Experimental Mathematics

Outer boundaries of self-similar tiles

Shawn Drenning, Judith Palagallo, Thomas Price, and Robert S. Strichartz

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Abstract

There are many examples of self-similar tiles that are connected, but whose interior is disconnected. For such tiles we show that the boundary of a component of the interior may be decomposed into a finite union of pieces, each similar to a subset of the outer boundary of the tile. This is significant because the outer boundary typically has lower dimension than the full boundary. We describe a method to realize the outer boundary as the invariant set of a graph-directed iterated function system. The method works under a certain "finiteness'' assumption. While it is not clear that this assumption always holds, and it is problematic to give a rigorous proof that it holds even in cases where it is "visually clear'' that it holds, we give some examples where the method yields clear and nontrivial results. Details concerning the algorithms may be found at the website www.math.cornell.edu/~$sld32/Tiles.html.

Article information

Source
Experiment. Math., Volume 14, Issue 2 (2005), 199-209.

Dates
First available in Project Euclid: 30 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.em/1128100132

Mathematical Reviews number (MathSciNet)
MR2169523

Zentralblatt MATH identifier
1082.52501

Subjects
Primary: 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20] 28A80: Fractals [See also 37Fxx]

Keywords
Self-similar tile outer boundary

Citation

Drenning, Shawn; Palagallo, Judith; Price, Thomas; Strichartz, Robert S. Outer boundaries of self-similar tiles. Experiment. Math. 14 (2005), no. 2, 199--209. https://projecteuclid.org/euclid.em/1128100132


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