Registered: Jan 2004
Posts: 350

Attached is a gif with samples of cellular automaton output in three colors.

They evolved from rules written as modulo 3 algebraic expressions having four variables: p, q, r, and s.

Values assigned to p came from the second nearest left neighbor of a cell.
Values assigned to q came from the nearest left neighbor.
Values assigned to r came from the nearest right neighbor.
And values assigned to s came from the second nearest right neighbor.
The cell itself did not enter into the calculation of its updated value.

Initial conditions were a set of 81 rule numbers whose digit expansion base 3 forms palindromic lists. There were 729 cells per step and the cellular automata evolved for 256 steps.

Colors in the samples correspond to the values of the cells as follows: ( 0 -> White ), ( 1 -> Blue ), and ( 2 -> Red ).

The first sample has symmetric bifurcating behavior against a blue and white horizontally stripped background.

The second sample has intersecting diagonals of inverted triangles. The white triangles travel from top and mid-left to the right. The red triangles travel from bottom and mid-right to the left. The blue triangles travel from top to bottom with an incline from right to left.

The third and fourth examples both self-organize into rows of inverted triangles. Then they repetitiously expand the pattern of rows.

Lawrence J. Thaden has attached this image:

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L. J. Thaden

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WOW ...

The first one reminds me of the picture of coral
or something on the cover of my old Dover copy
of Olaf Stapledon's 'Star Maker'.

Jon Awbrey

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Jon,

Quite a stretch, but I admire your capacity for associations.

I was taken by the symmetry of the first image. It must come from the algebraic rule itself since the initial conditions are not actually symmetric. They are composed of digit expansions, each of which is symmetric. But taken together, the digits are not symmetric.

The 81 rule numbers used as input to the initial conditions are:

{0,81,162,738,819,900,1476,1557,1638,2460,2541,2622,3198,3279,3360,3936,4017,4098,4920,5001,5082,5658,5739,5820,6396,6477,6558,6562,6643,6724,7300,7381,7462,8038,8119,8200,9022,9103,9184,9760,9841,9922,10498,10579,10660,11482,11563,11644,12220,12301,12382,12958,13039,13120,13124,13205,13286,13862,13943,14024,14600,14681,14762,15584,15665,15746,16322,16403,16484,17060,17141,17222,18044,18125,18206,18782,18863,18944,19520,19601,19682}.

Their digit expansions combine to make:

{0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,1,0,1,0,0,0,0,1,0,2,0,1,0,0,0,0,2,0,0,0,2,0,0,0,0,2,0,1,0,2,0,0,0,0,2,0,2,0,2,0,0,0,1,0,1,0,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,1,2,1,0,1,0,0,1,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,1,1,2,1,1,1,0,0,1,2,1,0,1,2,1,0,0,1,2,1,1,1,2,1,0,0,1,2,1,2,1,2,1,0,0,2,0,2,0,2,0,2,0,0,2,0,2,1,2,0,2,0,0,2,0,2,2,2,0,2,0,0,2,1,2,0,2,1,2,0,0,2,1,2,1,2,1,2,0,0,2,1,2,2,2,1,2,0,0,2,2,2,0,2,2,2,0,0,2,2,2,1,2,2,2,0,0,2,2,2,2,2,2,2,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,1,1,0,0,0,2,0,0,0,1,1,0,1,0,0,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,2,0,1,0,1,1,0,2,0,0,0,2,0,1,1,0,2,0,1,0,2,0,1,1,0,2,0,2,0,2,0,1,1,1,0,1,0,1,0,1,1,1,1,0,1,1,1,0,1,1,1,1,0,1,2,1,0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,0,1,2,1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,2,1,2,1,1,1,2,0,2,0,2,0,2,1,1,2,0,2,1,2,0,2,1,1,2,0,2,2,2,0,2,1,1,2,1,2,0,2,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,2,2,2,1,2,1,1,2,2,2,0,2,2,2,1,1,2,2,2,1,2,2,2,1,1,2,2,2,2,2,2,2,1,2,0,0,0,0,0,0,0,2,2,0,0,0,1,0,0,0,2,2,0,0,0,2,0,0,0,2,2,0,1,0,0,0,1,0,2,2,0,1,0,1,0,1,0,2,2,0,1,0,2,0,1,0,2,2,0,2,0,0,0,2,0,2,2,0,2,0,1,0,2,0,2,2,0,2,0,2,0,2,0,2,2,1,0,1,0,1,0,1,2,2,1,0,1,1,1,0,1,2,2,1,0,1,2,1,0,1,2,2,1,1,1,0,1,1,1,2,2,1,1,1,1,1,1,1,2,2,1,1,1,2,1,1,1,2,2,1,2,1,0,1,2,1,2,2,1,2,1,1,1,2,1,2,2,1,2,1,2,1,2,1,2,2,2,0,2,0,2,0,2,2,2,2,0,2,1,2,0,2,2,2,2,0,2,2,2,0,2,2,2,2,1,2,0,2,1,2,2,2,2,1,2,1,2,1,2,2,2,2,1,2,2,2,1,2,2,2,2,2,2,0,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2}

No symmetry here.

Here is another gif of an algebraic rule that has four variables and three colors.

This time the colors are: {0->CornflowerBlue, 1->MintCream, 2->LightPink}.

Initial conditions are a single LightPink cell just left of center on a row of 450 CornflowerBlue cells.

This rule has no number. It is just the algebraic expression 2 + q + s + 2 q s + p s + ( q + p ) r taken modulo 3.

The order of assignments from the values neighboring a cell to the variables in the algebraic expression is as follows:

Value of second nearest left neighbor is assigned to q.
Value of nearest left neighbor is assigned to r.
Value of nearest right neighbor is assigned to p.
Value of second nearest right neighbor is assigned to s.

The evolution is symmetric from the very beginning.

After step 821 it begins repeating a pattern 240 steps in length.

As Jason Cawley might say, it turns to “turtles all the way down”.

Lawrence J. Thaden has attached this image:

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L. J. Thaden

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Lawrence,

I'm afraid that after all these years
I'm still cutting my teeth bit by bit
on the B = GF(2) case, but in some of
the material I've read on differential
algebras appiled to discrete dynamics,
there's a tendency for folks to treat
any old finite field K as a parameter
that floats somewhere between B and
the real domain R.

There one thinks of a list of state variables,
starting with some initial set x_1, ..., x_n,
each of which holds a value in the field K,
and so the initial state space is K^n.

Then one thinks of the dual space that consists
of all functions of the type f : K^n -> K.
One notates this space as (K^n -> K).

The notion of something analogous to a CA rule
comes in when you think of the update functions
as members of (K^n -> K), that is, things like
<x_j>' = f_j <x_1, ..., x_n>.

For example, the ordinary ECAR's have K = B and
<x_j>' = f_j <x_(j-1), x_j, x_(j+1)>, for all j.

Generally speaking, it seems to be important to
consider the possibility of each cell's update
depending on its previous state, in other words,
to keep the update functions <x_j>' = f_j where
f_j depends non-trivially on x_j in the mix of
functions that one is considering.

That's as far as I get for the moment.

Jon Awbrey

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Thanks Jon.

I'm chewing on it.

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L. J. Thaden

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