Generación de curvas fractales a partir de
homomorfismos entre lenguajes
[con Mathematicar^®]

JOSÉ L. RAMÍREZ^{a *}, GUSTAVO N. RUBIANO^b

^a Universidad Sergio Arboleda, Escuela de Matemáticas, Bogotá, Colombia.

^b Universidad Nacional de Colombia, Depto. de Matemáticas, Bogotá, Colombia.

Resumen. En este artículo se hace una implementación con el software Mathematica 8.0 de algunas propiedades combinatorias de la cadena o palabra de Fibonacci, la cual se puede generar a partir de la iteración de un homomorfismo entre lenguajes. Asimismo se recopilan algunas propiedades gráficas de la curva fractal asociada a esta cadena de símbolos, la cual se puede generar a partir de unas reglas de dibujo análogas a las utilizadas en los L-Sistemas. Todos los códigos utilizados en el artículo se presentan en detalle y luego se aplican para generar nuevas curvas fractales. Finalizamos con una forma alternativa de generar la curva de Fibonacci y otras curvas a partir de cadenas características.
Palabras Claves: Combinatoria sobre cadenas, cadena infinita de Fibonacci, homomorfismos entre lenguajes, curvas fractales, L-sistemas, Mathematicar^®.
MSC2010: 11B39, 28A80, 68R15, 97N80.

Generating fractals curves from homomorphisms
between languages
[with Mathematicar^®]

Abstract. In this paper we implement with the software Mathematica 8.0 some combinatorial properties of Fibonacci Word, which can be generated from the iteration of a homomorphism between languages.We collect also some graphic properties of the fractal curve associated to this word, which can be generated from drawing rules similar to those used in the L-Systems. All codes used in this paper are presented in detail and then they are applied to generate new fractal curves. We conclude with an alternative way to generate the Fibonacci curve and other curves from characteristics words.

Keywords: Combinatorics on words, infinite Fibonacci word, homomorphism between languages, fractal curves, L-systems, Mathematicar^®.

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^*Autor para correspondencia: E-mail: hml@ciencias.unam.mx
Recibido: 01 de agosto de 2012, Aceptado: 14 de septiembre de 2012.