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Complex 2D Walks based on Context Independent L-Systems

Erik Jensen
Organization: University of California, Santa Cruz
Veikko Keränen
Organization: Rovaniemi Polytechnic
Department: School of Technology

2003 International Mathematica Symposium
Conference location

Imperial College, London

We report here a method of generating 2D walks based on context independent L-systems for two or four letter alphabets. Binary strings (words) are mapped to strings over a four letter alphabet by two separate techniques. 2D walks are generated from the resulting quaternary strings by assigning vectors to the characters as follows: a->(–1,0), b->(1,0), c->(0,–1), d->(0,1). In the case of long strings obtained by iterating endomorphisms, one may also use vectors for image words of letters. A walk is generated by adding up the vectors of sequential characters in a word through a regular grid and coloring the cells at which the vectors terminate. Plots of distance from the origin versus string position provide a rapid means for comparison of the strings resulting from different L-systems. Here one may also use matrix multiplication in a powerful way. We examine the 2D walks from a variety of L-systems and provide examples of considerably complex paths from simple L-systems. Square-free strings over four letters can result in both simple and highly complex 2D walks.

*Mathematics > Calculus and Analysis > Dynamical Systems

2D walks, context independent L-systems, binary strings, endomorphisms, matrix multiplication
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