Fractal Tilings Based on Dissections of Polyominoes

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Robert W. Fathauer: Fractal Tilings Based on Dissections of Polyominoes. In: Bridges 2006. Pages 293–300



Polyominoes, shapes made up of squares connected edge-to-edge, provide a rich source of prototiles for edge-to- edge fractal tilings. We give examples of fractal tilings with 2-fold and 4-fold rotational symmetry based on prototiles derived by dissecting polyominoes with 2-fold and 4-fold rotational symmetry, respectively. A systematic analysis is made of candidate prototiles based on lower-order polyominoes. In some of these fractal tilings, polyomino-shaped holes occur repeatedly with each new generation. We also give an example of a fractal knot created by marking such tiles with Celtic-knot-like graphics.

Extended Abstract


Used References

[1] Robert W. Fathauer, Fractal tilings based on kite- and dart-shaped prototiles, Computers & Graphics, Vol. 25, pp. 323-331, 2001.

[2] Robert W. Fathauer, Fractal tilings based on v-shaped prototiles, Computers & Graphics, Vol. 26, pp. 635-643, 2002.

[3] Robert W. Fathauer, Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons, in Proceedings of the 2000 Bridges Conference, edited by Reza Sarhangi, pp. 285-292, 2000.

[4] Robert W. Fathauer, Fractal Tilings Based on Dissections of Polyhexes, in Renaissance Banff, Mathematics, Music, Art, Culture Conference Proceedings, 2005, edited by Reza Sarhangi and Robert V. Moody, pp. 427-434, 2005.

[5] Robert W. Fathauer,

[6] Bruno Ernst, The Magic Mirror of M.C. Escher, Ballantine Books, New York, 1976.

[7] Peter Raedschelders, “Tilings and Other Unusual Escher-Related Prints,” in M.C. Escher’s Legacy, edited by Doris Schattschneider and Michele Emmer, Springer-Verlag, Berlin, 2003.

[8] Branko Grünbaum and G.C. Shephard, Tilings and Patterns, W.H. Freeman, New York, 1987.

[9] Solomon W. Golomb, Polyominoes, Princeton University Press, Princeton, New Jersey, 1994.

[10] The only pentomino with 4-fold rotational symmetry yields prototiles with 3 short edges. Adding 4 squares to this pentomino yields prototiles with either 3 or 5 short pseudo-edges. It can easily be seen that adding four squares to any 4-fold polyomino will either add 2 short pseudo-edges, leave the number of short pseudo-edges unchanged, or subtract 2 short pseudo-edges. The number of short pseudo-edges for prototiles is therefore always odd.

[11] H.-O. Peitgen, H. Jürgens, and D. Saupe, Fractals for the Classroom – Part One, Springer-Verlag, New York, 1992.


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