Dynamics of Urban Centre and Concepts of Symmetry: Centroid and Weighted Mean

The city is a kind of complex system being capable of auto-organization of its programs and adapts a principle of economy in its form generating process. A new concept of dynamic centre in urban system, called "the programmatic moving centre", can be used to represent the successive appearances of various programs based on collective facilities and their potential values. The absolute central point is substituted by a magnetic field composed of several interactions among the collective facilities and also by the changing value of programs through time. The center moves continually into this interactive field. Consequently, we introduce mathematical methods of analysis such as "the centroid" and "the weighted mean" to calculate and visualize the dynamics of the urban centre. These methods heavily depend upon symmetry. We will describe and establish the moving centre from a point of view of symmetric optimization that answers the question of the evolution and successive equilibrium of the city. In order to explain and represent dynamic transformations in urban area, we tested this programmatic moving center in unstable and new urban environments such as agglomeration areas around Lausanne in Switzerland.