"Three Elements 4-5-3"

Color print,11” by 11”, 2007.

This pattern contains lizards, fish, and bats representing the three classical elements, earth, water, and air. The pattern is inspired by M.C. Escher's Notebook Drawing Number 85. In this hyperbolic pattern, four blue lizards meet head-to-head, five red fish meet head-to-head, and three yellow bats meet head-to-head, unlike Escher's pattern in which three of each animal meet head-to-head. The symmetry group of this pattern is generated by reflections across the lines of bilateral symmetry of each of the animals; its symmetry group is the hyperbolic kaleidoscope group *543, in orbifold notation.

Doug Dunham, Professor of Computer Science, Department of Computer Science, University of Minnesota Duluth
Duluth, Minnesota, USA

"The goal of my art is to create repeating patterns in the hyperbolic plane. These patterns are drawn in the Poincare circle model of hyperbolic geometry, which has two useful properties: (1) it shows the entire hyperbolic plane in a finite area, and (2) it is conformal, i.e. angles have their Euclidean measure, so that copies of a motif retain their same approximate shape as they get smaller toward the bounding circle. Most of the patterns I create exhibit characteristics of Escher's patterns: they tile the plane without gaps or overlaps, and if colored, they are colored symmetrically and adhere to the map-coloring principle that adjacent copies of the motif are different colors. My patterns are rendered by a color printer. Two challenges are to design appealing motifs and to write programs that facilitate such design and replicate the complete pattern."