Limit-Periodic Tilings

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This Demonstration lets you explore the similarities between the Socolar–Taylor tiling [1], the Penrose tiling [2], and the relatively new viererbaum (or quadtree) tiling [3]. Each of these tilings is an areal inflation-factor four substitution tiling and also a limit-periodic tiling. Here we construct each uniquely displaced tiling as a sequence of periodic approximants. Each periodic approximant is determined by a sequence of branch permutations taken from the set . The set elements are also actions of the Klein four-group.


The limit periodic construction has a number of advantages over the more common construction using substitution rules. It lends itself to a graphical proof that the various matching rules enforce aperiodicity. It explicitly shows the limit periodic structure. Finally, it introduces sequences of branch permutations such as , which allow us to call tilings -close whenever branch-permutation sequences agree for the first terms.


Contributed by: Brad Klee (September 2015)
Open content licensed under CC BY-NC-SA




[1] J. E. S. Socolar and J. M. Taylor, "An Aperiodic Hexagonal Tile," Journal of Combinatorial Theory, Series A 118(8), 2011 pp. 2207–2231. doi:10.1016/j.jcta.2011.05.001.

[2] R. Penrose, "Remarks on Tiling: Details of a (1 + ϵ +)-aperiodic set," The Mathematics of Long-Range Aperiodic Order (R. V. Moody, ed.), NATO ASI series 489, Springer: Dordrecht, Netherlands, 1997 pp. 467–497.

[3] B. Klee, unpublished notes.

[4] M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, New York: Cambridge University Press, 2013.

[5] M. Baake, F. Gähler, and U. Grimm, "Hexagonal Inflation Tilings and Planar Monotiles," Symmetry 4(4), 2012 pp. 581–602. doi:10.3390/sym4040581.

[6] A. K. Dewdney, "Speichern von Bildern, Eine Katze im Viererbaum," Der Turing Omnibus, Heidelberg: Springer, 1995 pp. 340–346.

[7] D. Frettlöh, "Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor," Ph.D. thesis, Universität Dortmund, Dortmund, Germany, 2002.

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