2x2 4- rotation cyclic L-systems

2x2 4-rotation cyclic L-systems

2x2 4-symbol replacement 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.

Rotation cyclic systems
Rotated corner systems
Bar rotation motifs in cyclic systems
Bar rotation similarity motifs in rotation cyclic systems

Rotation cyclic systems

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 comparative statistics of these two systems, CW and CCW, are interesting. The CW case (above) has more 2x2 blocks of identical symbols, and also more "Brigid's octomino" patterns. The CCW case has more 2x1 blocks of symbols, and a pronounced vertical/horizontal banding of features. See the related corner systems (below) that highlight each symbol's independent distrubution.

Rotated corner systems

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

difference between second and fourth symbol patterns

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 rotation cyclic L-systems above. Note that the bar rotation 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 "Brigid's octomino" 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 caption's "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):

"If we start with a motif with no axes of symmetry, (or only bilatateral), and we put 90 degree rotations of it in order we generate a 2x2 with spiral or radial symmetry depending on whether it's chiral. Only 4 of these spirals are possible. One can arrange these 4 new spirals in a kind of order, but if one simply applies a 90 degree rotation to them as one did with the original, one gets back the same result. if we try to arrange these by simply rotating the whole set, one gets a symmetrical 16 x16 set with no more possibility of rotation.

I guess what I'm saying is, is that it's not possible to apply the same 90 degree rotation to an element, then rotate the result 90 degrees and get something unique, because eventually that process generates a 4 directional symmetry and makes all 90 degree rotations equivalent....so...while it is possible to work at sucessively higher scales and generate new stuff, the rule has to be different than simply 'place 90 degree rotations of current object next to it in clockwise fashion around the scale'. There may still be a way to keep the rule 'simple', but the rule itself may have to alter itself with increasing scales."

[2/6/08 Mark: This comment may be relevant only to compound rotate-rotate systems, not shuffle-rotate. I haven't worked out most of this topic yet.]

2/ 7/08 Ted Bell writes:

"I think i see how to scale the rotations now. However the very first step seems to have to be a rotation in the opposite direction to all subsequent rotations, else symmetries form that don't permit further rotation. The rule doesn't have to change after that though."

------------------------------------
There are no restrictions on use of the images on this page. Claiming to be the originator of the material, explicitly or implicitly, is bad karma. A link (if appropriate), a note to dow[at]uoregon.edu, and credit are appreciated but not required.

Comments are welcome (dow[at]uoregon.edu).