Enzymind Labs

2x2 4-rotation cyclic L-systems

 

Rotation cyclic systems

2x2 4-symbol systems with rules that are are related by rotations (n*pi/2, n = [ 0, 3 ] ) are highly symmetric. Tesselations that use these rules exhibit this symmetry at different scales.

With each rule the pi/2 clockwise rotation of the last rule:
Rows and columns are of two types, pairs that intersect 2 x 2 blocks alternating with pairs that don't.

Matlab command:
>> imOut = L_system_tiling( 'CW_rot', [n_g], 4, 1, 0, '', 0, [0 1; 3 2], [3 0; 2 1], [2 3; 1 0], [1 2; 0 3] );
where [n_g] is the number of generations.

With each rule the pi/2 counter-clockwise rotation of the last rule:
Alignments of pairs of 1 x 2, or 2 x 1, blocks give this pattern a perceptual horizontal and vertical bands.

Matlab command:
>> imOut = L_system_tiling( 'CCW_rot', [n_g], 4, 1, 0, '', 0, [0 1; 3 2], [1 2; 0 3], [2 3; 1 0], [3 0; 2 1] );
where [n_g] is the number of generations.

The comarative statistics of these two systems, CW and CCW, are interesting. The CW case (above) has more 2x2 blocks of identical symbols, and also more "houndstooth" patterns. The CCW case has more 2x1 blocks of symbols, and a pronounced vertical/horizontal banding of features.

The distributions of each symbol are different, but the patterns within a system are related:

Individual symbol distributions for the CW cyclic system. Note that the all the patterns are related by a half-width translations and/or horizontal or vertical mirroring.
All of these patterns are closely related to Thue-Morse patterns. In the fifth panel the absolute difference between the second and fourth pattern shows that the 2-D two-symbol Thue-Morse pattern appears as a logical union of the patterns of symbol one and three, or two and four. This corresponds to collapsing (treating as identical) these pairs of symbols. Individual symbol distributions for the CCW cyclic system. Note that the all the patterns are related by a half-width translations and/or a pi/2 rotation.

Bar rotation motifs in cyclic systems

These use the simplest and most symmetric possible motifs for which each rotation is unique, 2x2 black and white bars, in the 2x2 4-rotation cyclic L-systems above. Note that the bar rotations ordering is in the same or opposite sense as the rules:

Each rule the pi/2 CW rotation of the previous rule, and each motif the pi/2 CW rotation of the previous motif.
Each rule the pi/2 CCW rotation of the previous rule, and each motif the pi/2 CW rotation of the previous motif.
Each rule the pi/2 CW rotation of the previous rule, and each motif the pi/2 CCW rotation of the previous motif.	This pattern has a large fraction of 2x2 sets of "houndstooth" patterns
Each rule the pi/2 CCW rotation of the previous rule, and each motif the pi/2 CCW rotation of the previous motif.

Matlab commanda:
>> imOut = L_system_tiling( 'CW_rot', [n_g], 4, 1, 0, 'Bars_4-rotation_CW_motif.png', 0, [0 1; 3 2], [3 0; 2 1], [2 3; 1 0], [1 2; 0 3] );
>> imOut = L_system_tiling( 'CCW_rot', [n_g], 4, 1, 0, 'Bars_4-rotation_CW_motif.png', 0, [0 1; 3 2], [1 2; 0 3], [2 3; 1 0], [3 0; 2 1] );
>> imOut = L_system_tiling( 'CW_rot', [n_g], 4, 1, 0, 'Bars_4-rotation_CCW_motif.png', 0, [0 1; 3 2], [3 0; 2 1], [2 3; 1 0], [1 2; 0 3] );
>> imOut = L_system_tiling( 'CCW_rot', [n_g], 4, 1, 0, 'Bars_4-rotation_CCW_motif.png', 0, [0 1; 3 2], [1 2; 0 3], [2 3; 1 0], [3 0; 2 1] );
where [n_g] is the number of generations.

[ To Do: Do a table of the logical combinations of these bar rotation cyclic systems. ]

Bar rotation similarity motifs in rotation cyclic systems

There are several 4x4 motifs composed of bars that are interesting because the motifs are constructed using rules that are related to the rules of the L-system. The resuting symmetries lead to combinatorial patterns that often tessellate the plane.

[ To Do: Illustrate the diagonal a b pattern with an arrow diagram. ]
The "Dab order" of the bars refers to a "diagonal a b" pattern, like:
a b
b a
where a and b are related by a pi/2 rotation.

Each rule the pi/2 CW rotation of the previous rule, and each motif is in Dab order and is the pi/2 CW rotation of the previous motif.	(Full ninth generation pattern) Top-left corner of ninth generation pattern, with some (not all) unique contiguous elements marked in color. Black and white patterns are anti-symmetric.	Center of ninth generation pattern.
Each rule the pi/2 CCW rotation of the previous rule,and each motif is in Dab order and is the pi/2 CW rotation of the previous motif.	(Full ninth generation pattern) Top-left corner (fifth generation) of ninth generation pattern, with some (not all) unique contiguous elements marked in color. The black and white patterns are related by a pi or pi/2 rotation, depending on the generation.	The same top-left corner, but with several of one shape of contiguous region colored. These sets of four can tesselate in a square tiling, and are the basis for the "Flying Spaghetti Monster" (FSM) tesselations:
Each rule the pi/2 CW rotation of the previous rule, and each motif is in Dab order and is the pi/2 CCW rotation of the previous motif.	(Full ninth generation pattern) Top-left corner of ninth generation pattern, with some (not all) unique contiguous elements marked in color. The black and white patterns are related by a pi or pi/2 rotation, depending on the generation. The figures at the center of the top-left image (yellow, dark purple) form a complementary pair that tesselate in a square tiling.	Center of ninth generation pattern.
Each rule the pi/2 CCW rotation of the previous rule, and each motif is in Dab order and is the pi/2 CCW rotation of the previous motif.	(Full ninth generation pattern) Top-left corner of ninth generation pattern, with some (not all) unique contiguous elements marked in color. The black and white patterns are related by a pi or pi/2 rotation, depending on the generation.

2/6/08 Ted Bell writes with respect to integrating compound rotations of motifs into the rule system (a rule system that can rotate individual motifs as well as whole patterns with each generation):

2/ 7/08 Ted Bell writes:

Corner rotation cyclic systems

[ To Do ]



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