Scaling analysis of 2D fractal cellular structures

The correlations between topological and metric properties of fractal tessellations are analysed. To this end, Sierpinski cellular structures are constructed for different geometries related to Sierpinski gaskets and to the Apollonian packing of discs. For these geometries, the properties of the distribution of the cells' areas and topologies can be derived analytically. In all cases, an algebraic increase of the cell's average area with its number of neighbours is obtained. This property, unknown from natural cellular structures, confirms previous observations in numerical studies of Voronoi tessellations generated by fractal point sets. In addition, a simple rigorous scaling resp. multiscaling properties relating the shapes and the sizes of the cells are found.