Mark Dow

Geek art

Simple recursive systems and fractal patterns

Thue-Morse tilings of simple 3x3 motifs

    Examples of replacing the symbols of a 2-D Thue-Morse pattern with a symmetric pair of two-color, 3x3 pixel motifs. Also see Thue-Morse sequence tilings, with a wider range of examples.

Three bars motifs, link to Three bars motifs, link to Decimated three bars motifs, link to Decimated three bars motifs, link to Decimated three bars motifs, link to
Three bars motifs

 Plus motif, link to Plus motif decimated, link to Plus motif decimated, link to
Plus motif

Three bars motifs

TM with three bars motif algorithm graphic
Thue-Morse by generation graphic
 TM three bars 8th generation, link to TM three bars gray rectangles, link to
This system with all black and white rectangles replaced with a single gray tone. The surround contrast makes it appear to be a weave of two shades of gray.

[To Do: Describe relationship, and link, with Period-doubling weave, and show a direct comparison.]
Matlab command:
>> imOut = L_system_tiling(  'TM', [ng], 2, 1, 0, 'ThreeBars1.png', 0, [0 1; 1 0], [1 0; 0 1]  );
where [ng] is the number of generations.

TM three bars, decimated by 2 offset 0, link to
The same system (above), decimated by 2 (every second pixel) with an offset of 0.

The contiguous regions are diagonally oriented, similar to that of 2x2 bar motifs.

Are the black and white regions pi/2 equivalent after a pi/2 rotation?
 TM three bars, decimated by 4 offset 0, link to
The system decimated by 4 (every fourth pixel) with an offset of 0.

This pattern's contiguous regions are all bounded by parallelograms of various scales, elongate in different directions for black and white. [To Do: Color a few.]
 TM three bars, decimated by 4 offset 1, link to
The system decimated by 4 (every fourth pixel) with an offset of 1.
Matlab commands:
>> imOut = L_system_tiling(  'TM', [ng], 2, 2, 0, 'ThreeBars1.png', 0, [0 1; 1 0], [1 0; 0 1]  );
>> imOut = L_system_tiling(  'TM', [ng], 2, 4, 0, 'ThreeBars1.png', 0, [0 1; 1 0], [1 0; 0 1]  );
>> imOut = L_system_tiling(  'TM', [ng], 2, 4, 1, 'ThreeBars1.png', 0, [0 1; 1 0], [1 0; 0 1]  );
where [ng] is the number of generations.

Plus motif

TM with plus motif algorithm graphic
Thue-Morse by generation graphic
 TM plus motif 8th generation, link to
Matlab command:
>> imOut = L_system_tiling(  'TM', [ng], 2, 1, 0, '3x3Plus.png', 0, [0 1; 1 0], [1 0; 0 1]  );
where [ng] is the number of generations.

TM plus motif, decimated by 2 offset 0, link to
The same system (above), decimated by 2 (every second pixel) with an offset of 0.

The contiguous regions are diagonally oriented, similar to that of 2x2 bar motifs.

Are the black and white regions pi/2 equivalent after a pi/2 rotation?
 TM plus motif, decimated by 4 offset 0, link to
The system decimated by 4 (every fourth pixel) with an offset of 0.

 [To Do: How many unique contiguous shapes? Color a few.]
 TM plus motif, decimated by 4 offset 1, link to
The system decimated by 4 (every fourth pixel) with an offset of 1.
Matlab commands:
>> imOut = L_system_tiling(  'TM', [ng], 2, 2, 0, '3x3Plus.png', 0, [0 1; 1 0], [1 0; 0 1]  );
>> imOut = L_system_tiling(  'TM', [ng], 2, 4, 0, '3x3Plus.png', 0, [0 1; 1 0], [1 0; 0 1]  );
>> imOut = L_system_tiling(  'TM', [ng], 2, 4, 1, '3x3Plus.png', 0, [0 1; 1 0], [1 0; 0 1]  );
where [ng] is the number of generations.

To Do

[ To Do: L, cross motif systems ]



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