Thank you Johann-Tobias and Malthe for the interesting post. I rendered a video of Johann-Tobias answer in 3d:

2d grid model example video

Just to examine the layout and the causality of the model.

Thank you again. Tuomas

Heureka.

As far as I can see, the code does not include the idea of negations, which is why squares are created where they should not.

You are right. Negations would have prevented it.

If negations should be simulated, it would probably require some updating of the code, since the idea is not implemented yet. But they can still be used as analytical tools. And even though my example ended up being wrong, I hope that some of the ideas could still be of use. Especially the idea about subrules.

I will ask about implementing negations. It seems a good addition. Randomly generating rules with negations is harder though.

The idea about sub rules wasn't new. I already used it in previous designs, however, a few hours before you commented i learned how to express multiple rules independently: WolframModel also takes a list of rules. Your design was the correct one. My previous designs conflated the builder of "coordinate system" with the "filler of space" (r1 vs r2) which is i never ended up with a flat space.

However you still got the problem of long tails.

I found it nicer to pre generate a (partial) pattern using just r1 and then fill is seperatly.

The current rules:

My represenations are not chiral, so i ran into problems with red, which could be solved by understanding that the problem was created by chirality.

The dot always points away from the coordinate axis.

The code looks like that:

WolframModel[{
    {
     {5,6,9,8,5,6},
     {1,2},{2,1},{2,4},{4,2},{4,3},{3,4},{3,1},{1,3},
     {4,6},{6,4},{6,5},{5,6},{5,3},{3,5},
     {6,9},{9,6},{9,8},{8,9},{8,5},{5,8},
     {6,7},{7,6},{7,10},{10,7},{10,9},{9,10},
     {7,11},{11,7},{11,12},{12,11},{12,10},{10,12}
    } -> {
     {4,111,7,6,4,111,4},
     {7,111,4,6,7,111,7},
     {4,111,7,6,4,111},
     {1,2},{2,1},{2,4},{4,2},{4,3},{3,4},{3,1},{1,3},
     {4,6},{6,4},{6,5},{5,6},{5,3},{3,5},
     {6,9},{9,6},{9,8},{8,9},{8,5},{5,8},
     {6,7},{7,6},{7,10},{10,7},{10,9},{9,10},
     {7,11},{11,7},{11,12},{12,11},{12,10},{10,12},
     {4,111},{111,4},{111,7},{7,111}
    },
    {
     {6,7,10,9,6,7,6},
     {1,2},{2,1},{2,4},{4,2},{4,3},{3,4},{3,1},{1,3},
     {4,6},{6,4},{6,5},{5,6},{5,3},{3,5},
     {6,9},{9,6},{9,8},{8,9},{8,5},{5,8},
     {6,7},{7,6},{7,10},{10,7},{10,9},{9,10}
    } -> {
     {4,111,7,6,4,111,4},
     {1,2},{2,1},{2,4},{4,2},{4,3},{3,4},{3,1},{1,3},
     {4,6},{6,4},{6,5},{5,6},{5,3},{3,5},
     {6,9},{9,6},{9,8},{8,9},{8,5},{5,8},
     {6,7},{7,6},{7,10},{10,7},{10,9},{9,10},
     {4,111},{111,4},{111,7},{7,111}
    },
    {{1,2,3,4,1},{1,2},{2,1},{2,3},{3,2},{3,4},{4,3}}->
        {{11,12,2,1,11},{1,2},{2,1},{2,3},{3,2},{3,4},{4,3},{11,1},{1,11},{11,12},{12,11},{12,2},{2,12}}
},{
    {1,2,3,4,1},{2,3,4,1,2},{3,4,1,2,3},{4,1,2,3,4},
    {1,2,3,4,1,2},{2,3,4,1,2,3},{3,4,1,2,3,4},{4,1,2,3,4,1},
    {1,2},{2,1},{2,3},{3,2},{3,4},{4,3},{4,1},{1,4}
}
, 32, "FinalStatePlot"]

Thank You, Malthe Andersen,

after solving this toy problem, we can strive to attack bigger problems. The solution has more potential for optimization to run faster but now we can finaly say that we can build a 2d grid.

Look at the attached PDF for another proposal.

It is not as lean as your design, but getting it working should be the first priority. Optimization can happen later. Naturally we still run into the problem that filling out the square areas takes longer than growing a line but that will stay around since we are restricted by some notion of causality.

Attachments:

Proposal-1.pdf

I have an idea how i can salvage your design if you give me a few more minutes,

I have tried to reproduce your work and have written the rules as such. I hope they are free of careless error

WolframModel[{
    {{1,2},{2,1},{2,3},{3,2},{3,4},{4,3}}->
        {{1,2},{2,1},{2,3},{3,2},{3,4},{4,3},{22,2},{2,22},{22,23},{23,22},{23,3},{3,23}},
    {{1,2},{2,4},{4,2},{2,3},{3,2},{4,6},{6,4},{5,6},{6,5},{6,7},{7,6},{6,8},{8,6},{7,9},{9,7}}->
        {{1,2},{2,4},{4,2},{2,3},{3,2},{4,6},{6,4},{5,6},{6,5},{6,7},{7,6},{6,8},{8,6},{7,9},{9,7},{4,111},{111,4},{111,7},{7,111}},
    {{1,2},{2,4},{4,2},{2,3},{3,2},{4,6},{6,4},{5,6},{6,5},{6,7},{7,6},{6,8},{8,6},{7,9},{9,7},{7,10},{10,7},{11,10}}->
        {{1,2},{2,4},{4,2},{2,3},{3,2},{4,6},{6,4},{5,6},{6,5},{6,7},{7,6},{6,8},{8,6},{7,9},{9,7},{7,10},{10,7},{11,10},{4,111},{111,4},{111,7},{7,111}}
}


,{{1,2},{2,1},{2,3},{3,2},{3,4},{4,3},{4,1},{1,4}}
, 5, "FinalStatePlot"]

Those result in this rule plot:

It is distorted by the graph plotting style but the topology looks correct.

This rule with these initial conditions:

follows this cancerous evolution.

The problem it that r1 can make it grow in any direction and can start to grow from any square. I already devised an approach based on hyper graphs to trim the space of allowed extensions in something close to a plane. Do you want to revise your strategy?

Do you have a recent version of Mathematica available?

I agree that a lack of negation imposes certain restrictions.

I tried to transcribe your rules from the pictures but the results were very unsatisfactory, as nothing prevents it from growing multiple northward arms from the central piece. I will post results soon as the first renders are out.

You might want to look into my "Failing to make a grid" post to get an idea how HyperGraphs might be used as markers to prevent this ill.

Cool. I've been struggling to create a grid.

As far as i am aware it is not possible to create a negation in the WolframModel.

Do you have a code which produces those graphs using the WolframModel framework?

Why isn't the upper left point in r2 and r3 drawn?