Self-Assembly of Fractals | SpringerLink
Skip to main content
Springer Nature Link
Log in

Self-Assembly of Fractals

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 47 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 \(A + \mathit{kB} =\{\mathbf{ m} + k\mathbf{n}\vert \mathbf{m} \in A\)...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 142999
Price includes VAT (Japan)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
JPY 142999
Price includes VAT (Japan)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

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)

    Google Scholar 

  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

    Google Scholar 

  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

    Google Scholar 

  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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    MATH  Google Scholar 

  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)

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

Download references

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
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media New York

About this entry

Cite this entry

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

Download citation

Publish with us

Policies and ethics

Search

Navigation