Emily Gullerud and James Walker
After reviewing Frank Farris's book Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, published in 2015 by Princeton University Press, Dr. James Walker and his research student Emily Gullerud found a mutual interest in creating computer generated symmetric designs. They take stock images as color maps and transform them into pieces of art using algorithms written in Matlab. They also apply these algorithms toward the creation of artistic animations. Through these pieces of art, they strive to showcase the deep connections between art and mathematics.
The pixels for this hyperbolic wallpaper were taken from an image of blue and yellow buoys laying on a dock. We first used an algorithm to build a ternary tree of relatively prime integer pairs and their corresponding Bézout coefficients. These values were then used to construct Möbius transformations. A function built from a large sum of these transformations was composed with a trigonometric function in order to map pixels onto the upper half-plane. The result has translational symmetry at integer values along the real axis, along with cells of three-fold rotational symmetry.