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Indeed starting from a particular state it is for example not clear whether it is ever possible to reach all other states.
If I am correct that there is a simple underlying program for the universe, then this means that theoretical physics must at some level have only a very small amount of true physical input—and the rest must in a sense all just be mathematics.
With two possible colors, just 4 cases need to be specified in such rules, and there are 16 possible rules in all.
The smallest structures are stationary, but at initial condition 916 a structure is found that moves—all much the same as in the two other class 4 cellular automata that we have just discussed.
Yet as we have seen all of them can be captured by quite simple programs.
All the larger networks shown are approximately uniform, in the sense that similar results are obtained starting from any node.
Different runs [of initially random cellular automata]
The qualitative behavior seen with a given cellular automaton rule will normally look exactly the same for essentially all different large random initial conditions—just as it does for different parts of a single initial condition.
.)
• Is there a cuboid in which all edges and all diagonals are of integer length?
(Up to 8 nodes, all 8 trivalent networks have this property; up to 10 nodes 25 of 27 do.)
• Girth: the length of the shortest cycle in the network. … It turns out that for all trivalent networks the maximum is also 4, and is almost always 3.
But finally, when technology had advanced to the point where it became almost trivial for me to do so, I went back and generated some straightforward pages of pictures of all 256 elementary rules evolving from simple initial conditions. … But in all of these pictures I found nothing beyond repetitive and nested behavior.