangements, we capture the structures of urbanblocks in relation to the open spaces of streets and public squares. Indeed, the layout of the environment encodes moreinformation than two-dimensional configurations can express. However, we opted for an analytic approach able tosufficiently describe differences in built form – hence the reduction of 3-dimensions (3D) urban form to 2-dimensions(2D) cellular aggregations (figure 2).We characterise the spatial information encoded in two-dimensional configurations of buildings in the following terms.As mentioned, information will be quantified measuring Shannon entropy, operationally estimated by looking at thesequence of bits 1 and 0 representing built form cells and open space cells within sections of cities. Theoretically, thiscorresponds to measuring the Shannon entropy of a 2D symbolic sequence of 1 and 0. In this context, information findsa precise meaning: the entropy of the sequence, a measure of the surprise a source that produces the sequence causes inthe observer [65]. Physical arrangements characterised by higher levels of randomness, uncertainty or unpredictabilityare associated with high entropy. In contrast, the presence of regularities and patterns in urban structures corresponds tolower entropy, which means a higher predictability.The next step involves the preparation of our set of empirical cases, and the conversion of city maps into Nolli maps.We selected cities for their importance in their region or country. Selection also had to take into account the availabilityof information on built form. Many cities, particularly in Latin America, Africa and Asia, have incomplete informationregarding building footprints, i.e. their precise location, position and form.4
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29, 2020Figure 2:
Reduction from 3D to 2D configurations of built form (Manhattan, NYC).
For methodological reasons, we selected areas within these cities for the application of our measure. This selectionprocedure follows two critical considerations. The first and most important one observes that it is interesting to decouplethe analysis of urban structures between small-scale, detailed and denser urban areas, and large-scale regional andperipheral urban areas. In fact, the two areas are different, and for this reason, they can be naturally described usingdifferent methodologies. The first small-scale urban area is defined by specific features such as buildings and urbanblocks, which introduce typical characteristic scales. This means that there are some well-defined scales relatedto the distance above which configurations loose their correlations. These characteristic scales define sub-systemscharacterised by typical local patterns (urban blocks, individual buildings and possible neighborhoods). Here, humanaction is the principal vector defining shapes and patterns which generally appear in a stratified form, like the oneswe see in older and traditional central areas. In turn, large-scale regional and peripheral urban areas are likely toinclude sparse occupation, frequently with a scale-free character. This means that the characteristics of their patternsare independent of the scale we fix for analysing them. Looking at different scales, the underlying structure remains thesame. In these regions, physical features linked to topography, geographical formations and barriers (e.g.. water bodies,mountains, and valleys), along with the presence of very large infrastructures (e.g. highways) might play relevant rolesin the definition of the spatial patterns. In this work, we will focus only on small-scale areas with dense urban form.The second consideration takes into account that our method is well fitted for estimating entropy for dense andcontinuous urban areas. Fixing the density of built form cells allows us to obtain results independent from this parameter.The high continuity and homogeneity of built form allows us to use a specific extrapolation technique that will proveuseful for estimating the entropy of our 2D symbolic sequences. For these reasons, the selection of sections wasbased on the identification of dense areas, with a high spatial continuity in the fabric of built form. We will consideroccupation rates close to 50%, which means avoiding large empty areas or rarefied patterns of urbanisation.We prepared our sample extracting building footprints in sections of cities from the public map repository Google MapsAPI. We tested trade-offs between resolution and availability of data for distinct scales. We chose geographic areas of , , m , which were considered sufficient for representing the general spatial characteristics of dense urbanareas regarding the configuration of buildings, urban blocks and open spaces of 45 cities around the world (figure 3).Built form maps of the selected cities were then prepared and exported in high resolution, filtering layers and convertingentities representing buildings into solid raster cells. Images underwent a re-sizing process for cells and wereconverted to a monochrome system and then into a matrix of size × cells with binary numerical values(figure 4).Estimation of the Shannon entropy of the considered 2D cellular arrangements uses a method commonly applied forestimating the entropy of sequences of symbols encoded in one-dimensional strings [64]. For 1D data sets, the methodconsists of defining the block entropy of order n through H n = − (cid:88) k p n ( k ) log [ p n ( k )] , (1)where blocks are string segments of size n , and the sum runs over all the k possible n -blocks. Equation (1) correspondsto the Shannon entropy of the probability distribution p n ( k ) . The Shannon entropy of the considered system (the whole1D string) [64, 47], which we indicate with h , is obtained from the following limit:5 PREPRINT - J
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29, 2020Figure 3:
Location of 45 cities in our sample. Colours show amounts of Shannon entropy found by our method in cellularconfigurations of built form, from blue (low entropy) to red (high entropy). h = lim n →∞ H n /n, (2)which measures the average amount of randomness per symbol that persists after all correlations and constraints aretaken into account. The above limit exists for all spatial-translation invariant systems, as demonstrated in [23]. Moredetails about this method can be found in [64, 47].This approach can be generalized to sequences of symbols in two dimensions, which correspond to our situation. Wehave to define the n -blocks for a two-dimensional matrix [30]. The most intuitive idea is to consider a block of size n as a square which contains n cells. To obtain the sequence of H n also for n values that do not correspond to squares,we considered blocks that interpolate perfect squares, as described in figure 5. Note that there is no unique natural wayto scan a 2D matrix. We tested our approach for different reasonable forms of constructing the blocks, and the use ofdifferent paths does not seem to significantly influence the estimation of H n for the considered data set.Equation 2 gives precisely the entropy for a theoretical infinite set of data. In real situations, where the data set is finite,our method estimates the probabilities of distinct arrangements of cells within blocks up to a certain size n , countingtheir frequencies. For example, for H , it is sufficient to have knowledge of the symbol distribution p (2) , which isapproximated by the frequency of 0 and 1 present in the data set. It is important to note that it is common to find in theliterature of image processing, urban studies and ecological landscapes approaches that perform some entropy basedanalysis measuring our H or, at best, our H . Unfortunately, sometimes these quantities are wrongly referred to as theShannon entropy of the system, which, in contrast, is our h .If our data were a purely random set, h would coincide with H , and p (2) would give a full account of the spatialconfiguration. This is obviously not true for urban situations, where evident structures and strong long-range correlationsare present. In this case, estimating entropy is a difficult task, as taking correlations into account means computing H n for a large n . In fact, the estimation of h is good when the spatial range of correlations is smaller than the maximumsize of the block entropy we are able to compute. This estimation can be rendered difficult because of the exponentialincrease in the number of distinct cells arrangements in blocks with n . When working with two symbols, as in our case,the estimation of H n becomes not reasonable when n ≈ N , where N is the number of elements in our data set. Thus,in our case, this condition is verified for n ≈ . Even if this is a rough evaluation, it reasonably fixes the maximumsize of the blocks that can be investigated with sufficient statistical quality [47]. The limit taken in equation 2 can beempirically obtained fitting the set of H n /n points with an appropriate function and then taking its limit for n → ∞ .We found heuristically that, for all examined cases, the following ansatz provides an excellent fit: H n /n ≈ a + b/n c , b, c > . (3)6 PREPRINT - J
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29, 2020Figure 4:
Nolli maps with building footprint distributions in downtown areas of 45 analysed cities (9,000,000 m windows,1,000,000 cells), extracted from Google Maps. These sections are used to compute Shannon entropy. Rotation in grids and built formsystems does not affect results. PREPRINT - J
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29, 2020Figure 5: Areas in Rio de Janeiro and Manhattan, NYC (left). Examples of blocks with nine cells shown in red areamplified on the right. Configurations of the type (a) show great variation, like those found in Rio, while the type (b)shows regular arrangements frequently found in Manhattan. Blocks are constructed following the fixed path representedon the bottom right. Numbers indicate the order in which cells are added to blocks. The first block of size 1 correspondsto cell 1. Neighbouring cells are added in the corresponding order. Nolli maps are scanned with this set of different cellblocks.The fitted value of a gives a reasonable extrapolation of the Shannon Entropy h .Considering our database of 45 cities from North America, Europe, Asia, Oceania, Africa and South America, our goalis to develop a classification scheme based on the similarities and differences between the entropy levels of the sampledcities. In this sense, the next step consists in performing a proximity network analysis based on the measured entropyvalues, with the aim of identifying the presence of communities or clusters of cities sharing similar entropy levels. Inshort, entropy estimation will allow us to order our pool of cities and define a classification scheme. This scheme mayhelp us find similarities possibly consistent with same spatial cultures or world regions.Once we obtained the entropy h for all considered cities, we can quantify the levels of similarity defining a distancebetween cities i and j based on the values of h : d ij = | h i − h j | . We created a matrix of distances for the analysed citiesand then defined a network where cities are nodes, and edges (links between nodes i and j ) are present only if the valueof d ij is smaller than a fixed threshold value. The detection of clusters displayed by this network is a straightforwardtask considering the relatively small size of our data set.We further developed the cluster analysis applying a method for constructing a dendrogram representation of thedistance matrix. We used the unweighted pair group method with arithmetic mean (UPGMA). This method constructs adendrogram that reflects the structure present in the similarity matrix, building a hierarchy of clusters. The algorithmused in the analysis is part of the module Bio.Phylo in the Biopython package [16]. When this approach incorporatesa reliable dating of entities, it can be used to identify cultural phylogenies, like in the work of Barbrook et al. in thephylogenetic analysis of written texts [6, 43]. This is an interesting exception. In general, like in our case, culturalobjects are related in an involved form between themselves, and dating is a major challenge for long standing livingentities, whether they are cities or languages [18, 19, 14]. The use of the empirical functions of equation 3 provides an excellent fit for all the considered cities. The values of theparameters c are contained in the interval [0 . , . . These values are consistent with the entropy convergence foundin written texts, where c ranges from 0.4 to 0.6 [27, 26, 28], and with a result for a Beethoven sonata where an exponent0.75 was found [5]. These results seem typical of language-like systems, where the presence of long-range order ischaracterised by a slowly decaying contribution to the asymptotics of the entropy for large n . Despite the relative slow8 PREPRINT - J
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29, 2020convergence, the fine quality of the fits allows a good extrapolation of the Shannon Entropy h . As an example, theresults for the estimation of H n / n and the corresponding fitting procedure for the city of Los Angeles are displayed infigure 6. Results for the estimation of entropy h for the sampled cities can also be seen on a horizontal axis in figure 6,showing how this measure introduces a clear sorting among our data.Figure 6: Left : An example of the estimated values of H n /n for the city of Los Angeles. The continuous line represents the bestfitting of our data using the function of equation 3. All the analysed cities present a very similar behaviour. Right:
Estimated valuesof h for the 45 cities under analysis. The similarity networks were constructed fixing the threshold value to . , which corresponds to the 90% confidenceinterval of the extrapolated values of h . We chose to implement the clustering analysis in increasing subsets of our poolof cities, starting within a same region. This way, it was easier to extract and visualise potential patterns or clusters ofcities sharing similar entropy levels.We started by looking into European cities (figure 7). Selected cities in Europe cluster in two main groups in theproximity network and corresponding dendrogram. The first one includes predominantly cities in Northern Europe,along with Barcelona and Madrid, which present lower levels of entropy. The second one includes mostly cities inSouthern Europe, with higher entropy levels. The clustering displayed by the proximity network shows how Madridand Amsterdam lie at the connection between both communities.Figure 7: Left:
Proximity network of the considered European cities based on the value of h . The edge lengths are not proportionalto the levels of proximity between entropy values. Right:
Dendrogram constructed using the UPGMA method applied to the distancematrix obtained in terms of the values of h . The important information displayed by the tree is its topology. The tree featuresessentially two main morphological groups corresponding to cities in Northern and Southern Europe. Next, we analysed the cities of Europe and the Americas along with Lagos in Africa (figure 8). We can distinguishdifferent clusters in the proximity network. While Brazilian cities São Paulo, Rio de Janeiro and Fortaleza remain asisolated clusters, other Latin American cities Mexico City, Ecatepec and Lagos in Africa form a small cluster joint to a9
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29, 2020large cluster dominated by cities of Southern Europe. Another major cluster aggregates cities of Northern Europe andCanada, along with US city San Francisco, Spanish cities Barcelona and Madrid, and South American cities BuenosAires and Santiago de Chile. A smaller connected cluster is formed by major cities in the United States, whereasChicago stands as an outlier. The dendrogram further clarifies these relations: Mexico City and Ecatepec along withLagos share a common branch with the cluster comprised of Brazilian cities. Major US cities Chicago, Los Angeles,New York and Washington are placed in related branches, close to other North American cities (except San Francisco).Birmingham, Santiago and Buenos Aires relate to a same branch, as cities with the lowest entropy levels in theirrespective regions. There is also a branch relating Northern European cities, Madrid and Barcelona, and a major clusterdominated by Southern European cities.Figure 8:
Proximity network and dendrogram of the analysed European, North and South American cities, and Lagos in Africa.
The concluding analysis joins together all the considered cities, adding the Asian and Oceanian data. The numberof clusters in the proximity network is similar to the previous analysis, with the addition of a new one with themost ordered cities, Beijing and Chicago. Apart from this fact, the community structure seems unchanged. OtherAsian cities distribute themselves among pre-existing clusters: most Asian cities join either the cluster dominated bySouthern European cities or the cluster with most Latin American cities. Furthermore, the network shows an interestingconnectivity, from the most ordered cities Beijing and Chicago to major North American cities, then to a mixed clusterformed by cities from different regions sharing relatively low entropy levels, connected through Shanghai to a largecluster dominated by Southern European cities. This cluster in turn connects to the highest entropy groups, from MexicoCity to Tokyo and Brazilian cities. The complete dendrogram can be seen in figure 9.
What can the proximity networks and hierarchical clusters based on entropy values tell us about the cultural hypothesis?
That means examining the possibility that similarities in the ways of ordering built form can be explained either by (i)regional proximity, as cities from a geographically defined culture and identity (e.g. ‘American cities’, ‘Italian cities’,‘Islamic cities’); or by (ii) similarities in the form of producing patterns historically shaped by tradition in self-organised,bottom-up processes or by top-down agencies of self-regulation, allowing us to find elements in common even betweendifferent regions. Of course, our analysis brings no value judgement in the sense of pointing out a certain level ofentropy as desirable. We may start by interpreting these differences in the light of the ‘planned versus unplanned’dichotomy so persistent in the urban imagination [46]. 10
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29, 2020Figure 9:
Proximity network and dendrogram of North American, European, Asian, Oceanian, African and South American citiesunder analysis.
Beginning with the analysis of European cities, we found a subtle difference in levels of entropy between Northernand Southern cities. Northern European (i.e. Anglo-Saxon, Germanic and Russian) cities in our sample displayed ingeneral lower levels of entropy – from Birmingham (0.209) and Munich (0.225) to Amsterdam (0.254) and Vienna(0.263) – than Southern (i.e. Latin European) cities, from Rome (0.260) to Paris (0.286) and Marseille (0.292), with theexception of Spanish cities Barcelona (0.227) and Madrid (0.240). While the analysed area of Madrid is composed as apatchwork, its parts are mostly regular in themselves. In turn, a large part of contemporary Barcelona was notoriouslybuilt according to Ildefonso Cerdà’s 1859
Eixample orthogonal plan. These features echo the Spanish tradition ofregular grids deployed in colonized regions in Latin America, coupled with a strict alignment of buildings frontalfacades, and contribute to set them apart from other Southern European cities. Potential common traces in Northerncities include grids usually composed like a patchwork of partially regular areas (e.g. London, Munich, Amsterdam).This development pattern is frequently related to prior rural ownership and property boundaries. Regular grid sectionsrelate to resources like land survey and delimitation based on measurement, prior to subdivision into building plots [46].These cities also display considerable consistency in the building type adopted, leading to regularity in urban blocksurfaces. Even though Munich’s historical core shows curved urban blocks, frontal and back facades are predominantlyaligned. Geometric variation in the position of rear facades may be intense, combined with frequently sinuous urbanblocks (e.g. Moscow, Vienna, Brussels).In turn, frequent curves in streets and block systems may follow medieval footpaths of previous open fields and ruralfield divisions related to landscape features (e.g. historic cores of Milan, Lisbon and Athens). Practical modes of plotdivision and building seem to closely relate to topography (e.g. Lisbon) and watercourses (e.g. Nice, Toulouse andZaragoza). In these areas, buildings can be frequently strung along topographic lines and watercourses. Despite suchirregular features, there is considerable consistency in the position of frontal facades aligned along streets and openspaces.To be sure, bottom-up processes of cellular aggregation take morphogenetic paths involving randomness [41, 11], trialand error [2, 3], and path dependence [53]. These are processes where location decisions may influence the directionof subsequent decisions. If an urban system shows positive feedback from a particular configuration, an increasingproportion of that choice increases the probability of another building being added in a similar way to the system,favouring the dominant pattern. This means that the built form system can phase-lock in a specific, path-dependentconfiguration. Geometric consistencies resulting from trial and error processes and urban advantages triggered byincreasing densities and decreasing internal distances [55] can be reproduced as traditional modes of building. Thisprocess may eventually lead to institutionalised rules, like those prescribing particular building types, facade alignmentsor uniform setbacks even along originally unplanned street networks.When we take the 45 cities into account, we notice three main branches in the hierarchical clusters (figure 9, at abranch length around 0.075). A first cluster clearly emerges with cities with the lowest levels of entropy in the sample11
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29, 2020lower entropy cases. It is further divided into three initial branches. Beijing ( h = runs counter the first aspect of the cultural hypothesis: the similarity in entropy levels for cities within a same culture or region. This might have to do with the evolutionof these cities in comparison to others in the Latin American region. Cities founded in the Sixteenth century bySpanish colonizers in the Americas were often created in a rigid orthogonal pattern, following the 1573 Ordenanzas dePoblaciones , the first code of urbanism of the early modern period in the West. This was the case for Santiago andparticularly Buenos Aires, with its plain topography [63]. These areas became the historical and economic core of thesecities, with high density and compact patterns of built form. As these cities expanded, patchworks were added aroundthe core’s regular structure, adding entropy to the mix. Nevertheless, the levels of order in those central configurationsare felt in the analysis, bringing them to closer to cities with higher levels of order in built form, like Toronto andPhiladelphia.The second cluster highlights the highest entropy group in the sample, comprised of Brazilian cities Rio de Janeiroand São Paulo, in Latin America ( h = we cannot associate particular levels of entropy exclusively with particular regions,a first possibility of verifying the cultural hypothesis . We have to ask ourselves what in different regions could havetriggered similar entropy levels. The idea of a planned-unplanned dichotomy suggests that we should look into the actualevolution and planning conditions existing (or not) in these different cities, many of them having faced considerablegrowth in the twentieth century. We checked the existence of modern planning rules that act specifically upon builtform, namely: (1) Land parcelling: how land is divided into urban plots, and whether there are rules guiding the shapeand regularity of plots. (2) The layout of urban blocks and streets: what are the rules for layouts – say, whether theyimpose orthogonal systems or ‘planned picturesque’ systems like intentionally curved and varied block shapes andstreet networks. (3) Regulations on building design and location: whether there are rules that specify the positionof buildings in plots (e.g. frontal and lateral setbacks), and in relation to neighbouring buildings. We examined thelegislation in emblematic cases in Turkey, Nigeria, China, Brazil, Mexico, United States, England and The Netherlands.We found something that goes counter the planned-unplanned account of ordered and disordered cities: cities whichhave top-down planning may also exhibit high built form entropy. They do have rules and government agencies thatregulate building and urbanisation.But how can high entropy in built form be somehow influenced by top-down rules? We found that cities from differentregions – namely, Brazil, Nigeria, Mexico and Turkey – may have certain aspects of planning in common, which allowgreat variation in built form to come into being. For instance, these cities share emphases on parcel-based, piecemealdevelopments. New urbanised areas are mostly exempt from requirements to keep connections to neighbouring areas,including street continuity and grid alignment. Another crucial instance here is how individual buildings can bepositioned in their plots. Some regulations may enforce frontal and lateral setbacks, and define rules like increasingsetbacks as buildings grow taller. Simple local rules focused exclusively on individual buildings rather than coordinatedconstruction among nearest neighbours lead to a high level of fragmentation in built form.12
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29, 2020Going a step further, whole areas in these cities are urbanised and built by people’s own hands in informal settlements,hence apart from planning regulations. This is especially the case throughout the Twentieth century, when cities indeveloping countries experienced fast growth. We are likely to find high entropy mostly associated with variation in theshape of urban blocks (related to angular variation in surrounding streets) in those settlements. In short, parcel-based,piecemeal developments, patchworks of diverse blocks and street networks, and fragmented built form are key featuresof highly entropic urban landscapes.Figure 10:
Urban sections (500x500m) with similar entropy levels, different spatial configurations: Toronto ( h = h = PREPRINT - J
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29, 2020All this shows that cities from distinct regions may share similar entropy levels as far as built form is concerned. Theirtypical combinations of cells might be different, and they might share neither geographical proximity nor commonhistorical roots, but they still can contain similar levels of disorder, as captured by our measure (figure 10). This suggestscertain common traits between different regional cultures shaping how built form is ordered.That said, even though regions do not have entropy values necessarily different from others, individual regions do seemto converge around certain values.
This interesting pattern emerges once we visually distribute a classification of the 45cities according to increasing entropy values on a global map (figure 3). Some regions show higher levels of regularityand predictability in built form systems than others. We suggest that our measure seems to capture spatial informationpotentially related to different emphases on order and coordination latent in different planning cultures , the secondaspect of the cultural hypothesis seen above.How do these findings on regional differences compare with previous studies, based on different spatial entitiesand methods? We have seen that Medeiros’s [50] analysis of street networks based on betweenness centrality andtopological depth identified clusters of US/Canadian cities with the highest levels of accessibility, in contrast withBrazilian cities in South America, followed by European cities. Largely echoing Medeiros’s findings, Boeing [17]explored angular orientation entropy and grid order indicators to identify US/Canadian cities with the lowest orientationentropy. European cities also exhibit higher orientation entropy than Latin American cities. Louf and Barthelemy’s [48]classification based on block areas also identifies differences between most American cities and European cities.US/Canadian cities display low entropy in our analysis as well, but our results on European cities differ from thosestudies. Consistency in built form in European cities brings entropy to lower levels. Our approach was also ableto identify differences between Northern and Southern European cities. In their turn, São Paulo and Rome exhibitthe highest entropy levels in Boeing’s study. In our approach, despite the varied shapes in its block system, Rome’sconsistency around aligned buildings lowers its entropy to a level far from São Paulo. Interestingly, these differentapproaches converge about Brazilian cities: they exhibit the lowest average betweenness centrality, and highestorientation entropy and built form entropy in these different samples – probably due to an extraordinary variation in theposition of buildings in plots, coupled with fragmented grid patchworks. Nevertheless, the differences between findingsare clearly related to differences between the morphologies of street networks and built form systems: the fact that asame street network can support endlessly different configurations of buildings. Levels of order in street networks donot necessary cause low entropy in built form.
In this paper, we developed an approach to spatial information based on Shannon entropy. The approach was designedto (1) measure the entropy characterising levels of order and disorder in cellular configurations present in 45 citiesaround the world. (2) we applied the method to investigate the hypothesis of ‘spatial cultures’ as ways of ordering urbanform. Put another way, we verified whether the entropy measure could accurately grasp features and differences inbuilt form systems; then we looked for traces of ‘information signatures’ potentially consistent with specific regionsor cultures. This method is intended as a step towards a more precise understanding of spatial cultures as emergentpatterns – i.e. how typical configurations of built form emerge from local rules of aggregation active at the scale ofcellular configurations.Of course, any search for ‘information signatures’ of spatial cultures embodied in the tangible spatiality of cities facescertain risks: (a) Different cultures or regions may not have distinct ways of ordering space. In other words, there couldbe no ‘spatial cultures’ related to regions or even enough differences between cities to be associated with a particularculture. (b) Spatial cultures may well have specific information signatures, but these may not be encoded at the scale oflocal cellular configurations of built form systems. (c) In case the possibilities above were wrong, a measure of spatialinformation based on Shannon’s entropy in cellular configurations may not be precise enough to capture informationsignatures or even qualitative differences in configurations.In the research process, our method allowed us to find the distribution and clustering of cities around certain valuesof built form entropy. We would like to conclude our work discussing such entropy values and clusters in connectionwith characteristics of these cities, including aspects particularly related to what we called ‘cultural hypothesis’: theidea that similarities in the ways of ordering built form can be explained by (i) regional proximity, as geographicallydefined cultures and identities, or by (ii) common features in urban morphogenesis shared by distinct regions, say rulesin similar ‘planning cultures’. That meant looking for reasons of non-contingent similarities and differences betweencities.The usual association of bottom-up processes of spatial production in disordered, unplanned cities, as opposed totop-down processes of spatial production in ordered, planned cities, suggested that we should look specifically into14
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29, 2020planning rules guiding built form. We found that the ‘planned/unplanned’ dichotomy in urban studies may have beenvalid in pre-modern periods of certain urban cultures, but it seems of limited explanatory power once we considermodern and contemporary planning. Cities with top-down planning may also have high built form entropy.A key difference lies in the kind of rules applied and how they deal with buildings. Cities with high entropy in differentregions in our sample seem to have in common rules that focus mostly on individual buildings, allowing great variationin how they are placed in plots and blocks, including increasing lateral and frontal setbacks as buildings grow taller.This focus may happen for specific periods of their histories – long enough to shape the evolution and morphologyof large portions of these cities. This is the case of planning in countries like Brazil, Mexico, Nigeria and Turkey,especially when the analysed cities faced fast growth in the Twentieth century. In short, we found that simple localrules centred on individual buildings rather than coordinated construction lead to high fragmentation in ensembles ofbuilt form . These rules are frequently coupled with piecemeal developments and grid patchworks, including informalsettlements, shaping a visible fragmentation of urban landscapes.Our analysis brings other findings. First, proximity networks and hierarchical clusters show similarities in cities fromdifferent regions (e.g. high entropy cities including São Paulo, Tokyo, Istanbul and Lagos), with close entropy valueseven if they have geometrically distinct arrangements (figure 10). This suggests that the measure does not necessarilygenerate specific values as exclusive ‘information signatures’ for each region, a first possibility of verifying the culturalhypothesis.Second, despite that fact, the measure seems to capture something of the ‘planning culture’ of these regions. We foundhigher frequencies of certain regular arrangements in cities with top-down planning coupled with a strong focus on rulesfor coordinated modular construction, each building adjusting and aligning to those around, taking into account systemicconsequences of ensembles of built form. The high frequency of certain arrangements can also be found in cases ofbottom-up processes of cellular aggregation potentially involving path dependence, i.e. built form systems locked intospecific configurations, reproduced in traditional modes of building – patterns that can be eventually institutionalisedinto formal planning rules. This seems to be the case especially in the passage from pre-modern to modern urbanisationof European cities. On the other hand, we found plenty of variation in cellular aggregations in urban cultures thatallow the construction of buildings in uncoordinated actions between individual developers. This clearly leads to lessregularity and higher unpredictability in what surrounding built forms will be like as cities grow. Summing up, in bothtop-down and bottom-up form-making processes, local rules guiding how to position buildings in relation to others seemto trigger bifurcated developments as the built form system evolves in size and complexity, leading either into greaterconsistency or into greater fragmentation. But that is not the whole story, of course. We may find many possibilities inbetween those archetypal paths, or combinations of them in different parts of cities, like patchworks, or intermingled inlayers of ordered and disordered aggregations – say, the iconic case of Manhattan, based on the regularity of a gridironstreet layout, and planning rules that made room for enormous variation in built form.Third, although regions do not necessarily have exclusive values of built form entropy, individual regions do seemto converge around certain values.
Our results show certain consistencies, grouping cities from a same region (e.g.Brazilian cities, American cities). To use Hillier’s words [38], this echoes the idea that societies create their own spatialcultures – their distinctive ways of ordering space and shaping cities. Such finding needs to be further examined througha larger sample of cities and comparisons with other approaches, along the lines we explored above. Of course, deephistorical conditions and local contingencies are likely be at play, and must be carefully taken into account.Finally, differences between results obtained from street network-based measures and our measure of entropy shed lighton the potential dissociation between the morphology of streets and the morphology of built form systems in every city:the endless combinatorial possibilities of configurations of buildings, missing from street network approaches, addcomplexity to urban phenomena and suggest the need for a renewed interest in built form systems.Our sample is not a random set, which would be impossible due to the lack of information on building footprints in manycities and countries. Methodologically, at the present stage, our approach takes account of the spatial information latentin the arrangements of cells capturing relations of proximity, but eventually missing some correlations at large distances.On the one hand, cellular growth shapes larger structures as fundamental features of cities – a subject explored in otherworks [35, 40, 12, 11]. On the other hand, humans have a clear hierarchical reading [1], and structures at larger scalesseem to have more weight than structures at smaller scales to differentiate objects. Further development of this researchwill look into broader spatial structures in cities by introducing measures of statistical complexity. In addition, wewish to expand this approach to other forms of physical information, such as three-dimensional differences betweenbuildings, physical cues and landmarks [22, 67]. 15
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29, 2020Even though there is no value judgement in our work or claims of particular levels of entropy as desirable, differentlevels of built form entropy may well trigger different cognitive and practical responses from people. Higher degreesof entropy may be associated with spatial and visual surprises in navigation. Surprises can be considered desirableby some, as famously suggested by Camillo Sitte [66] and explored by Gordon Cullen’s [24] concept of ‘serialvision’. Notwithstanding, empirical studies in spatial cognition and neuroscience have shown that certain regularitiesand alignment effects (say, between paths or objects like buildings, or triggered by cardinal directions) improve ourjudgement of relative direction in navigation and our capacity to determine the position of objects in a surrounding area,affecting intelligibility and our memory of the built environment [49, 31, 29]. Effects of urban form on cognition are ahot research topic and could benefit from explorations into entropy and regularity in physical space as informationalfeatures in navigation [35, 36, 39]. Furthermore, human knowledge of spatial properties and patterns goes beyondphysical information and can integrate configurational, visual and semantic aspects of an urban environment [52, 36, 54].More work is needed to understand how physical information is associated with non-physical information, and isenacted by social agents making decisions and cooperating in cities.
We would like to thank the following researchers, architects and urban designers for their support in the analysisof planning regulations in a number of cities around the world: Cynthia Adeokun (Lagos and London), Ilgi Toprak(Istanbul), Mayra Gamboa Gonzáles, Juan Ángel Demerutis Arenas and Claudia Ortiz-Chao (Mexico City and Ecatepec),Tatiana Rivera Pabón (Buenos Aires and Santiago), Akkie Van Ness (Oslo and Amsterdam) and Chaogui Kang (Chinesecities). We also thank Lilian Laranja for discussions on culture and built form entropy. Any errors in interpretation arethe authors’ responsibility.
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