"Twice Iterated Knot No. 1"

Digital print, 9" x 15", 2008.

The starting point for this knot is a nine-crossing knot that has been carefully arranged to allow seamless iteration. Four regions of this starting knot are replaced with a scaled-down copy of the full starting knot, incorporated in such a way that the iterated knot is still unicursal. These same four regions are then replaced with a scaled-down copy of the iterated knot, resulting in a complex knot possessing self similarity.

"127-crossing Self-similar Knot"

Digital print, 12" x 10", 2009.




The starting point for this knot is a three-crossing knot that has been carefully arranged to allow seamless iteration. The strands, smoothly varying for the most part, also contain sharp corners, a design esthetic borrowed from Celtic knots. Two of the three crossings in the starting knot are replaced with a scaled-down copy of the group of three crossings, incorporated in such a way that the iterated knot is still unicursal. The scaling factor is the inverse of the square root of 3, and there is a 90 degree rotation with each iteration. Five iterations are performed in similar fashion, resulting in a complex knot possessing self similarity.

"Fractal Ripples No. 1"

Digital print, 15" x 10", 2009.

The starting point for this graphically-constructed fractal is a group of concentric rings (a "ripple pattern"), with each larger ring being increasingly transparent. This ripple pattern is designed to mimic the waves that would form around a stone dropped in water. Three half-scale copies of the ripple pattern are positioned around the starting ripple pattern. The resulting group of four ripple patterns is then scaled by half, and three copies are positioned around the starting ripple pattern. This process is repeated until the new ripple patterns are so small as to be invisible to the eye at the scale of the print. Dark areas are created by the overlap of the waves, similar to constructive interference patterns. The final fractal reveals a skewed Sierpinski triangle. The print paradoxically presents a collection of straight lines to the eye, even though it is entirely comprised of circles.

"Fractal Ripples No. 3"

Digital print, 15" x 10", 2009.




The starting point for this graphically-constructed fractal is a group of concentric rings (a "ripple pattern"), with each larger ring being increasingly transparent. This ripple pattern is designed to mimic the waves that would form around a stone dropped in water. Four one-third-scale copies of the ripple pattern are positioned around the starting ripple pattern. The positions were intentionally chosen as the midpoints of the line segments in a Koch Curve construction. The resulting group of five ripple patterns is then scaled by one third, and four copies are positioned around the starting ripple pattern. This process is repeated until the new ripple patterns are so small as to be invisible to the eye at the scale of the print. Dark areas are created by the overlap of the waves, similar to constructive interference patterns. The final fractal reveals a Koch Curve, even though the ripple patterns are centered in what would be empty spaces in the curve.

Robert Fathauer, Small business owner, puzzle designer, and artist
Tessellations, Phoenix, Arizona

Robert Fathauer makes limited-edition prints inspired by tiling, fractals, and knots. He employs mathematics in his art to express his fascination with certain aspects of our world, such as symmetry, complexity, chaos, and infinity. His artworks are created on a Macintosh computer, primarily using the commercial programs FreeHand and Photoshop.