Chapter 9: Fundamental Physics

Section 12: Evolution of Networks

Planar networks

One feature of a planar network is that it is always possible to identify definite regions or faces bounded by connections in the network. And from Euler's formula f + n = e + 2, it then follows that the average number of edges of each face is always 6(1 - 2/f), where f is the total number of faces. Note that with my definition of dimension for networks, the fact that a network is planar does not necessarily mean that it has be two-dimensional—and for example the networks on page 509 are not.

From Stephen Wolfram: A New Kind of Science [citation]