Enzymind Labs

Progressive 2x2 rule systems

Constant rule systems:

Progressive 1

Progressive 1b

Progressive 2

Progressive 3

Logical combinations of constant progressive 2x2 rule systems

These systems all have algorithms for which every symbol is replaced by a combination of itself and the next symbol. The algorithms are of this form:

a; a -> [ f( a, b ) ], b -> [ f( b, c ) ], c -> [ f( c, d ) ] ...

For n_g generations the number of symbols n_s required is n_s = (n_g + 1); the last symbol's rule is not applied.

These systems have a direct correspondance with 2x2 n_s symbol systems (where n_s is fixed, and not dependent on the number of generations n_g). If these progressive systems are collapsed modulo n, they are equivalent to an n_s symbol system. For example:

   0; 0 -> [ 0 1 ], 1 -> [ 1 2 ], 2 -> [ 2 3 ] ... modulo 2

= 0; 0 -> [ 0 1 ], 1 -> [ 1 0 ], 0 -> [ 0 1 ] ...

= 0; 0 -> [ 0 1 ], 1 -> [ 1 0 ]

which is the Thue-Morse L-system. So the fixed point (resulting pattern) of the Progressive 2 system (below) modulo 2 will be the 2-D Thue-Morse pattern.

The form of these systems (the progression of rules) is closely related to the shift operator. For example a pattern that is invariant under the action of a Baker's map , such as this "Ising tartan ":

Ising model probability density for the two-sided lattice using the dyadic mapping

has the same characteristics (self-symmetry and non-periodicity) as the other progressive systems on this page. What is the L-system that creates this pattern? Is it what I call a progressive system? I suspect so, with a mirror  in the rule system. 

[To Do: Work this out.]

The Baker's map is an exactly solvable model of deterministic chaos, in that the eigenfunctions and eigenvalues of the transfer operator can be explicitly determined. The folded Baker's map is a two-dimensional analog of the tent map.

Constant rule systems

There are only three rules that maintain a constant relationship from one rule to the next and that produce unique patterns (not related by rotation of the image or replacement rules and/or mirroring of the ordered rule pallette).

Also see Logical combinations of constant progressive 2x2 rule systems

Progressive 1:

s₀; s_n -> [ s_n+1, s_n; s_n, s_n ]

High resolution Progressive 1 system image

The histogram (above right) of the relative number of symbols/values in the image shows that the distribution is skewed toward the first symbol. This distribution was for a tenth generation image, but the shape of the distribution is the same for other generations. (Pascal distribution?)

Matlab command:
>> imOut = L_system_tiling( 'Progressive_1_[n_g]', [n_g], 12, 1, 0, '', 0, 'Progressive_1_12_rules.png' );
where [n_g] = [1, 11] for a 12 rule set. (see attached 'Progressive_1_12_rules.png' argument file)

Progressive 1b:

s₀; s_n -> [ s_n, s_n+1; s_n+1, s_n+1 ]

High resolution Progressive 1b system image

The histogram (above right) of the relative number of symbols/values in the image shows that the distribution is skewed toward the high symbol. This distribution is a symbol-wise mirror of the distribution for the Progressive 1 image. (Pascal distribution?)

This system has a pattern identical to that of the Progressive 1 system (see above), after a 180 degree rotation and mirroring of the rules/pallete. As a system, where the '-' sign indicates element-wise subtraction of an ordered symbol set:

In words, the Progressive 1b system is identical to the high symbol's constant system minus the Progressive 1 system rotated by pi radians. This corresponds to a half rotation of each replacement rule and swapping of the two symbols within a rule.

Matlab command:
>> imOut = L_system_tiling( 'Progressive_1b_[n_g]', [n_g], 12, 1, 0, '', 0, 'Progressive_1b_12_rules.png' );
where [n_g] = [1, 11] for a 12 rule set. (see attached 'Progressive_1b_12_rules.png' argument file)

Progressive 2:

s₀; s_n -> [ s_n, s_n+1; s_n+1, s_n ]

High resolution Progressive 2 system image

The histogram (above right) of the relative number of symbols/values in the image shows that the distribution is symmetric about the middle symbol. (A binomial distribution?)

The 1-D L-system analog (corresponding to the to row) is the sequence 0, 01, 0112, 01121223 ..., which is Sloane's EIS A000120, "1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n). ".  If the values or coded as binary numbers, and the sum of these number's bits modulo 2 is a 2-D Thue-Morse sequence.

Matlab command:
>> imOut = L_system_tiling( 'Progressive_2_[n_g]', [n_g], 12, 1, 0, '', 0, 'Progressive_2_12_rules.png' );
where [n_g] = [1, 11] for the attached 12 rule set.

Progressive 3:

s₀; s_n -> [ s_n, s_n+1; s_n, s_n+1 ]
High resolution Progressive 3 system image

This system is aperiodic in one direction (horizontally) and constant in the second direction. The relative distribution of the number of each symbol (above right) is the same as that of the Progressive 2 system -- symmetric about the middle symbol.

Matlab command:
>> imOut = L_system_tiling( 'Progressive_3_[n_g]', [n_g], 12, 1, 0, '', 0, 'Progressive_3_12_rules.png' );
where [n_g] = [1, 11] for the attached 12 rule set.

[ To Do: The material below is not complete ]

(right click, View Image for large image)

Progressive 4:

(right click, View Image for large image)

Progressive 5:

(right click, View Image for large image)

The progressive 2x2 rule systems on this page can be constructed with operations on the binary digits of the indices of each pixel. The reason these rule systems correspond with operations on the digits of numbers is because numeral systems use an isomorphic progressive rule system, resulting in the same nested, redundant, and aperiodic patterns when numbers are in an ordered set.

(Also see similar patterns, generated from a related rules at:

[Patterns from] bitwise functions, Chapter 2 Notes > Section 1 > Page 871 > Note (a)

[Patterns from] arbitrary digit operations, Chapter 10 Notes > Section 11 > Page 1091 > Note (e),

(Wofram 2002)

For example a decimal or binary number is constructed from a progressive rule in the same sense....

See "[To Do: Recurrent sequences and numeral systems]".

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