To figure out the fundamental theory of physics, we need to explore network systems (see chapter 9 of NKS).
In this project I studied a few rules for network systems which preserve planarity:
starting from a hexagonal grid:
Assuming that networks come into an equilibrium after a certain amount of steps, the goal was to compute some properties of this equilibrium, such as the distribution of face sizes, and local dimensionalities.
Evolution after the first few steps (for the first rule):
Two properties of equilibrium distributions were computed: distribution of face sizes, and distribution of local dimensionalities.
A surprizing thing is that for the first 5 rules, the distributions of face sizes turned out to be almost the same:
However, all the distributions for local dimensionalities are different:
If one looks at local dimensionalities on a network, one sees that high dimensionality corresponds to negative curvature:
Also, regions with large dimensionality occur where face sizes are large.
Networks, planar, fundamental physics, faces, dimensionality, curvature.
Subgraphs of hexagonal grid (were used to construct rules):
It turned out that different rules produce different graph distributions, which are stable in equilibrium. All the rules produce networks with varying curvature, including regions with very low curvature (~1), which one might identify with local structures (particles).
It is interesting to note, that distributions of face sizes turned out to be much more similar than distributions of local dimensionalities.
Make systematic enumeration of rules
Figure out whether the equilibrium network is a manifold
Study particles (as non planarities in a network)
Analyze how much steps are required until the equilibrium is reached
Study different initial conditions (torus, for example)
Compute the number of possible ways to apply the evolution rule as a function of time
Chapter 9 of NKS, in particular “Random Replacements” note on p. 1038.