Planar Network Systems

Name: Maxim Piskunov
Instructor: Todd Rowland
Wolfram Science Summer School 2014

Homework Solution

OuterCA[rule_, init_, steps_] := CellularAutomaton[{rule, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, init, {{{steps}}, All, All}]
InitCA[] := ReplacePart[Array[0&, {1000, 1000}], {{500, 500}  1, {501, 500}  1, {500, 501}  1, {501, 502}  1}];
PlotRule28503[] := ArrayPlot[OuterCA[28503, InitCA[], 4000], PixelConstrained  1]
PlotRule28503[]

Project Description

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:
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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.

Code

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Rules

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Hexagonal Grid

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Balls

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Enumeration

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Visualization

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Evolution

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Faces

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Dimentionality

Visualizations

Evolution after the first few steps (for the first rule):

Main Results

Two properties of equilibrium distributions were computed: distribution of face sizes, and distribution of local dimensionalities.
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Face sizes

A surprizing thing is that for the first 5 rules, the distributions of face sizes turned out to be almost the same:
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Local dimensionalities

CDFs:
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.

Demonstrations

Subgraphs of hexagonal grid (were used to construct rules):

Conclusions

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.
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Make systematic enumeration of rules

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Figure out whether the equilibrium network is a manifold

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Study particles (as non planarities in a network)

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Analyze how much steps are required until the equilibrium is reached

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Study geodesics

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Study different initial conditions (torus, for example)

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Compute the number of possible ways to apply the evolution rule as a function of time

Links/References

Chapter 9 of NKS, in particular “Random Replacements” note on p. 1038.
Network system
Planar
Fundamental physics
Graph faces
Dimensionality

Other

Last Modified: Thursday, July 17, 2014