Abstract
It is well known that the discrete Sierpinski triangle can be defined as the nonzero residues modulo 2 of Pascal’s triangle, and that from this definition one can construct a tileset with which the discrete Sierpinski triangle self-assembles in Winfree’s tile assembly model. In this paper we introduce an infinite class of discrete self-similar fractals (a class that includes both the Sierpinski triangle and the Sierpinski carpet) that are defined by the residues modulo a prime p of the entries in a two-dimensional matrix obtained from a simple recursive equation. We prove that every fractal in this class self-assembles and that there is a uniform procedure that generates the corresponding tilesets. As a special case we show that the discrete Sierpinski carpet self-assembles using a set of 30 tiles.
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Kautz, S.M., Lathrop, J.I. (2009). Self-assembly of the Discrete Sierpinski Carpet and Related Fractals. In: Deaton, R., Suyama, A. (eds) DNA Computing and Molecular Programming. DNA 2009. Lecture Notes in Computer Science, vol 5877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10604-0_8
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DOI: https://doi.org/10.1007/978-3-642-10604-0_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10603-3
Online ISBN: 978-3-642-10604-0
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