Swans Square Fractal by Tis Veugen

About "Swans Square Fractal"

by Tis Veugen

About the artwork

The inspiration for this image comes from Escher's famous Square Limit, where part of the fish tiles are mirrored. In our image, the triangles of the tessellation are deformed into swans, in such a way that all swans have the same shape; they are only rotated and/or scaled, but not mirrored. Another major difference with Escher's artwork is the fractal structure towards the center.
There are many more variations on Escher's Square Limit, for example: Fathauer [1] shows that Escher's deformation scheme is not consistent at the diagonals. Escher uses sibling tiles on the diagonals, while Fathauer uses child tiles, which results in an octagonal boundary of the fractal construction. In addition, a hexagon or other polygons can be used instead of a square. My tool Tissellator contains over 40 variations of the Square Limit artwork.
The most surprising thing is that Escher did not combine his Square Limit with a fractal center. He investigated fractal behavior from a single central point, as shown for example in [2, p. 91]. He also made the fractal "Smaller and Smaller". His lecture illustration for showing fractal geometry, see [2, p. 252], actually contains exactly the same scheme of decreasing triangles as the scheme around the center in our figure! So Escher could have made a dual fractal image, but we will never know why he didn't.

[1] Robert Fathauer. Tessellations – Mathematics, Art, and Recreation. CRC Press, 2021
[2] Doris Schattschneider. Visions of Symmetry - Notebooks, Periodic Drawings, and Related Work of M. C. Escher. W. H. Freeman and Company, 1990