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2013 12: 335 originally published online 25 June 2013Planning TheoryKang Cao and Yan Zhang

Urban planning in generalized non-Euclidean space  

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Planning Theory12(4) 335 –350

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Corresponding author:Kang Cao, Department of Regional and Urban Planning, Zhejiang University, Hangzhou, 310058, China. Email: k_cao@126.comYan Zhang, College of Architecture and Urban Planning, Shenzhen University, Shenzhen, 518060, China. Email: zhyan607@126.com

Urban planning in generalized non-Euclidean space

Kang CaoZhejiang University, China

Yan ZhangShenzhen University, China

AbstractThe concepts of space and time have dramatically changed with the development of modern sciences, such as geometry, physics, and astronomy. This radical shift also changes every aspect of our perspectives, including those on urban space and urban planning. With a brief introduction to the evolution of geometry, this article discusses the development of the way we think about and evaluate urban space. Such development is divided into two phases: homogeneous urban space and generalized non-Euclidean urban space, each with corresponding patterns of urban planning. Some key issues in the non-Euclidean pattern of planning are then explored intensively, such as varied measurements of urban land, geodesics in the city, and urban planning in spacetime. While emphasizing the way of thinking in urban planning process, this article provides some proposals that differ from those related to absolute space.

KeywordsHeterogeneous space, non-Euclidean geometry, nonhomogeneous space, urban planning, urban space

Introduction

Human beings live in uneven or nonhomogeneous space given varied natural and human factors. Such unevenness in space was recognized by ancient Greek mathematicians, such as Archimedes, but it was not until the 17th century that scientists, including Newton and Leibnitz, described this unevenness through calculus or considered the essential sig-nificance of topological considerations on calculus (Plotnitsky, 2002: 136). The system-atic study of spatial unevenness in the 19th century formed part of the concrete foundation of non-Euclidean thinking. Scholars and planners began comprehending and integrating

487967 PLT12410.1177/1473095213487967Planning TheoryCao and Zhang2013

Special issue article

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urban planning in the perspectives of uneven or nonhomogeneous spaces in the late 20th century because of the inevitable time lag between original thinking and its widespread development. The “spatial turn,” which emerged in the social sciences and humanities in the last 20 years or so (Thrift, 2006), increased this tendency and deepened people’s understanding of urban space and, consequently, urban planning.

Roughly speaking, nonhomogeneity of the space in mathematics is expressed by a richness of the line,1 which is decided by the variable metric of the space. By contrast, the concept of nonhomogeneity or heterogeneity in urban space is viewed as a double-edged sword because apart from being the reason why cities are diversified, multiple, and pluralistic, it also accounts for various problems in cities. The efforts of scholars and practitioners to alleviate these problems gave birth to modern urban planning in the late 19th and early 20th centuries. In some sense, it is the nonhomogeneity in and of urban space that entail urban planning. It is natural, then, to introduce and apply non-Euclidean thinking that focuses on nonhomogeneous space to planning theory and practice.

This article aims to think and argue urban planning in the context of non-classical, non-Euclidean thinking. In the first part of the article, we attempt to provide a general sketch of the revolution of geometry from Euclidean to non-Euclidean geometry as a background for the understanding of non-Euclidean space. In the second part, we com-pare homogeneous space and generalized non-Euclidean space in cities. This process leads to the detailed exploration of three key issues in the non-Euclidean pattern of plan-ning in the third part: varied measurements of urban land, geodesics in the city, and urban planning in spacetime. We conclude that our exploration is only a minute part of the realm of non-Euclidean thinking in planning, and could be extended to the discussion of perhaps the more important concept of planning logic.

Revolution in geometry

The paradigm shift from Euclidean to non-Euclidean geometry is an outcome of continu-ous inquiries by many scientists about fundamental geometry concepts, such as line, plane, and space. Along with the rise and emergence of complex numbers, this shift is part of the non-Euclidean revolution (Plotnitsky, 2002: 133).

Euclid, a celebrated ancient Greek mathematician and educator in around 300 BC, is called the “father of geometry” for his remarkable contributions. He integrated and sys-tematized fragmented knowledge and scattered axioms in ancient geometry by using the pure deductive principles of logic. The Euclidean geometry that bore his name dominated the field for more than 2000 years until it was superseded by non-Euclidean geometry in the last 200 years. Euclidean geometry is currently grounded on homogeneous planar and three-dimensional space. As Aristotle stated in the 4th century BC, the literature of sci-ence should be based on terms and postulates (in Heath, 1993: 8). The root and kernel of Euclidean geometry were its 23 defined terms and 5 postulates in Book I of Euclid’s masterpiece, Elements. The first four Euclidean postulates were clear, perspicuous, and self-evident. Postulate 5, however, was controversial for its indirectness and dependence. It was also called the “Parallel Postulate,” and stated that “for every line l and for every point P that does not lie on l there exists a unique line m through P that do not intersect with l” (Greenberg, 1980: 16–17).2 This controversy prompted mathematicians in the

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following centuries to demote it incessantly as a theorem deduced from postulates or expressed in an objectionable way, usually by appealing to the concept of equidistance (Gray, 1989: 34). Nonetheless, their efforts were in vain. The encyclopedist Jean le Rond d’Alembert even called it “le scandale des elements de géométrie” (d’Alembert, 1759, in Trudeau, 1987: 154). The deadlock was broken in the 19th century, with tremendous con-sequence. Non-Euclidean geometry caused huge waves not only in the realms of geome-try and mathematics but also in the entire domain of human thinking because of the unremitting attempt to deny Postulate 5.

The concept of non-Euclidean geometry was established in the 19th century by C.F. Gauss (1777–1855), F. Schweikart (1780–1859), J. Bólyai (1802–1860), N.I. Lobachevsky (1792–1856), and B. Riemann (1826–1866) (Trudeau, 1987: 157–159). All were aware of the independence of Postulate 5 and made modifications to it by deny-ing either the existence or uniqueness of parallels (Gray, 1989: 34). Among them, Lobachevsky and Riemann’s contributions were more important. Lobachevsky pro-claimed that according to the Parallel Postulate, “for every line l and every point P not on l there pass through P at least two distinct parallels to l,” which points to more than one or innumerable parallel lines to the known line. On the contrary, Riemann assumed that in the above-mentioned premise (every line l and every point P that does not lie on l), every line through P intersects with l, that is, parallel lines do not exist (Greenberg, 1980: 149, 365). These three seemingly conflicting axioms were actually not contradic-tory when adapted to a different situation (Figure 1). From then on, neutral geometry diverged into three branches: (1) the classical branch, that is, Euclidean geometry, was called “parabolic geometry” (plane geometry, planar space with a curvature of exactly 0); (2) Lobachevskian geometry or “hyperbolic geometry” (pseudo-spherical geometry, open space, with a curvature greater than 0 or positive); and (3) Riemannian geometry3 or “elliptic geometry” (spherical geometry, with a curvature less than 0 or negative). The last two were collectively known as non-Euclidean geometry. Furthermore, Gauss, Riemann, and others discovered that this divergence was ultimately reconstructed into a new kind of geometry in which the three branches of geometry were special cases. Once a metric was defined, a type of geometry and its space or manifold would be identified.

Figure 1. Three different geometries and geometric space.Source: Drawn by Jin Zhu.

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Scientists then envisioned an infinite number of spaces in motion with respect to each other and thus opened up a relativist plurality of spaces (Shields, 2006). The universal geometry that embraced both Euclidean geometry and non-Euclidean geometry is identi-fied as general Riemannian geometry4 because Riemann extended three-dimensional space to manifolds of higher dimensions. Solid geometry is hereby expanded to n- dimensional or higher dimensional geometry (Katz, 1998).

Homogeneous space and generalized non-Euclidean space in cities

In sum, Euclidean geometry is the geometry of homogeneous space, whereas non-Euclidean geometry is the geometry of nonhomogeneous curved space, which can be labeled as non-Euclidean space. In the study of geometry in abstract non-Euclidean space, the degree of nonhomogeneity of space is manifested in the variable curvature in the manifold, the foundation of Riemannian geometry. “Manifold” or “manifoldness” (mannifgaltigkeit), a concept put forward by Riemann, transformed our mathematical, physical, and philosophical understanding of spatiality (Plotnitsky, 2009). The signifi-cance of the manifold lies in that it is much more than a new term, it is conceptual. This revolutionary concept defines a unity of space, which is composed or conglomerated by local (Euclidean or flat) spaces and the relationships among them (Plotnitsky, 2009). A manifold can be described by determined dimensions or parameters. Thinking of space as a “three-dimension manifold” of variable curvature provides us with a unique angle from which to comprehend space, as well as a platform from which to propose new physical theories (such as Einstein’s Theory of Relativity) and develop potential plan-ning theory.

The concept of Euclidean space has deeply affected people’s perception of space for two millennia or so. The 19th century and the second half of the 20th century under the cultural turn saw scholars developing numerous topological approaches to recognizing space as a differential concept (Woodward et al., 2010) other than a static, isotropic, and homogeneous idea. Ideologists and, in particular, geographers, who hold postmodern perceptions, profoundly criticize previously proposed planning treatments that take urban space as a single Euclidean space. For example, Henri Lefebvre (1984: 1) indicates that space is seen as an empty arena on which the city’s shaping force of spatial relations and processes acted. David Harvey (1989) argues that Euclidean geometry provides the basic language of discourse for those involved in the “conquest and control of space” (pp. 252–254): builders, architects, engineers, land managers, and so on. Space is regarded as abstract, universal, homogeneous, absolute, and continuous. Edward Soja (1989) points out that geography has treated the space “as the domain of the dead, the fixed, the undialectic, the immobile—a world of passivity and measurement rather than action and meaning” (p. 37). Doreen Massey shows a long history of understanding space as “the dead, the fixed” (Massey, 2005: 13), and argues that space should be rec-ognized as the product of interrelations, the sphere of coexisting heterogeneity, and as always under construction (Massey, 2005: 9). Nigel Thrift (2006) concludes from ana-lyzing Julie Mehretu’s artwork that space is, by principle, in constant motion; it is every-where (“everything is spatially distributed”) and with many guises.

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Mathematical non-Euclidean space contains only one nonhomogeneous factor—the variable curvature, which is formed by matter that fills the space. Physically, a space may be co-extensive with matter (Plotnitsky, 2009). Matter engenders gravitational field and determines metric structure, and finally determines the three-dimensional manifold (the space) and its curvature. Urban space possesses mathematical spatiality and varies and changes according to the “things” it involves. However, the factors that engender hetero-geneity for a city are more intricate and diverse in physical urban space than matter is in mathematical space. This aspect makes urban space more complex and multifaceted than mathematical space. Shields (2006) argues that social space emphasizes qualitative het-erogeneity; it “is not locked within one topology” and is “a produced order of difference that can be heterogeneous in and of itself” (pp. 148–149). As one form of social space, urban space is characterized as fragmented, splintered, or retribalized (Ellin, 1997).

We argue that the causes of urban and regional space heterogeneity are varied and can be classified into broad categories, such as natural factors and human factors. Natural factors or physical impediments, including parks, nature reserves, and areas devoted to sports, can result in discontinuities on the urban surface (Parr, 2007). Human factors make urban space both material (lived) and imagined (ideological) shaped by a consoli-dation of (social, cultural, economic, and political) power (Yeoh, 2006). Moreover, some factors are material, visible, and touchable, and others may be ideological, psychologi-cal, social, ethical, religious, and so on (see Table 1). The conception of “generalized non-Euclidean space in cities” in this article, borrowed from, but extending that found in, geometry, means urban space is under the restriction and constraint of more sophisticated natural and human factors. By affecting these factors, urban planning may contribute to changing the curvature of urban space: its degree of heterogeneity.

Some key issues of urban planning in generalized non-Euclidean space

A brief overview of the phylogeny of geometry in the first part follows the trajectory of the gradual evolution of the concepts of line and space. Likewise, it looks into our under-standing of space as pivotal to our Weltanschauung and to urban planning because plan-ning focuses on urban space. The relationship between urban planning, which emerged thousands of years before in the construction of cities of the Middle East and China (Hall, 2000: 20), and geometry is extremely close and long because both focus on space. The key distinction is the fact that urban planning focuses on concrete or physical urban space and geometry on abstract geometrical space. Linearly, the evolution of modern urban planning can be divided into two phases, according to the development of geom-etry: Euclidean urban planning and non-Euclidean urban planning.

We believe that it is necessary to introduce a non-Euclidean mode of thinking to plan-ning because it provides us with a new perspective to approach the true nature of urban space, and hence with different measures and methods to plan it. There are at least three specific reasons for this. First, most ancient people thought that they lived on flat ground. This perspective corresponded with many of their daily experiences and is the basis of Euclidean geometry. We now know that the Earth is approximately a spheroid, which means our living space is a heterogeneous space, roughly speaking. Euclidean thinking

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has its limits of applicability and is insufficient in such context. The concepts and ways of thinking contributed by non-Euclidean thinking are comparatively more applicable in comprehending the heterogeneous, pluralistic, and multifaceted urban world in which we live. Scholars have also studied how cities should be treated as fractals, based on fractal geometry (Batty, 1995; Batty and Longley, 1994; White and Engelen, 1993). This shows that people try to approach the true nature of space from different paths, whether non-Euclidean geometry or fractal geometry.

Second, in the phase of Euclidean urban planning, urban region and space are regarded as single, integrated, unitary, and material objects that can be addressed by rational plan-ning instruments. Time and space are assumed by planners as external containers to urban life (Graham and Healey, 1999). Such Euclidean ideology inevitably led to a Euclidean way of thinking and practice in urban planning, which held the reins of urban planning for decades. Its disregard for or neglect of higher dimensions of the spacetime continuum makes it difficult for planners to consider the city in a relational, nonlinear, heterogeneous way. However, rather than a background or “residential flat” (Weyl, 1952, in Plotnitsky, 2009) of matter, a concept in the Euclidean mode of thinking, space is co-extensive with matter (as mentioned above) and is “determined” by matter. A non-Euclidean mode of thinking embraces these conceptions and is, therefore, a useful instrument for compre-hending the relativity of space.

Third, according to Friedmann (1994), the Euclidean mode of planning or plan-making equates to “setting out a desired future along with the practical measures necessary to achieve it.” It is characterized by instrumental rationality, which limits planning to finding the optimal means to achieve given ends, and blueprint planning, which limits planning to an end-state prescription. Representations of planning projects are also in the form of two-dimensional blueprints that “replace the discontinuous patchy space of practical paths by the homogeneous, continuous space of geometry” (Bourdieu, 1984, in Harvey, 1989:

Table 1. Constraint factors in generalized urban non-Euclidean space.

Natural factors

Topographical Hill, river, sloping field, lake, pond, wetlands, and so on Typological Wasteland, brownfield, greenfield, cultivated land, sands, woodland, garden plot, and so on

Human factors

Material Dynamic: street, sidewalk, plaza, subway, railway, light rail, skyway, airport, harbor, and so on Static: varied communities, inner city, CBD, suburb, historical and cultural heritage, and so on Ideological Social, economic,a political, religious, cultural, and so on

CBD: central business district.a Parr (2007) provides four spatial definitions of the city, of which the Built City (the city as a physical entity) is the core or basis of the other three “economic” definitions: the Consumption City, Employment City, and Workforce City.

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253). An advocate of a non-Euclidean mode of planning, John Friedmann indicates that many of the characteristics of non-Euclidean planning are described in Donald Schon’s (1971) work and his own (Friedmann, 1973) on transactive planning in the early 1970s (Friedmann, 1994). Non-Euclidean planning, he argues, takes place in actual or real time, with face-to-face communication in everyday life. It involves interaction on local and regional levels (even neighborhood and block levels) rather than abstract, Euclidean spa-tial strategies in imagined future time. Hence, planning should embrace substantive rationality and intermediation mechanisms. Most planning issues bring fragments of an “organized civil society” to the negotiating table, so that planning becomes a political and negotiable process (Alexander, 1994; Friedmann, 1993, 1994; Graham and Healey, 1999). We argue that the concept of “non-Euclidean” planning has a very broad and multifaceted sense. Non-Euclidean planning not only includes the Friedmannian real-time-interaction mode of planning but plays an active role in the fields of cost–benefit analysis for land use, urban transport management and route design, and planning in spacetime, to mention just a few.

Consequently, it is natural to think urban spaces and make urban planning in a non-Euclidean way. Traditional urban planning thoughts and practices that are based on the perspectives of an even Euclidean space should be modified on the basis of heterogene-ous non-Euclidean space.

Varied measurements of urban land

The concept of the manifold of variable curvature, in general Riemannian geometry, makes it possible for a plurality of spaces and alternative spaces. This concept can be used in understanding urban space and alleviating urban problems. Various natural and human factors (see Table 1) engender the unevenness of the measurement in urban space, which correspondingly brings about dispersion in planning cost. For example, urban land can be roughly classified into two types according to whether it is developed: (1) ecological, organic, and unused greenfield and (2) abandoned and underused brownfield (US Environmental Protection Agency, 1997). The costs and likely returns of investment in these lands are widely divergent. Industrial districts, commercial properties, storage land, and so on can all be possible brownfields, with the costs and benefits differing consider-ably and being hard to evaluate and calculate. But the brownfield can be fractionized to varied single areas each with the same nature, based on differential geometry. This is like using different atlases to describe a differentiable manifold,5 which is locally similar enough to Euclidean space. Rules of calculus can be applied to the individual space of the manifold. The description, measurement, and planning of urban space also need the col-lection of charts, such as various categories and programs for the land. As we divide and subdivide urban land in varied ways, urban space is multiply covered by these classifica-tions, which thereby lead to multiple measurements and calculable costs and benefits.

In most cases, the cost of urban land development can be measured monetarily, but areas such as protected historical sites cannot be assessed for monetary value. These areas contain heritage on and under the ground and are heterogeneous compared with other areas in a city. However, in the Urban Renewal programs in the 1950s, some neigh-borhoods and historic areas in the downtown or old parts of the cities were demolished

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entirely, and the heterogeneity of urban space was wiped out at the same time. This is like using a single chart to describe a nonhomogeneous space, which would inevitably lead to flawed consequences. Nevertheless, the charts and the maps we make for urban space are open and can be reworked by an individual, group, or social formation (Deleuze and Guattari, 1987: 33). Sweeping urban renewal programs, for instance, were opposed and repudiated publicly in the 1970s on recognition of their problems and new ways to understand and regenerate the historical areas had emerged, such as “safeguarding.” In the “Nairobi Recommendation” in 1976,6 the concept of safeguarding was defined as “containing the meaning of identification, protection, conservation, restoration, renova-tion, maintenance and revitalization of historic or traditional areas and their environ-ment” (UNESCO, 1977: 21). Implied in the “Recommendation” is the acknowledgment of the multiplicity of urban space and the diverse measures to identify, preserve, and revitalize it. It also indicates that mapping urban space is not just description; it enters the relations between elements (the local spaces) to get a sense of what may emerge (Hillier, 2010), in terms of the reuse, rearrange, and redevelopment of the areas. This is because, as mentioned above, manifold is not only conglomerated by local spaces but by the rela-tionships among them. With the concept of manifold in non-Euclidean geometry, the relationships between a certain local urban space and other spaces can be understood more comprehensively. For one thing, it indicates that the chart we use to describe and “tweak” (Hillier, 2010) the local space is part of the whole atlas; it is an approximate description. It also reveals that the overlapping part (see Figure 2) between this space and its “neighborhood” is of vital importance because they are both keys to understanding the global properties of the manifold, and to defining the status of the local space in the manifold. For instance, the Nairobi Recommendation stated that “every historic area and its surroundings should be considered in their totality as a coherent whole” (UNESCO, 1977: 21). The surroundings of historic areas are the overlapping areas between historic areas and the city. They act as the spatial, social, and cultural ties between the overall city and these areas. Without their surroundings, the historic areas will lose their sense and influence even though they are historically significant, and likewise, the idea that sur-roundings are not the scope of certain core regions, but the overlapping areas may help us safeguard the visible and intrinsic continuity of cities in a more open way.

Geodesics in the city

As a special area of both urban planning and civil engineering, coordinated with land use planning, urban transportation planning evaluates and designs transport facilities. It should equally take into account the heterogeneous factor. Engineers and planners have different viewpoints and considerations as they undertake transportation plan-ning. For example, when making subway route selections, geologic conditions would be a main factor that engineers consider; while for planners, human factors are perhaps the most important: where the density of population is higher, where the site should be set up. But one of the key issues that probably all of them may consider is to plan urban geodesics, which should satisfy integrated social, economic, and environmental laws; that is, it should result in the shortest distance, least time, or minimum economic and environmental cost between beginnings and ends. When light moves in space, it

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invariably takes the shortest path called geodesic in space. Geodesic is constrained by curvature, and curvature is related to the spatial distribution of matter. In the more complex urban space, factors that constrain urban geodesics, roads, streets, footpaths, cycle lanes, boulevards, highways, and so forth are considerably more complicated than those that constrain a purely geometric space. Planners actually could decide at their discretion on how to plan traffic on the premise that they obey integrated social, economic, and environmental laws.

It should be noted that in some cases, there might be infinite geodesics between points (Figure 3(a)). The shortest driving route between two points on a city map, for instance, is a fold line rather than a straight line. This is because the city map is a constrained space that is restricted to the “materials” in a city. In this case, the driving route is restricted to roadways. The urban geodesic will be altered again if someone chooses bus, bike, or other mixed trip patterns to go from one stop to the other. In other words, the urban geo-desic might differ according to our trip patterns. As light follows the gravitation law, people should also obey social, economic, environmental, and even psychological laws and make optimal choice on routes and trip patterns such as by walk or by car, by public transport or by private car, straight ahead or passing round, and so on. In abstract, vac-uum space, light movement is decided only by curvature, but in urban space, residential densities (Barrett, 1996), accessibility (Farthing et al., 1996), and so on also influence travel behaviors.

Figure 2. An overlapping area in a manifold.Source: The definition of transition map in Wikipedia, redrawn by Hui Wang.

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Geodesic is also locally the shortest distance between points (Figure 3(b)). Suppose one is traveling between two places in a city. If the shortest path is bound to pass through the downtown where it is crowded, buildings dense, and traffic jammed, the quickest way to get to the destination is perhaps a detour. A body with a giant mass curves the gravity field, and thereby the geodesic. Similarly, a high-density area, in this case, the downtown, curves the alternative route. Thus, we have two urban geodesics: one is the shortest in distance and the other is the shortest in time. A geodesic can be replaced in some special cases by a hypothetical traversable wormhole (Figure 3(c)). It is possible because the

Figure 3. The locality of the geodesic: (a) there might be infinite geodesics between points, (b) geodesic is locally the shortest distance between points, and (c) a hypothetical wormhole between points.Source: Drawn by Jin Zhu.

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connections between two places (points) in a city can be both physical and nonphysical, such as informational. Today, the world is digitized, and “tunnel” systems such as the Internet and other electronic communications networks are used to convey and exchange information between different places instantaneously. Urban planning plays an important role in facilitating the construction of informational hyperspace bridges between points in spacetime: the cities. It is beneficial to reduce transport demands and thereby curb pollu-tion and material and energy consumption.

Urban planning in spacetime

As previously stated, Euclidean Postulate 5 was challenged successfully, but the jugger-naut of revolution never stopped. Time has long been identified as linear and isolated with space, and temporal relations between events are represented by the relations between points on a straight line (Grosz, 1995: 95). However, Einstein advanced his remarkable Theory of Relativity on the basis of Riemann’s thoughts (especially the con-cept of manifold) at the beginning of the 20th century. Einstein clearly described the lived world as a four-dimensional spacetime manifold. The space of this manifold is not homogeneous and isotropic, but curved by the presence of objects; time is not invariant but shifts with relative accelerations. Both time and space are relative or not absolute. The long-standing belief that time and space are invariant—time moves linearly forward and space is even, single, and universal—has crumbled. “Space and Time are to fade away into the shadows, and only a world [of space-time] in itself will subsist” (Minkowski, 1952, in Gray, 1989: 185).

Time is a very subtle and occasionally differently interpreted concept in planning theory debates. According to Friedmann (1993), there should be an awareness of the existence of space–time geographies within cities and places. Friedmann’s interpretation is as follows: planning is everyday events in real or actual time, happens “here and now” because planners can be effective “only in the evanescent and still undecided present” (Friedmann, 1993: 482). Alexander (1994) believes that whether in Euclidean planning or non-Euclidean planning, planning’s intrinsic nature and its future-orientation differen-tiates planning from similar activities such as decision making, administration, and man-agement. There was a tendency to emphasize “the present” in and after the “Communicative Turn” (Healey, 1992; Huxley and Yiftachel, 2000) in planning theory in the last two decades. However, as the concept of “emergence” and “becoming” began to emerge in the contemporary literature of planning theory in the context of poststructuralism, it seemed that the focus of planning would be between “now” and “then,” the future. As the redefinition of the line has led to a collapse of classical geometry, discussions about time—a fundamental issue in planning—may yield unexpected results that will deepen our understanding of the substance of planning. We argue that the concept of time should be discussed in a relativistic context with space; urban planning should be based on the views of relational spacetime, which is constitutive of events.

In the Euclidean model of planning, time is a linear, even-speed and one-directional flow. Time and space are irrelative and independent elements in planning. The fixed time-based strategy, normally the 15- or 5-year plan, is a typical instance of this perspec-tive, and such common time schedules are assumed to be in accordance with the complex

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relations of multiple development processes (Graham and Healey, 1999). This approach or practice disregards the dynamic movement of time (the most essential temporality) and misunderstands the equally dynamic nature of space (Grosz, 2004; Thrift, 2006). It strips time from the spacetime continuum and regards time as a constant and even pro-cess. Events in this process are like points on the line. We assume that events in planning could be any steps in a given planning procedure or any real-time events in planning process and practice. We have a linear flow chart for rational procedural planning (see, Taylor, 1998: 67–68), which mainly concerns the “procedure” of planning. However, the locations of events in spacetime are described by the coordinate system, namely, three spatial dimensions and one temporal dimension. Events, therefore, are like points in the four-dimensional spacetime. They are not the points on the one-dimensional time dimen-sion of typical planning practice.

The coordinate system depends on the observers or reference frames. In other word, coordinate choice is free; it is relatively independent of the spacetime. Dimensions are understood as components of the coordinate system. Gudmundsson and Höjer (1996) depict three fundamentally different intellectual conceptions of sustainable development (Figure 4), where sustainability and development represent different dimensions in Figure 4(c). It implies the importance of mutual balance between different dimensions when dealing with social issues. In the domain of planning, we are able to put multiple aspects that are related to an event into a coordinate system and consider them holisti-cally (Figure 5(a)). In practice, however, we are still not satisfied because more often than not advocators of each of the aspects complain about the scant attention they received. Through reviewing such kind of actualities, we realize that the coordinate sys-tem in Figure 4(a) is a static structure, like Euclidean geometry. In Figure 5(a), objective factors are overly abstracted to single dimensions, while interior relationships inside the dimensions and the complexity of reality cannot be reflected. Unconsciously, for exam-ple, we are used to regarding one of the factors as a number (the scale on the dimension). The conditions of modernity in which we find ourselves are like the spacetime described by Einstein rather than Euclid: there is no single prioritized coordinate system. Different coordinate systems and observers should be “present” simultaneously and understood

Figure 4. The transformation of the coordinate system of the concept of sustainability: (a) single directional concept of sustainable development; (b) dichotomous concept of sustainable development, indicating inherent contradiction; and (c) multidirectional concept of sustainable development.Source: Figure 1 from Gudmundsson and Höjer (1996), redrawn by Jin Zhu.

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comprehensively (Figure 5(b)). Only then can we grasp the complete meaning of an event (the point in Figure 5) and the varied influences engendered by its advancement. In this case, the number of dimensions that we can grasp in the new spacetime perspective is nine rather than three.

Conclusion

Non-Euclidean urban planning is a huge issue that cannot be fully demonstrated in a single article. In this article, we try to explore the inspiration of non-Euclidean thinking on planning thoughts, theories, and methodologies. We discuss three key issues, namely, varied measurements of urban land, geodesics in cities, and urban planning in spacetime. This discussion represents preliminary work on non-Euclidean planning, and we recog-nize that other issues, such as planning logic, are worth exploring in the future. The fal-sification of Postulate 5 results in a “domino effect.” Spacetime is proved to be not absolute by the Theory of Relativity. Logic is also proved to be not absolute by the Incompleteness Theorem proposed by mathematical logician Kurt Gödel. What this sig-nifies to urban planning remains to be discussed.

People through the ages have subconsciously used conceptions of non-Euclidean space in city construction and urban planning. The diversity of cities generates the het-erogeneity of urban space. Urban diversity and heterogeneity for one person enriches the spatiality and landscape of cities, and avoids stereotyping. For another, it engenders urban problems. The planner’s task is to reach a form of equilibrium between homogene-ity and heterogeneity. The Euclidean model of planning, which treats urban space as even space and as an “external space container,” should shift to engage perspectives of generalized non-Euclidean space. It is natural and inevitable; closer to our actual life, daily experiences, and real spaces; closer to the evolution of science; and closer to the general shift of the perspective and model of urban planning.

Acknowledgments

The authors express their sincere gratitude to Prof. Jean Hillier and Prof. Cewen Cao, the journal’s referees, and Dr Hui Wang and Mr Jin Zhu for their enormous help in writing this article.

Figure 5. Single and multiple coordinate systems for describing an event in planning.Source: Drawn by Hui Wang.

348 Planning Theory 12(4)

Funding

This paper is supported by the National Natural Science Foundation of China (NNSFC) (51008268), The Ministry of education of Humanities and Social Sciences Fund (13YJC770002), The Natural Science Fund of Zhejiang Province (Y12E080086), and The Project of Zhejiang Provincial Department of Education (Y201016458).

Notes

1. Or we may call it “geodesic” in cases of curved surface and space. The line is the shortest dis-tance between two points. This is easy to imagine and understand in a flat and even space, but when the space is curved, or in terms of geometry when curvature is not 0, the line or the geo-detic is also curved and changes according to the curvature of the space. Technically, “nonhomo-geneity” in modern differential geometry refers to manifolds of variable curvature. If a curved space has a constant curvature, such as a sphere, it is globally homogeneous as well.

2. The original text of Euclid is

if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (Heath, 1956)

3. In 1854, Riemann presented his famous memoir “On the Hypotheses Which Lie at the Foundations of Geometry” in Göttingen, which showed his profound and creative ideas on geometry. Riemann developed some unified principles that not only classified all the then exist-ing forms of geometry but also created a number of new types of space (Ritchey, 1991). Riemann’s principles substantially influenced Einstein as the latter formulated his Theory of Relativity and theory of four-dimensional spacetime manifold.

4. The former is, thus, a special Riemannian geometry, but this name is seldom used. Riemannian geometry refers to the general study of manifolds with reasonable metrics (Gray, 1989: 155).

5. A differentiable manifold can be infinitely subdivided.6. The recommendation was entitled “Recommendation Concerning the Safeguarding and

Contemporary Role of Historic Areas,” which was passed by the general conference of United Nations Educational, Scientific and Cultural Organization (UNESCO) on 26 November 1976.

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Author biographies

Kang Cao is currently Associate Professor of Urban Planning at Zhejiang University, China. Her research focuses primarily on the philosophical traditions and evolution of planning thoughts and theory, and Western planning history in ancient times and since the 19th century. She is the author of A Brief History of Western Modern Urban Planning History (2010, Southeast University Press, in Chinese) and several articles published in journals such as City Planning Review and Urban Planning Forum.

Yan Zhang received her PhD in urban planning from Tongji University and is now a lecturer at Shenzhen University, China. Her research interests are the transformation of development zones in China and the urban growth management of Chinese cities.