# Doug Dunham

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.

This is a hyperbolic pattern of butterflies, seven of which meet at left front

wing tips and three of which meet at their right rear wings. The pattern is inspired by M.C. Escher's Euclidean image Regular Division Drawing Number 70. Disregarding color, the symmetry group of this pattern is generated by 7-fold and 3-fold rotations about the respective meeting points of the wings, and is 732 in orbifold notation (or [3,7]+ in Coxeter notation). This pattern exhibits perfect color symmetry and its color group is the simple group of order 168 (represented as a subgroup of the alternating group A8).

This is another hyperbolic pattern of butterflies, six of which meet at left front wing tips and four of which meet at their right rear wings. The pattern is inspired by M.C. Escher's Euclidean image Regular Division Drawing Number 70, and is colored similarly. Disregarding color, the symmetry group of this pattern is generated by 6-fold and 4-fold rotations about the respective meeting points of the wings, and is 642 in orbifold notation (or [4,6]+ in Coxeter notation). This pattern exhibits perfect color symmetry and its color group is S3, the symmetric group on three objects.