Topological Maps and Models of Shapes

We study the topological properties of physical froths. They are cellular networks with minimum incidence numbers (D+1 edges incident on a vertex in D-dimensions), and with cells with homogeneous shapes and sizes. We present a method where the structure of froths is analyzed as organized in concentric layers of cells around a given, arbitrary, central cell. A map gives, by recursion, the number of cells in successive layer. In 2 and 3 dimensions this map (logistic map) has one parameter, given as a function of the average topological properties of the cells in the layers. From the behaviour of the number of cells per layer with the topological distance, one obtains the curvature of the space tiled by the froth. By using the map it is therefore possible to characterize the shape of the manifold tiled by the froth in term of the topological arrangements of its tiles. We propose a topological method to compute the Gaussian curvature of a surface from a sample of points.