# Doug Dunham

The goal of my art is to create repeating patterns in the hyperbolic plane. These patterns are drawn in the PoincarĂ© 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 fish, lizards, and bats, in the style of M.C. Escher's Regular Division Drawing Number 85, with three fish, three lizards, and four bats meeting head-to-head, respectively. In general if p fish, q lizards, and r bats meet head-to-head, and 1/p + 1/q + 1/r < 1, as is the case here, then the pattern must be hyperbolic.

This is a hyperbolic pattern of fish, lizards, and bats, in the style of M.C. Escher's Regular Division Drawing Number 85, with three fish, three lizards, and four bats meeting head-to-head, respectively. In general if p fish, q lizards, and r bats meet head-to-head, and 1/p + 1/q + 1/r < 1, as is the case here, then the pattern must be hyperbolic.

This is a hyperbolic pattern of fish, lizards, and bats, as in M.C. Escher's Regular Division Drawing Number 85, with four fish, five lizards, and three bats meeting head-to-head, respectively. Note that the numbers of animals meeting head-to-head is different for each animal. In general if p fish, q lizards, and r bats meet head-to-head, and 1/p + 1/q + 1/r < 1, as in this case, then the pattern must be hyperbolic.