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Paths in a Rectangular Integer Lattice

Jaime Rangel-Mondragón
Organization: Universidad Autonoma de Querétaro
Department: Facultad de Informatica
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The generation of all paths from a given lattice point to another one is considered. The paths are constrained within the limits of a rectangle and are described by unit steps in orthogonal directions. The paths follow a prescribed number of self-intersections and a given length. Several particular cases extend the basic problem. Paths that do not leave holes, i.e., paths that touch each of the square forming the underlying grid are generated. Some paths can give rise to fractals. A bijection is established between the segments of a path and the squares forming the grid in such a way that by succesively mapping of the original path into itself we obtain self-intersecting space-filling open curves.

The file paths.gz is a compressed file containing the notebook paths.nb.

*Mathematics > Discrete Mathematics > Graph Theory
*Mathematics > Geometry > Plane Geometry
*Wolfram Technology > Programming > 2D Graphics

Fractals, paths, length of a path, number of crossings, fractalized paths, space filling curve
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fractals (6.1 KB) - data text file
paths.gz (481 KB) - gzip compressed file
paths.nb (6.1 MB) - Mathematica notebook