Reverse Fractal Design via Lindenmayer Systems Based on Box-Counting Dimension

We present a systematic framework for the reverse fractal design of Lindenmayer (L-) systems from target box-counting dimensions. By inverting the self-similarity formula D=lnnln(1/r) to r=n−1/D, we analytically determine L-system parameters (copy count n and scaling factor r) that generate fractals with prescribed dimensions. Our key innovation is a topology template library encoding eight spatial arrangements that separate dimensional constraints from geometric structure, enabling systematic design space exploration. For a single target dimension D=1.5, the framework generates 21 topologically distinct L-systems (spanning binary, ternary, quaternary, and pentagonal patterns), all achieving machine-precision accuracy (error <10−15) with sub-millisecond efficiency. Comprehensive validation through 25 unit tests and standard fractals (Koch snowflake and Sierpinski triangle) confirms mathematical correctness. This template-based approach provides unprecedented flexibility: users control complexity through n selection, visual density through r, and esthetics through topology choice. Applications span fractal art generation, natural structure simulation, procedural content generation, and texture synthesis.