Parametric 2-dimensional L systems and recursive fractal images: Mandelbrot set, Julia sets and biomorphs

This is the author’s version of a work that was accepted for publication in Computers & Graphics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computers & Graphics 26, 1, (2002) DOI: 10.1016/S0097-8493(01)00162-5L Systems have proved their expressive power. They have been used to represent the class of the initiator/iterator fractal curves (such as Sierpinski's gasket and von Koch's snowflake curve). Parametric L Systems, introduced by Prusinkiewicz and Lindenmayer, link real valued parameters to the symbols. In this paper, parametric 0L systems are extended to n dimensions and used to represent a different class of classic fractals that includes objects such the Mandelbrot and Julia sets, or Pickover’s biomorphs

Grammatical evolution to design fractal curves with a given dimension

Original paper in http://ieeexplore.ieee.org/Lindenmayer grammars have frequently been applied to represent fractal curves. In this work, the ideas behind grammar evolution are used to automatically generate and evolve Lindenmayer grammars which represent fractal curves with a fractal dimension that approximates a predefined required value. For many dimensions, this is a nontrivial task to be performed manually. The procedure we propose closely parallels biological evolution because it acts through three different levels: a genotype (a vector of integers), a protein-like intermediate level (the Lindenmayer grammar), and a phenotype (the fractal curve). Variation acts at the genotype level, while selection is performed at the phenotype level (by comparing the dimensions of the fractal curves to the desired value).This paper has been sponsored by the Spanish Ministry of Science and Technology (MCYT), project numbers TIC2002-01948 and TIC2001-0685-C02-01

Grammatical Evolution with Restarts for Fast Fractal Generation

In a previous work, the authors proposed a Grammatical Evolution algorithm to automatically generate Lindenmayer Systems which represent fractal curves with a pre-determined fractal dimension. This paper gives strong statistical evidence that the probability distributions of the execution time of that algorithm exhibits a heavy tail with an hyperbolic probability decay for long executions, which explains the erratic performance of different executions of the algorithm. Three different restart strategies have been incorporated in the algorithm to mitigate the problems associated to heavy tail distributions: the first assumes full knowledge of the execution time probability distribution, the second and third assume no knowledge. These strategies exploit the fact that the probability of finding a solution in short executions is non-negligible and yield a severe reduction, both in the expected execution time (up to one order of magnitude) and in its variance, which is reduced from an infinite to a finite value.Comment: 26 pages, 13 figures, Extended version of the paper presented at ANNIE'0