# Doug Dunham

The goal of my art is to create aesthetically pleasing 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, they are colored symmetrically, and they adhere to the map-coloring principle that no adjacent copies of the motif are the same color. These patterns are designed using an interactive drawing program and then rendered by a color printer. The two major challenges in creating these patterns are (1) to design appealing motifs and (2) to write programs that facilitate such design and replicate the complete pattern.

This is a hyperbolic pattern of bordered squares, in the style of Victor Vasarely's square grid patterns. It is based on the regular tessellation {4,5} of the hyperbolic plane, with five squares meeting at each vertex. Vasarely's square grid patterns were based on the familiar {4,4} tiling of the Euclidean plane.

This is a hyperbolic pattern of randomly colored bordered squares, in the style of some of Victor Vasarely's randomly colored square patterns. It is based on the regular tessellation {4,6} of the hyperbolic plane, with six squares meeting at each vertex. Some of Vasarely's related patterns were based on the distorted {4,4} Euclidean grids.

This is a hyperbolic pattern of randomly colored circles within randomly colored squares, in the style of Victor Vasarely's similar patterns in the Euclidean plane. It is based on the regular tessellation {4,6} of the hyperbolic plane, with six squares meeting at each vertex. Vasarely's related patterns were based on the familiar {4,4} tiling of the Euclidean plane.