I have been working with various types of needlecrafts since graduate school. I have also been interested in fractals and fractal geometry for more than 15 years. Only recently, however, have I combined these two interests to create counted cross stitch pieces to illustrate the mathematics of fractals associated with iterated function systems. A computer program I wrote to draw such fractals can generate the cross stitch diagrams that I use to create my designs.
The Sierpinski Triangle is a fractal that can be generated by dividing a square into four equal subsquares, removing the upper right subsquare, and then iterating the construction on each of the three remaining subsquares. That is our “Theme”, shown in the upper left. The “Variations” arise by exploiting symmetries of the square. The three variations in this piece were generated by rotating the upper left and lower right subsquares at each iteration by 90 or 180 degrees, either clockwise or counterclockwise. The self-similarity of the fractals, illustrated by the use of three colors, means that you can read off which rotations were used from the final image. Each design shows the construction through seven iterations, the limit that could be obtained for the size of canvas used.