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. I have combined these mathematical and artistic interests to create cross stitch and back stitch 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.
The traditional Pythagorean Tree fractal is constructed by starting with a square and constructing two small squares that enclose a right triangle with the original square along the hypotenuse. This construction is then applied recursively to each of the two smaller squares. This back stitch design shows the first ten iterations of this construction. The piece was stitched so that the new squares added at each iteration appear in a different shade of green. The last set of squares (those smallest in size) uses a dark green color to show the development of the Levy dragon fractal in the limit of the iterations.
A dragon curve is a type of self-similar fractal obtained by recursively replacing line segments with a motif of two or more line segments. This back stitch piece uses an original symmetric motif consisting of eight segments, the middle four of which form a square, to approximate a dragon curve fractal obtained after four iterations. The colors are used to illustrate the motif. The segments making up the square are in gold and the two segments before and two segments after the square are dark red. Because of the way the iteration is done, the entire piece from the beginning point to the end point can be traced by a continuous path (which overlaps itself in places). That was how the stitching was done, alternating the red and gold threads.