## 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.