# Lisa Shier and Doug Dunham

Several artists have used different fiber arts to portray hyperbolic geometry. Some have knitted hyperbolic surfaces embedded in ordinary 3-space, and Tony Bomford created patterns using the Poincare circle model of hyperbolic geometry. We want to explore the possibility of using embroidery to render hyperbolic patterns in this model. Such patterns present considerable challenges for the embroiderer. The features become smaller as the bounding circle is approached and not all features can be included. The detail level drives a desire for the largest possible scale of the embroidery, which comes with a cost in complexity of implementation. The stitches must also be oriented to maintain the symmetry of the original design.

This pattern of fish was inspired by M.C. Escher's "Circle Limit III" print which was based on the regular hyperbolic tessellation {8,3} of octagons meeting three at a vertex. In contrast, our pattern is based on the {10,3} tessellation, and thus has five fish meeting at right fin-tips instead of four as in Escher's print. Each backbone arc is an equidistant curve - a circular arc that is a constant hyperbolic distance from the hyperbolic line with the same endpoints on the bounding circle. The arcs curve to the left since there are more fish on the right. Our pattern requires six colors instead of the four that Escher used in order to achieve (perfect) color symmetry. In fact, the color symmetry group is the symmetric group A(5).