Colorings of Plane Patterns Defined by Sequences and Arrays, Inspired by Weaving Designs

Shelley L. Rasmussen

This article describes schemes for coloring an $m \times n$ grid in two or more colors. Such a grid is based on the ``product'' of two q-ary sequences, one of length m and the other of length n, each built from an alphabet A containing q distinct symbols and a $q \times q$ product matrix that defines the product of two elements a_i and a_j of A. With this product matrix and a color set containing up to q^2 colors, we define the $m \times n$ product of two such q-ary sequences, along with its associated $m \times n$ colored grid. Considered in detail is the special case when the product matrix is a one-step left (or right) circulant latin square and the number of colors is from 2 to q, as results when the operation is addition (or subtraction) with modular arithmetic.