"Generating fractals can be an artistic endeavor, a mathematical pursuit, or just a soothing diversion."
Kerry Mitchell, from The Fractal Art Manifesto
Click on images to enlarge. All images are copyright © Larry Riddle. They are low resolution jpeg files and may not be used without permission.
This series of four counted cross-stitch pieces was inspired by a chapter from Chaos and Fractals: New Frontiers of Science. 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 repeating the construction on each of the three remaining subsquares. That is our "theme". The "variations" arise by exploiting symmetries of the square. The chapter in Chaos and Fractals describes 232 different images that can be made this way, of which I did 16 in counted cross stitch. The first images I chose to do are the only eight symmetric fractals (in this case a reflective symmetry across a 45° diagonal). The other eight were chosen either because they represented related variations or simply because they looked interesting. Each cross stitch piece is 13.5" x 13.5" and consists of 7 iterations of the fractal construction on 25 count per inch fabric. The first piece was displayed in the 2011 Exhibition of Mathematical Art at the Joint Mathematics Meeting and featured in the 2012 Calendar of Mathematical Imagery from the American Mathematical Society. For more information about the mathematical construction of these fractals, see my section on Sierpinski Relatives that is part of my website on Classic Iterated Function Systems.
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. The back stitch design on the left shows the first 10 iterations of this construction (14" x 11" on 14 point canvas). The piece was stitched so that the new squares added at each iteration were done in different shades of green. The last set of squares (smallest in size) were done in dark green to show the development of the Lévy dragon (see next item) in the limit of the iterations.
When the Pythagorean tree is viewed as an iterated function system, one can start the construction with any image. For the digital print on the right I wanted to start with an actual picture of Pythagoras. I tweaked the functions in the iterated function system to use a rotation by 45° and a second rotation by 45° with a reflection. This gave a reflective symmetry for the trunk with two scaled images of Pythagoras looking at each other. I created the “leaves” with 500,000 points plotted using the random chaos game algorithm and colored based on the Barnsley "color stealing" algorithm. To give the leaves more realistic shading, the colors were stolen from a digital photograph of a field of green and yellow grass. This image was part of the 2012 Exhibition of Mathematical Art at the Joint Mathematics Meeting and is included in the 2014 Calendar of Mathematical Imagery from the American Mathematical Society. See the section on the Pythagorean Tree for more information.
The fractal now known as the Levy Dragon was first studied by Paul Levy in 1938, well before computers were able to draw the image or the name "fractal" had even been invented! The first image is that of the dragon done in counted cross stitch on 16 point canvas (10" x 8"). The Levy Tapestry consists of four copies of the Levy Dragon built from the edges of a square. There are two choices as one can orient the initial triangles used in the dragon construction to point inside the square or to point outside the square. The back stitch design in the middle (12.5" x 12.5" on 10 point canvas) shows the twelfth iteration for the inside construction while the one on the right (12.5" x 12.5" on 18 point canvas) shows the twelfth iteration for the outside construction. The outside tapestry was done in two shades of blue to better show each of the four copies of the Levy dragon. See Levy Dragon for more information.
A symmetric binary tree is constructed by starting with a trunk that branches into two segments at a given angle, one to the left and one to the right, with each branch scaled by the same amount. Each branch is then split again according to the same angle and scaling ratio. This construction is iterated as many times as desired. This back stitch design (12.5" x 12.5" on 18 count canvas) consists of four copies of 8 iterations of a symmetric binary tree with angle 45 ° and scaling ratio 1/sqrt(2), with the copies placed at 90° around a common center point. The branch tips of this binary tree (the points at the end of an infinite iteration of branches) is the Levy Dragon shown above.
The Heighway Dragon involves only a slight modification to the formulas that generate the Levy Dragon, yet the image is completely different. It was first investigated in 1966 by three physicists but was introduced to mathematicians by Martin Gardner in his Mathematical Games column of Scientific American. One of the amazing properties of the Heighway Dragon is that despite its boundary being extremely non-smooth, four copies of the dragon fit exactly together around a central point. I tried to illustrate this mathematical idea in my back stitch design on the left (10" by 10" on 18 count canvas) that shows four copies of 12 iterations of the dragon in red, green, gold, and blue, each rotated by 90° as you go clockwise around the image. 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 design in the middle (14" by 11" on 18 count canvas, 14 iterations) illustrates that the Heighway Dragon is composed on an infinite union of geometrically similar components (shown in different shades of blue), any two of which intersect at no more than one cut point. In theory one could start stitching this design at the "tail" on the left and alternate horizontal and vertical stitches until reaching the "head" at the right. Two copies of a Heighway dragon placed back to back (so one is rotated 180°) will form a twindragon. The design on the right (10" by 10" on 28 count canvas) shows how the twindragon can form a periodic tiling of the plane with translational symmetry. See Heighway Dragon for more information.
A space filling curve is a continuous curve that passes through every point in a square. It is usually constructed mathematically as the limit of a sequence of piecewise linear curves. This back stitch piece (12.5" x 12.5" on 14 point canvas) shows the sixth iteration in the construction of a space filling curve described in the topology textbook by James Munkres. It can also be described as the sixth iteration of an iterated function system using four functions (reflected by the four colors in the design). The curve starts in the lower left corner and traces a continuous path through the red section into the blue section, then the orange section, and finally the green section, ending in the lower right corner. The curve intersects itself at many points but never crosses itself. The piece was stitched to follow this continuous path. Each of the four sections uses two alternating colors to illustrate the basic design of the iterated function system based on a triangular motif of 8 line segments turning at 45° angles. The motif is repeated over and over throughout the design with varying horizontal and vertical alignments. The "gaps" in the design would be filled in as more iterations are taken until in the limit all gaps would disappear and the square would be completely filled with four solid sections.
This algorithm for constructing this image starts with a horizontal line. That line is replaced by a motif consisting of 8 line segments as shown in this picture
where the original line is in black and the 8 segments of the motif are in red. Each of the individual segments in the motif is now replaced by a scaled version of the motif to produce 64 new segments. This process is continued forever to produce a fractal, with the motif being scaled increasingly smaller at each step. This back stitch piece (14" x 11" on 22 point canvas) is what is obtained after 4 iterations. The colors are used to show the motif. The segments making up the square are in gold and the two segments before and two segments after the square are a dark red. Because of the way the iteration is done, the entire piece from the left beginning point to the right ending point can be traced by a continuous path (which overlaps itself in places). This is how the stitching was done, alternating the red and gold threads.
The Koch Snowflake, based on the Koch curve that appeared in 1904, is an example of a space filling curve. This means that the inside of the snowflake can be completely filled without any gaps by a mathematical curve that never crosses itself. It takes an infinite number of iterations to do this, however, so no computer image can ever show the actual curve! I tried to illustrate this mathematical idea with the digital print on the left that shows the fourth iteration of the construction using an iterated function system with 13 functions. The curve starts in the lower left corner and ends in the lower right corner. If you look closely you can trace the continuous path of this curve from beginning to end. The print shown on the right is an image of the Koch snowflake that was drawn by plotting millions of points using the random chaos game algorithm and Michael Barnsley’s color stealing algorithm. The colors for the snowflake were stolen from a photograph of a fuchsia plant taken in Highlands, NC. The snowflake has been rotated to better fit the flowers. See Koch Snowflake for more information.
The Sierpinski carpet is constructed by first starting with a solid square, subdividing the square into nine congruent subsquares, and then removing the middle subsquare. The same procedure is then applied recursively to the remaining eight subsquares. The recursive construction is continued ad infinitum. The limiting set is the Sierpinski carpet. Rather than always removing the middle square, however, I wanted to replace that square with an image that would be scaled smaller and smaller at each iteration, in essence creating the mathematical complement of the Sierpinski carpet. After some experimentation, I found that round logos made perfect images for such a design. The Agnes Scott logos were a natural fit. Each image was made using five iterations, thus resulting in 1+8+82+83+84 = 4681 copies of the logo in each picture.
Symmetric fractals are formed using a single affine transformation or a given iterated function system, then modified using a cyclic or dihedral symmetry group. The image on the left is constructed by applying the elements of the D4 dihedral symmetry group to the iterated function system for the Koch curve. The second image was obtained by starting with a single affine transformation consisting of a scaling by 1/2, a rotation by 180°, and a translation by (0.5, 0.866), then given Z5 cyclic symmetry (rotations by 60°). The third image started with an iterated function system for a symmetric binary tree of angle 45° to which was then applied the elements of the Z4 cyclic group (rotations by 90°) to create a new iterated function system (the image also exhibits D4 symmetry). The last image comes from composing the elements of the Z2 cyclic group with the Golden Dragon fractal. The four images were colored using pixel counting in which the colors represent how many times a particular pixel in the image was plotted. See Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature by Michael Field and Martin Golubitsky, Oxford University Press (Edition 1, 1992/95) or SIAM (Edition 2, 2009) for more information about symmetric fractals constructed from a single affine transformation.
Mathematics Department | Agnes Scott College