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 Heighway Dragon fractal was introduced to mathematicians by Martin Gardner in his Mathematical Games column of Scientific American. One of the amazing properties of the dragon is that despite its boundary being extremely non-smooth, four copies of the dragon fit exactly together around a central point. This back stitch design illustrates that mathematical idea by showing 4 copies of 12 iterations in the construction of the dragon via line segments, each rotated in succession by 90° around the center. The entire image can then be repeated to tile the plane. What might not be obvious from the design is that each dragon can be traced starting at the center point as one continuous alternating sequence of vertical and horizontal stitches.
The twindragon is formed by placing two Heighway dragons back-to-back with the tail of one corresponding to the head of the other (so one is rotated by 180°). This design shows how the twindragon forms a periodic tiling of the plane with translational symmetry. Each of the nine copies consists of 12 iterations in the construction of the twindragon via line segments. The use of a cross stitch on a 28 count canvas essentially fills the interior of the twindragon, illustrating how both it and the Heighway dragon from which it comes are space-filling curves.