Doug Dunham
Professor of Computer Science, Department of Computer Science, University of
Minnesota Duluth
http://www.d.umn.edu/~ddunham/
"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, they are colored symmetrically, and they adhere to the map-coloring principle that no adjacent copies of the motif are the same color. 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."
“Fish Pattern 3-4 with Triangle ”
2006, Color printer, 11 by 11 inches
This pattern is the "reverse" of Escher's "Circle Limit III" pattern where four fish meet at right fins and three fish meet at left fins. The backbone circular arcs are equidistant curves in hyperbolic geometry - a constant hyperbolic distance from the hyperbolic line (orthogonal circular arc) having the same endpoints on the bounding circle. The special case where the backbone arcs "straighten out" to become chords occurs when three fish meet at the center and at their noses. This pattern exhibits perfect color symmetry with the minimum number of colors, fish along each backbone arc being the same color.
“ Fish Pattern 3-5 with Triangle ”
2006, Color printer, 11 by 11 inches
This pattern is in the "Circle Limit III" family of patterns, with three fish meeting at right fins and five fish meeting at left fins. The backbone circular arcs are equidistant curves in hyperbolic geometry - a constant hyperbolic distance from the hyperbolic line (orthogonal circular arc) having the same endpoints on the bounding circle. The special case where the backbone arcs "straighten out" to become chords occurs when three fish meet at the center and at their noses. This pattern has the alternating group A(5) as its color symmetry group, fish along each backbone arc being the same color.