3x3 boundary L-systems

The simplest 3x3 replacement systems that result in connected curves.

Three symbol systems

The more symmetric 3-symbol 3x3 systems that form a boundary.

Boundary 6, 6b

Boundary 6 system pattern

Boundary 6 average pattern, link to 7th generation

average across generations

self-symmetries

Boundary 6 pattern Fourier transform, link to 7th generation

pattern DFT amplitudes, colored by phase
hue-phase scale bar

average pattern DFT

Boundary 6b system pattern

Boundary 6b average pattern, link to 7th generation

average across generations

self-symmetries

Boundary 6b pattern Fourier transform, link to 7th generation

pattern DFT amplitudes, colored by phase
hue-phase scale bar

average pattern DFT

MATLAB command:
>> imOut = L_system_tiling( 'BD6', [n_g], 3, 1, 0, '', 0, [1 2 2; 2 0 2; 2 2 21], [0 10 0; 2 2 30; 2 2 0], [2 2 2; 2 2 2; 2 2 2] );
>> imOut = L_system_tiling( 'BD6b', [n_g], 3, 1, 0, '', 1, [1 2 2; 2 0 2; 2 2 21], [0 10 0; 2 2 30; 2 2 0], [2 2 2; 2 2 2; 2 2 2] );
where [n_g] is the number of generation

	Boundary 6 array 1 system pattern
	Boundary 6 array 2 system pattern
	Boundary 6barray 1 system pattern
	Boundary 6barray 2 system pattern
	Boundary 6barray 3 system pattern

MATLAB command:
>> imOut = L_system_tiling( 'BD6arr1', [n_g], 3, 1, 0, '', [0 0 0 0; 0 0 0 0; 0 0 0 0; 0 0 0 0], [1 2 2; 2 0 2; 2 2 21], [0 10 0; 2 2 30; 2 2 0], [2 2 2; 2 2 2; 2 2 2] );
>> imOut = L_system_tiling( 'BD6arr2', [n_g], 3, 1, 0, '', [0 110 0 110; 110 0 110 0; 0 110 0 110; 110 0 110 0], [1 2 2; 2 0 2; 2 2 21], [0 10 0; 2 2 30; 2 2 0], [2 2 2; 2 2 2; 2 2 2] );
>> imOut = L_system_tiling( 'BD6barr1', [n_g], 3, 1, 0, '', [1 1 1 1; 1 1 1 1; 1 1 1 1; 1 1 1 1], [1 2 2; 2 0 2; 2 2 21], [0 10 0; 2 2 30; 2 2 0], [2 2 2; 2 2 2; 2 2 2] );
>> imOut = L_system_tiling( 'BD6barr2', [n_g], 3, 1, 0, '', [1 121 1 121; 121 1 121 1; 1 121 1 121; 121 1 121 1], [1 2 2; 2 0 2; 2 2 21], [0 10 0; 2 2 30; 2 2 0], [2 2 2; 2 2 2; 2 2 2] );
>> imOut = L_system_tiling( 'BD6barr3', [n_g], 3, 1, 0, '', [1 111 1 111; 111 1 111 1; 1 111 1 111; 111 1 111 1], [1 2 2; 2 0 2; 2 2 21], [0 10 0; 2 2 30; 2 2 0], [2 2 2; 2 2 2; 2 2 2] );
where [n_g] is the number of generation

Boundary 7, 7b

[To Do: Show combined symmetries.]

Boundary 7 system pattern

Boundary 7 average pattern, link to 7th generation

average across generations

self-symmetries

Boundary 7 pattern Fourier transform, link to 7th generation

pattern DFT amplitudes, colored by phase
hue-phase scale bar

average pattern DFT

Boundary 7b system pattern, link to 7th generation

Boundary 7b system pattern
[To Do: The variations with horizontal mirror symmetry.]

Boundary 7b average pattern, link to 7th generation

average across generations

self-symmetries
[To Do: Show combined symmetries.]

Boundary 7b pattern Fourier transform, link to 7th generation

pattern DFT amplitudes, colored by phase
hue-phase scale bar

average pattern DFT

MATLAB command:
>> imOut = L_system_tiling( 'BD7', [n_g], 3, 1, 0, '', 0, [1 2 2; 11 0 31; 2 2 21], [0 2 2; 2 1 2; 10 2 2], [2 2 2; 2 2 2; 2 2 2] );
>> imOut = L_system_tiling( 'BD7b', [n_g], 3, 1, 0, '', 1, [1 2 2; 11 0 31; 2 2 21], [0 2 2; 2 1 2; 10 2 2], [2 2 2; 2 2 2; 2 2 2] );
where [n_g] is the number of generation

To Do:
L_system_tiling( 'BD7c', 5, 3, 1, 0, '', 0, [1 2 2; 11 110 31; 2 2 21], [0 2 2; 2 1 2; 10 2 2], [2 2 2; 2 2 2; 2 2 2] );
L_system_tiling( 'BD8', 5, 3, 1, 0, '', 0, [1 31 2; 11 10 31; 2 11 21], [0 2 2; 2 1 2; 10 2 2], [2 2 2; 2 2 2; 2 2 2] );

(touching boundary)
L_system_tiling( 'TB1', 6, 3, 1, 0, '', 0, [1 31 2; 11 10 31; 2 11 21], [0 31 2; 2 2 1; 10 11 2], [2 2 2; 2 2 2; 2 2 2] );
L_system_tiling( 'TB1', 6, 3, 1, 0, ''dz_zig_bl-18x18_motif.png'', 0, [1 31 2; 11 10 31; 2 11 21], [0 31 2; 2 2 1; 10 11 2], [2 2 2; 2 2 2; 2 2 2] );
L_system_tiling( 'TBps1', 4, 3, 1, 0, 'dz_zig_bl-18x18_motif_d.png', 0, [1 31 2; 11 10 31; 2 11 21], [0 31 2; 2 2 1; 10 11 2], [2 2 2; 2 2 2; 2 2 2] );
L_system_tiling( 'TB2', 6, 3, 1, 0, '', 0, [1 10 1; 11 10 31; 21 30 21], [0 31 2; 2 2 1; 10 11 2], [2 2 2; 2 2 2; 2 2 2] );
L_system_tiling( 'TB3', 7, 3, 1, 0, '', 0, [1 10 1; 11 10 31; 21 30 21], [0 31 2; 2 2 1; 10 11 2], [2 2 2; 2 2 2; 2 2 2] );
L_system_tiling( 'TB4', 4, 3, 1, 0, 'diag_key_bl-9x9_motif.png', 0, [1 2 2; 2 0 2; 2 2 21], [0 10 0; 10 2 30; 0 10 0], [2 2 2; 2 2 2; 2 2 2] );

(space filling)
L_system_tiling( 'SF1', 2, 2, 1, 0, 'SF1_2s_motif.png', 0, [1 10 1; 11 10 31; 21 30 21], [1 30 1; 0 1 0; 31 10 11] );

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