An Algorithm for Random Fractal Filling of Space

• Published in: Computer graphics forum Vol. 32; no. 8; pp. 89 - 97
• Main Authors: Shier, John, Bourke, Paul
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.12.2013
• Subjects: Algorithms; Analysis; and systems; Computer graphics; dimension; fractal; Fractals; G.3 [Mathematics of Computing]: Probability and Statistics-Stochastic processes; geometry; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling–Geometric algorithms, languages, and systems; I.3.m [Computer Graphics]: Miscellaneous; languages; packing; space filling; Studies
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.12163

Computational experiments with a simple algorithm show that it is possible to fill any spatial region with a random fractalization of any shape, with a continuous range of pre‐specified fractal dimensions D. The algorithm is presented here in 1, 2 or 3 physical dimensions. The size power‐law exponent c or the fractal dimension D can be specified ab initio over a substantial range. The method creates an infinite set of shapes whose areas (lengths, volumes) obey a power law and sum to the area (length and volume) to be filled. The algorithm begins by randomly placing the largest shape and continues using random search to place each smaller shape where it does not overlap or touch any previously placed shape. The resulting gasket is a single connected object. Computational experiments with a simple algorithm show that it is possible to fill any spatial region with a random fractalization Q1 of any shape, with a continuous range of pre‐specified fractal dimensions D. The algorithm is presented here in 1, 2 or 3 physical dimensions. The size power‐law exponent c or the fractal dimension D can be specified ab initio over a substantial range. The method creates an infinite set of shapes whose areas (lengths, volumes) obey a power law and sum to the area (length and volume) to be filled.