A Fractal Perspective on Scale in Geography

The Mathematics of Urban Morphology

2019

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It is possible that one day computer programs designed for cognition might be able to pick out these centers and rank order them by their degrees of life." Alexander (2002-2005 Abstract A city is a whole, as are all cities in a country. Within a whole, individual cities possess different degrees of wholeness, defined by Christopher Alexander as a life-giving order or simply a living structure. To characterize the wholeness and in particular to advocate for wholeness as an effective design principle, this paper develops a geographic representation that views cities as a whole. This geographic representation is topology-oriented, so fundamentally differs from existing geometrybased geographic representations. With the topological representation, all cities are abstracted as individual points and put into different hierarchical levels, according to their sizes and based on head/tail breaks -a classification and visualization tool for data with a heavy tailed distribution. These points of different hierarchical levels are respectively used to create Thiessen polygons. Based on polygon-polygon relationships, we set up a complex network. In this network, small polygons point to adjacent large polygons at the same hierarchical level and contained polygons point to containing polygons across two consecutive hierarchical levels. We computed the degrees of wholeness for individual cities, and subsequently found that the degrees of wholeness possess both properties of differentiation and adaptation. To demonstrate, we developed four case studies of all China and UK natural cities, as well as Beijing and London natural cities, using massive amounts of street nodes and Tweet locations. The topological representation and the kind of topological analysis in general can be applied to any design or pattern, such as carpets, Baroque architecture and artifacts, and fractals in order to assess their beauty, echoing the introductory quote from Christopher Alexander.

Section: Discussion On the Topological Representation And Analysismentioning

confidence: 99%

A Topological Representation for Taking Cities as a Coherent Whole

The Mathematics of Urban Morphology

2019

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“…For example, Euclidean geometric thinking tends to see things individually (rather than holistically) and non-recursively (rather than recursively). The work by Jiang and Brandt [8] is a good reference for a more detailed comparison about these two geometric ways of thinking. The shift from Euclidean to fractal or living geometry implies that the fractal or living geometry needs to be the dominant way of thinking.…”

Section: Scaling Lawmentioning

confidence: 99%

“…The concept of spatial heterogeneity, as conceived in current geography literature, is mistaken because it does not recognize the fact of far more smalls than larges. This notion of far more smalls than larges adds a fourth meaning of scale: a series of scales ranging from the smallest to the largest that forms the scaling hierarchy [8]. The scaling hierarchy can be further rephrased as: numerous smallest, very few largest and some in between the smallest and the largest.…”

mentioning

confidence: 99%

Spatial Heterogeneity, Scale, Data Character and Sustainable Transport in the Big Data Era

2018

IJGI

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“…Euclidean geometry has served as the foundation of cartography, ever since human beings began to measure the magnitude of the Earth, if not even earlier (Robinson et al 1995, Slocum et al 2008, Anson and Ormeling 2013. We cartographers tend to see geographic featuressuch as rivers, cities, streets and buildingindividually rather than holistically, non-recursively rather than recursively; we tend to focus on individual scales rather than on all scales or the underlying scaling hierarchy ranging from the smallest to the largest (Jiang and Brandt 2016); we tend to believe inconsciously or subconsciouslymore or less similar things, as reflected in Tobler's law (Tobler 1970), rather than far more small things than large ones, which is formulated as scaling law (Jiang 2015a). This Euclidean geometric perspective is so stubborn that makes some maps or mappingfor example automatic map generalizationdifficult or virtually impossible.…”

Section: Introductionmentioning

confidence: 99%

“…A cartographic curve is traditionally viewed as a collection of more-or-less similar line segmentsa non-recursive perspective. From a recursive perspective, a cartographic curve consists of far more small bends than large ones, and small bends are embedded in large ones (Jiang and Brandt 2016). Inspired by the living geometry of Christopher Alexander (2002Alexander ( -2005, a cartographic curve is a coherent whole, in which nested bends constitute coherent sub-wholes at different levels of scale.…”

Section: Introductionmentioning

confidence: 99%

New Paradigm in Mapping: A Critique on Cartography and GIS

2019

Preprint

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As noted in the introductory quotation, an ideal map was long ago seen as the map of the map, the map of the map, of the map, and so on endlessly. This recursive perspective on maps, however, has received little attention in cartography. Cartography, as a scientific discipline, is essentially founded on Euclidean geometry and Gaussian statistics, which deal with respectively regular shapes, and more or less similar things. It is commonly accepted that geographic features – such as rivers, cities, streets and building – are not regular and that the Earth’s surface is full of fractal or scaling or living phenomena with far more small things than large ones at different levels of scale. This paper argues for a new paradigm in mapping, based on fractal or living geometry and Paretian statistics, and – more critically – on the new conception of space, conceived and developed by Christopher Alexander, that space is neither lifeless nor neutral, but a living structure capable of being more living or less living. The fractal geometry is not limited to Benoit Mandelbrot’s framework, but is extended towards Christopher Alexander’s living geometry and based upon the third definition of fractal: A set or pattern is fractal if the scaling of far more small things than large ones recurs multiple times. Paretian statistics deals with far more small things than large ones, so it differs fundamentally from Gaussian statistics, which deals with more or less similar things. Under the new paradigm, I make several claims about maps and mapping: (1) Topology of geometrically coherent things – in addition to that of geometric primitives – enables us to see a scaling or fractal or living structure; (2) Under the third definition, all geographic features are fractal or living, given the right perspective and scope; (3) Exactitude is not truth – to paraphrase Henri Matisse – but the living structure is; and (4) Töpfer’s law is not universal, but scaling law is. All these assertions are supported by evidence, drawn from a series of previous studies. This paper demands a monumental shift in perspective and thinking from what we are used to on the legacy of cartography and GIS.