Two-color Fractal Tilings
Inhaltsverzeichnis
Reference
Robert W. Fathauer: Two-color Fractal Tilings. In: Bridges 2012. Pages 199–206
DOI
Abstract
A variety of two-color fractal tilings (f-tilings) are described, in which no two adjacent tiles have the same color. Two-colorable examples from f-tilings that have been described previously are identified, and two techniques are used for converting f-tilings that are not two-colorable into new f-tilings that can be so colored. In the first of these, tiles are combined in order to change the valence of vertices to all be even, ensuring two-colorability. This technique is applicable to a limited number of f-tilings and can result in prototiles with an infinite number of edges and corners. In the second technique, tiles are divided into two or more smaller tiles such that all vertices of the new f-tiling have even valence.
Extended Abstract
Bibtex
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, Fractal Tilings Based on Dissections of Polyominoes, in Bridges London, Mathematics, Music, Art, Architecture, Culture Conference Proceedings, 2006, edited by Reza Sarhangi and John Sharp, pp. 293-300, 2006.
[7] Robert W. Fathauer, http://www.mathartfun.com/shopsite_sc/store/html/Compendium/encyclopedia.html.
[6] Robert W. Fathauer, “Fractal Tilings Based on Dissections,” in Homage to a Pied Piper, edited by Ed Pegg Jr., Alan Schoen, and Tom Rodgers, AK Peters, Wellesley, MA, 2009.
[8] Bruno Ernst, The Magic Mirror of M.C. Escher, Ballantine Books, New York, 1976.
[9] 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.
[10] K.W. Chung and H.M. Ma, Automatic generation of aesthetic patterns on fractal tilings by means of dynamical systems,” Chaos, Solitons, and Fractals, Vol. 24, pp. 1145-1158, 2005.
[11] Branko Grünbaum and G.C. Shephard, Tilings and Patterns, W.H. Freeman, New York, 1987.
[12] König, D., Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.
[13] Robert W. Fathauer, unpublished.
Links
Full Text
http://archive.bridgesmathart.org/2012/bridges2012-199.pdf
