I have been working with needle crafts since graduate school. I have also been interested in fractals and fractal geometry for more than 20 years. Only recently, however, have I combined these mathematical and artistic interests to create cross stitch and backstitch pieces to illustrate the beauty and mathematics of fractals associated with iterated function systems. As a mathematician I like to seek fractal images that have symmetry or illustrate some interesting mathematical idea. I must be sure that the fractal can be represented accurately on a canvas that permits only vertical, horizontal, and diagonal stitches of a fixed size. Fractals that are built from squares or from lines rotated by multiples of 45° work particularly well.
This back stitch piece shows the sixth iteration of an iterated function system for a space filling curve using four functions (indicated by the scaling and rotations of the four sections in the design). The curve starts in the lower left corner and traces a continuous path through the red section into the blue, orange, and green sections, ending in the lower right corner. The curve intersects itself at many points but never crosses itself. The piece was stitched along this continuous path. Each of the four sections uses two alternating shades of color to illustrate the basic design of the IFS based on a motif of 8 line segments at right angles. The motif is repeated throughout the design in horizontal and vertical orientations.