Enzymind Labs

I recently wrote a 3-D L-system program, a simple extension of the 2-D L-system program.

[ To Do: More documentation and material planned.]

Thue-Morse cube, 105_150

For comparison, these are the corresponding two symbol Thue-Morse systems in one and two dimensions:

Note that all rules alternate in all directions, and each rule is the comlement of the other.

Thue-Morse cube, surface rendered.

Above: Cross sections of 3-D Thue-Morse patterns, with the slice plane centered on a long diagonal of the pattern. The long range self-similar structure of the Thue-Morse pattern is more easily seen in this slice than in the 2-D square system's pattern. Notice the slightly lighter/darker trapezoids that repeat at different scales; it is a discrete similarity tiling.

Ted Bell showed me another to make these patterns with a triangular replacement system: it can be considered as the Thue-Morse sequence on a triangular tesselation. [To Do: Diagram the rules for a triangular construction.]

Matlab command:
>> volOut = L_System_3D_tiling( '', [n_g], 2, 1, 0, '', 0, [0 1; 1 0], [1 0; 0 1], [1 0; 0 1], [0 1; 1 0] );
where [n_g] is the number of generations.
Oversampled and sliced with Space Software.

Serpinski corner, 0_127

Large version

Matlab command:
>> volOut = L_System_3D_tiling( '', [n_g], 2, 1, 0, '', 1, [0 0; 0 0], [0 0; 0 0], [0 1; 1 1], [1 1; 1 1] );
where [n_g] is the number of generations.
Oversampled and rendered with Space Software.

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