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

Ortega, Alfonso; Cruz, Marina de la; Alfonseca, Manuel

doi:10.1016/s0097-8493(01)00162-5

Cited by 4 publications

(4 citation statements)

References 7 publications

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“…Many techniques and modifications were introduced to obtain these fractals. The techniques include: the inverse iteration method [7], the boundary scanning method [29], the level set method [22], the continuous potential method [6] and even the L-systems [21]. One of the widely used technique of obtaining the Julia sets is the escape time algorithm [3], and it is used in this paper.…”

Section: Introductionmentioning

confidence: 99%

Biomorphs via modified iterations

Gdawiec¹,

Kotarski²,

Lisowska³

2016

J. Nonlinear Sci. Appl.

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The aim of this paper is to present some modifications of the biomorphs generation algorithm introduced by Pickover in 1986. A biomorph stands for biological morphologies. It is obtained by a modified Julia set generation algorithm. The biomorph algorithm can be used in the creation of diverse and complicated forms resembling invertebrate organisms. In this paper the modifications of the biomorph algorithm in two directions are proposed. The first one uses different types of iterations (Picard, Mann, Ishikawa). The second one uses a sequence of parameters instead of one fixed parameter used in the original biomorph algorithm. Biomorphs generated by the modified algorithm are essentially different in comparison to those obtained by the standard biomorph algorithm, i.e., the algorithm with Picard iteration and one fixed constant.

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Section: Introductionmentioning

confidence: 99%

Biomorphs via modified iterations

Gdawiec¹,

Kotarski²,

Lisowska³

2016

J. Nonlinear Sci. Appl.

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“…Ochoa [32] evolves D0L systems with a single rule, that generate shapes similar to plants. Other authors [33][34] [35] evolve parametric L systems (an extension of Lindenmayer grammars) [36], which means that they are faced with the important problem that parametric systems are not closed under the action of genetic algorithms. This problem would be solved easily by our approach, because Grammatical Evolution goes through an intermediate grammar (in this case it would describe a valid parametric system), which ensures that the actual L system generated is syntactically correct.…”

Section: Grammatical Evolutionmentioning

confidence: 99%

Grammatical evolution to design fractal curves with a given dimension

2003

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Lindenmayer Grammars have been applied frequently to represent fractal curves. In this work, the ideas behind Grammar Evolution are used to automatically generate and evolve Lindenmayer Grammars that represent fractal curves with a fractal dimension that approximates a pre-defined required value. For many dimensions, this is a non trivial task to be performed manually. The procedure we are proposing here 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).

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“…In a recent paper [9] we have proposed a way in which an extended definicion of Lindenmayer grammars (parametric two-dimensional L systems) is capable of representing fractals of the first type. In this paper we describe how this set of algorithms may be programmed in APL2 to easily generate pictures showing selected sections of the Mandelbrot set, Julia sets, biomorphs and other fractals in the same family.…”

Section: Fractalsmentioning

confidence: 99%

Complex systems in APL

Alfonseca

Ortega

Cruz³

2002

SIGAPL APL Quote Quad

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Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in: Marina de la CruzCIEMAT e-mail: marina@ciemat.es AbstractWe have been working for several years on the representation, study and simulation of complex systems by means of formal methods. APL2 and other programming languages have been used to develop our tools and experiments. This paper summarizes our APL2 works on fractal sets, automatic programming of cellular automata and simulation of multi agent systems .

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

Cited by 4 publications

References 7 publications

Biomorphs via modified iterations

Biomorphs via modified iterations

Grammatical evolution to design fractal curves with a given dimension

Complex systems in APL

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