Enzymind Labs

Also see Tilings with 2x2 3-symbol L-systems

These 3 symbol L-systems have at least one terminating rule, a rule that refers only to itself. This often results in large contiguous regions, sometimes with a thin fractal boundary. In these examples a boundary is labeled black and the final (third) terminating symbol is labeled white.

Bound 2x2 a5u

This algorithm has two terminating rules; both rules 1 and 3 only refer to themselves so those regions don't change after scaling. The initial condition is the second rule.

The fraction of white in the final image is the infinite series:

SUM_n=0->oo( 1/4^n ) = ?

Matlab command:
>> imOut = L_System_tiling( 'Bound_2x2_3s_a5u_10', 10, 3, 1, 0, '', 1, 'Bound_2x2_3s_a5u_rules.png' );
(see attached "rules" argument file)

Step 1

A tiling of Bound 2x2 a5u. This illustrates how a recurrent system (Bound 2x2 a5u) can be combined with a simple periodic tiling (also see other tilings with 2x2 3-symbol L-systems). The initial condition is applied only once, at the first generation, setting up a particular 4x4 tilling.

[ To Do: Illustrate: Another way to express the rules for the same pattern involves a sequential application of L-systems. This has the advantage of separating generation of the tiling and the recurrent 'step' pattern, allowing any number of tiling generations with any other number of pattern generations.]

Rules 2-5 are the four rotations of Bound 2x2 a5u's rule 2, 1 and 5 are Bound 2x2 a5u's terminating rules 1 and 3, and the initial condition (all rule 7) and rule 7 (each of the rule 2-5 rotations) generate the 4x4 tiling of rotations.

This tiling, and related tilings, have interesting perceptual features due to allignment of nearby boundaries. While it is formed from rotations of Bound 2x2 a5, the neighborhood combinations lead to the percept of fully black square regions interspersed with nearly fully white square regions with symmetries across both diagonals. [To Do: Illustrate with thumbnails of the original, black and largely white squares.]

22_30_80

This system's pattern is the motif for a similarity tiling, Wings, that uses a single fractal prototile.

Matlab command:
>> imOut = L_System_tiling( '22_30_80_10', 10, 3, 1, 0, '', 0, [0 2; 1 1], [1 0; 2 0], [2 2; 2 2] );

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