Notes

Chapter 9: Fundamental Physics

Section 12: Evolution of Networks


[Generating] arbitrary transformations [between networks]

By applying the string transformation rules on page 1035 at appropriate locations, it is possible to transform any string of A's and B's to any other. And the analog of this for networks is that by applying the rules shown below at appropriate locations it is possible to transform any network into any other. These rules correspond to the moves invented by James Alexander in 1923 in connection with transforming one knot into another. (Note that the first two rules suffice for all planar networks, and are sometimes called respectively T2 and T1.)

As an example, the pictures below show how a tetrahedron network can be transformed into a cube.

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