Chapter 9: Fundamental Physics

Section 12: Evolution of Networks

Cellular structures [in nature]

There are many systems in nature that consist of assemblies of discrete regions—and the lines that define the interfaces between these regions form networks. In many cases the regions are fixed once established (compare page 988). But in other cases there is continuing evolution, as for example in soap and other foams and froths, grains in metals and perhaps some biological tissues. In 2D situations the lines between regions generically form a trivalent planar network. In a soap foam, the geometrical layout of this network is determined by surface tension forces—with connections meeting at 120° at each node, though being slightly curved and of different lengths. Pressure differences lead to diffusion of gas and on average to von Neumann's Law that the area of an n-sided region changes linearly with time, at a rate proportional to n - 6. Typically the network topology of a foam continually rearranges itself through cascades of seemingly random T1 processes (rule (b) from page 511), with regions that reach zero size disappearing through T2 processes (reversed rule (a)). And as noted for example by Cyril Smith in the early 1950s there is a characteristic coarsening that occurs. Something similar is already visible in the pure T1 pictures in the note above. But results such as the so-called Aboav–Weaire law that m[n] from the note above is in practice about 5n + c suggest that T2 processes are also important. (Processes like cell division in 2D biological tissue in effect directly add connections to a network. But this can again be thought of as a combination of T1 and T2 processes, and in appropriate idealizations can lead to very similar results.)

Image Source Notebooks:

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