Planar Network Systems
Planar Network Systems
Name: Maxim Piskunov
Instructor: Todd Rowland
Wolfram Science Summer School 2014
Homework Solution
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
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:
 
 
 
 
 
 

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
Code
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Rules
Rules
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Hexagonal Grid
Hexagonal Grid
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Balls
Balls
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Enumeration
Enumeration
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Visualization
Visualization
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Evolution
Evolution
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Faces
Faces
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Dimentionality
Dimentionality
Visualizations
Visualizations
Evolution after the first few steps (for the first rule):
Main Results
Main Results
Two properties of equilibrium distributions were computed: distribution of face sizes, and distribution of local dimensionalities.
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Face sizes
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
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
Demonstrations
Subgraphs of hexagonal grid (were used to construct rules):
Conclusions
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
Make systematic enumeration of rules
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Figure out whether the equilibrium network is a manifold
Figure out whether the equilibrium network is a manifold
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Study particles (as non planarities in a network)
Study particles (as non planarities in a network)
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Analyze how much steps are required until the equilibrium is reached
Analyze how much steps are required until the equilibrium is reached
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Study geodesics
Study geodesics
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Study different initial conditions (torus, for example)
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
Compute the number of possible ways to apply the evolution rule as a function of time
Links/References
Links/References
Chapter 9 of NKS, in particular “Random Replacements” note on p. 1038.
Network system
Planar
Fundamental physics
Graph faces
Dimensionality
Other
Other
Last Modified: Thursday, July 17, 2014