Mathematics and morphogenesis of cities: A geometrical approach

Cities are living organisms. They are out of equilibrium, open systems that never stop developing and sometimes die. The local geography can be compared to a shell constraining its development. In brief, a city’s current layout is a step in a running morphogenesis process. Thus cities display a huge diversity of shapes and none of the traditional models, from random graphs, complex networks theory, or stochastic geometry, takes into account the geometrical, functional, and dynamical aspects of a city in the same framework. We present here a global mathematical model dedicated to cities that permits describing, manipulating, and explaining cities’ overall shape and layout of their street systems. This street-based framework conciliates the topological and geometrical sides of the problem. From the static analysis of several French towns (topology of first and second order, anisotropy, streets scaling) we make the hypothesis that the development of a city follows a logic of division or extension of space. We propose a dynamical model that mimics this logic and that, from simple general rules and a few parameters, succeeds in generating a large diversity of cities and in reproducing the general features the static analysis has pointed out.

Figure 1

The representation of a graph by its symmetric adjacency matrix (a). This graph is planar: It admits at least two geometrical graphs as drawings (b).

Figure 2

A geometrical graph (a) and two of its rectifications (b). The upper one is not straight because two edges intersect outside of the vertices set.

Figure 3

A straight graph (a) and its hypergraph structure (b) deduced from $R_{\pi /20}$. Viewed as a city’s map, this graph contains seven streets segments but three streets.

Figure 4

The histograms of degrees’ distribution in Amiens (entire city and center). We observe that both distributions are peaked in 3 even in the more regular city center, where the number of connections 4 is still half. The large number of 2 comes from straightening, especially in the suburbs’ curved streets, and should not be taken into account.

Figure 5

. The “topological maps” of Amiens (left-hand side) and its center (right-hand side). In each map, the red street is the topological center and the color of each street refers to its distance to that center. The maximum distance to the center is the radius of the map (18 in the entire city and 16 in the center). It is striking that this radius increases slowly with the size of the considered street system. We can even infer that the construction of the surrounding highway belt (the found center) is made precisely to keep the topological radius of the city small.

Figure 6

Angular distribution of Amiens (top left: entire city; top right: center). We notice in both distributions a fourfold symmetry. The doubled angular distributions (bottom) appear as ellipsoïds, which allows to define the anisotropy coefficient as the ratio of the eigenvalues of the inertia matrix: $\mathcal{A}=0.42$ in the entire city and $\mathcal{A}=0.71$ in the center.

Figure 7

The histogram represents the distribution of the logarithm of street length in the center of Amiens. The red plot is the fitting of this histogram with a gaussian mixture whose parameters are $p_{-}=0.5$, $m_{-}=2.2$, ${\sigma }_{-}=0.3$; $m_{+}=4.3$, ${\sigma }_{+}=1.2$.

Figure 8

The level lines of the potential field for a city reduced to a single segment of length 1 with ${\lambda }_0=1$ and $\beta =10$.

Figure 9

Simulations of the morphogenesis model with constant parameters with variation of the organization $P_e$ and the construction $\omega$. In each thumbnail, the rejecting radius is ${\lambda }_0=10\text{m}$, and there is no sprawling: $k=0$, the number of settlements is $N=80$, and the available space is bound in a square with sides of 400 m. The red and bold segment represents an initial street segment and a scale of 20 m.

Figure 10

The Chinese town of Xi’an in 1949, whose various subdivision patterns inside a regular grid recall variations in the parameters of the morphogenesis model.

Figure 11

(a), (b), (c), (d) Four steps in the development of the city (e) with 600 urban operations. For this simulation, ${\lambda }_0=10\hspace{0.5em}\text{m}$, $K=10$, $K_f=0.1$, $P_e=0.8$, $\omega =0.7$, and the windows are adapted to the size of the current city. The main phenomenon at work is the dynamics between the inner development and the extension of the city that creates two hierarchical scales.

Figure 12

The distribution of the logarithm of street lengths for two synthetic cities (top) and the second-order topology (bottom) for the resulting city of simulation (see Fig. 11). As for real cities (Amiens), the second-order topology presents a bounded hierarchical representation of the city and the street length is well fitted by a mixture of log-normal random variables.

Figure 13

A city developing with varying parameters from two perpendicular, long structuring elements. The historic center of the city has been built with a low construction parameter and no sprawling ($\omega =0.2$ and $f_{\text{ext}}=0$) to recall the tree aspect of a central souk in the Kasbah. Then parameters are changed to $\omega =0.8$ and $f_{\text{ext}}=0.15$, which produces an industrial crown.