Bounce Mapping: The Effects of the Range Coordinates Sharing a Greatest Common Divisor on Path Behavior in an Integer Lattice

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Computer Sciences | Physical Sciences and Mathematics


Michael Heroux, Computer Science


In an integer lattice, patterns can be seen in the number of paths and the number of directions they take in relation to the width and height of the lattice. This phenomenon can also be seen in 3-dimensional lattices – with the numbers and directions of paths relating to the width, height, and depth of the lattice – and extended beyond that into any dimension, where the maximum value for each coordinate is given by a set of range coordinates. Given the range coordinates for a lattice, mathematical formulae can be written to describe these patterns within the lattice, predicting the number of paths and the number of directions each path takes. These path formulae will not directly relate to the set of range coordinates, but to the set of greatest common divisors of different subsets of the set of range coordinates.