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Self-Assembly of Fractals

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 43 Accesses

Years and Authors of Summarized Original Work

  • 2009; Kautz, Lathrop

  • 2009; Lathrop, Lutz, Summers

  • 2010; Patitz, Summers

  • 2012; Lutz, Shutters

  • 2013; Kautz, Shutters

  • 2014; Barth, Furcy, Summers, Totzke

Problem Definition

This problem is concerned with the self-assembly fractal patterns and structures. More specifically, it deals with discrete self-similar fractals and different notions of them self-assembling from tiles in the abstract Tile Assembly Model (aTAM) and derivative models. The self-assembly of fractals and fractal-like structures is particularly interesting due to their pervasiveness in nature, as well their complex aperiodic structures which result in them occupying less dimensional space than the space they are embedded within.

Using the terminology from [1], we define \(\mathbb{N}_{g}\) as the subset {0, 1, …, g − 1} of \(\mathbb{N}\), and if \(A,B \subseteq \mathbb{N}^{2}\) and \(k \in \mathbb{N}\), then \(A + \mathit{kB} =\{\mathbf{ m} + k\mathbf{n}\vert \mathbf{m} \in A\)...

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Recommended Reading

  1. Barth K, Furcy D, Summers SM, Totzke P (2014) Scaled tree fractals do not strictly self-assemble. In: Unconventional computation & natural computation (UCNC) 2014, University of Western Ontario, London, 14–18 July 2014 (to appear)

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  2. Cannon S, Demaine ED, Demaine ML, Eisenstat S, Patitz MJ, Schweller R, Summers SM, Winslow A (2012) Two hands are better than one (up to constant factors). Technical report 1201.1650, Computing Research Repository. http://arxiv.org/abs/1201.1650

  3. Doty D, Gu X, Lutz JH, Mayordomo E, Moser P (2005) Zeta-dimension. In: Proceedings of the thirtieth international symposium on mathematical foundations of computer science, Gdansk. Springer, pp 283–294

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  4. Kautz SM, Lathrop JI (2009) Self-assembly of the Sierpinski carpet and related fractals. In: Proceedings of the fifteenth international meeting on DNA computing and molecular programming, Fayetteville, 8–11 June 2009, pp 78–87

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  5. Kautz S, Shutters B (2013) Self-assembling rulers for approximating generalized Sierpinski carpets. Algorithmica 67(2):207–233. doi:10.1007/s00453-012-9691-x, http://dx.doi.org/10.1007/s00453-012-9691-x

  6. Lathrop JI, Lutz JH, Summers SM (2007) Strict self-assembly of discrete Sierpinski triangles. In: Proceedings of the third conference on computability in Europe, Siena, 18–23 June 2007

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  7. Lathrop JI, Lutz JH, Summers SM (2009) Strict self-assembly of discrete Sierpinski triangles. Theor Comput Sci 410:384–405

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  8. Lutz JH, Shutters B (2012) Approximate self-assembly of the Sierpinski triangle. Theory Comput Syst 51(3):372–400

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  9. Padilla JE, Patitz MJ, Pena R, Schweller RT, Seeman NC, Sheline R, Summers SM, Zhong X (2013) Asynchronous signal passing for tile self-assembly: fuel efficient computation and efficient assembly of shapes. In: UCNC, Milan, pp 174–185

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  10. Patitz MJ, Summers SM (2008) Self-assembly of discrete self-similar fractals (extended abstract). In: Proceedings of the fourteenth international meeting on DNA computing, Prague, 2–6 June 2008 (to appear)

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  11. Rothemund PWK, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol 2(12):e424. doi:10.1371/journal.pbio.0020424, http://dx.doi.org/10.1371

  12. Soloveichik D, Winfree E (2007) Complexity of self-assembled shapes. SIAM J Comput 36(6):1544–1569

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Author information

Authors and Affiliations

  1. Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA

    Matthew J. Patitz

Authors
  1. Matthew J. Patitz
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Corresponding author

Correspondence to Matthew J. Patitz .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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© 2016 Springer Science+Business Media New York

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Patitz, M.J. (2016). Self-Assembly of Fractals. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_662

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