2009 How to make fractal tilings and fractal reptiles

Research Overview

This paper develops systematic methodologies for constructing two important classes of fractal objects: fractal tilings (self-similar patterns that tile the plane) and fractal reptiles (self-replicating geometric shapes), advancing both theoretical understanding and computational techniques in fractal geometry.

Key Contributions

Construction Algorithms:

• Develops step-by-step algorithmic procedures for generating fractal tilings with specified self-similarity properties
• Presents novel methods for creating fractal reptiles that maintain geometric coherence while exhibiting self-replication at multiple scales
• Provides computational complexity analysis for the construction processes

Mathematical Framework:

• Establishes rigorous mathematical foundations for fractal tiling theory, including scaling relationships and symmetry properties
• Derives necessary and sufficient conditions for fractal reptile existence and uniqueness
• Proves convergence properties of the iterative construction processes

Geometric Analysis:

• Investigates the relationship between initial generator shapes and resulting fractal structures
• Analyzes how scaling factors and rotation angles affect the final fractal patterns
• Studies the topological properties of the constructed fractals including connectivity and boundary behavior

Computational Implementation:

• Provides practical algorithms that can be implemented in computer graphics software
• Discusses numerical precision requirements and computational efficiency optimizations
• Demonstrates the construction process through detailed examples and visual illustrations

Impact and Significance

Published in the journal Fractals, this research has contributed fundamental techniques to the field of fractal geometry with applications extending beyond pure mathematics into computer graphics, architectural design, and pattern analysis. The systematic construction methods have enabled researchers and practitioners to create complex fractal structures with precise control over their geometric properties, opening new possibilities for fractal-based design and analysis.

Recommended citation: Peng-Jen Lai. (2009). "How to make fractal tilings and fractal reptiles." Fractals. Volume 17, Issue 4, Pages 493-504.
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