# Doug Dunham

The goal of my art is to create repeating patterns in the hyperbolic plane. These patterns are drawn in the Poincaré 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 consisting of fish, lizards, and bats in the style of M.C.~Escher's Euclidean Regular Division Drawing Number 85. In Escher's drawing, three animals of each type meet head-to-head. The three animals each represent one of the three elements that they live in: the fish, lizards, and bats live in water, on land, and in the air respectively. In my pattern, four of each of the animals meet head-to-head. In general if p fish, q lizards, and r bats meet head-to-head, and

1/p + 1/q + 1/r < 1, then the pattern must be hyperbolic; if 1/p + 1/q + 1/r = 1 (as in Escher's Drawing 85), the pattern is Euclidean.

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.

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.