Figure 11: Building blocks, lattice where we embed and .
We chose building blocks so that they generate homogeneous grids and with fixed length edges. Both constraints are necessary for the volumetric argument we want to use. Homogeneity is similar to the requirement for planar graphs in normal graph theory that edges do not cross. Edge-crossing would generate inhomogeneities with fixed length edges on 2D grids.
If is even, we naturally choose edges perpendicular to each other. Two counter-directed edges of the building block form an independent axis in the dimensional axis. Therefore, we set
Furthermore, we know that the boundary of a subgraph has a dimension less (for a sphere , but ). In general, if the subgraph is -dimensional, the volume is proportional to and the boundary to . Size of the random subgraph with nodes is then:
We now start in the middle of our subgraph, and we subsequently add nodes until we reach the boundary. In the analogous mechanism of adding nodes means putting them into the bin, , representing the height of the tree and the boundary is reached until all nodes are into the bin.
Resuming, we consider embedded in , both in a homogeneous -dimensional grid composed by myriads of the above mentioned building blocks. We let grow until it reaches the same size of through summation and we infer .
Results are very similar to the Mandelbrot's formula