# 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 M.C. 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.

This work was inspired by M.C. Escher's hyperbolic print `Circle Limit IV'. Escher's print only has the dihedral group D3 as its symmetry group, generated by reflections across the three diameters that are axes of symmetry of the central angels and devils. This is because the interior details of some of the angles and devils have been filled in, whereas others are only partially filled in, and some have no interior details at all. In our pattern, we have filled in the interior details of all the angels and devils, giving the pattern many more symmetries. Thus there are 4-fold rotations about wingtip meeting points and reflections across the center lines of the angels and devils. Thus the symmetry group is 4*3 in orbifold notation.