Skip to main content
SpringerLink
Log in

Approximate Self-Assembly of the Sierpinski Triangle

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz, and Summers proved that the Sierpinski triangle cannot self-assemble in the “strict” sense in which tiles are not allowed to appear at positions outside the target structure. Here we investigate the strict self-assembly of sets that approximate the Sierpinski triangle. We show that every set that does strictly self-assemble disagrees with the Sierpinski triangle on a set with fractal dimension at least that of the Sierpinski triangle (≈1.585), and that no subset of the Sierpinski triangle with fractal dimension greater than 1 strictly self-assembles. We show that our bounds are tight, even when restricted to supersets of the Sierpinski triangle, by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it. To verify this strict self-assembly we develop a generalization of the local determinism method of Soloveichik and Winfree.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barlow, M.T., Taylor, S.J.: Defining fractal subsets of ℤd. In: Proc. London Math. Soc., vol. 64, pp. 125–152 (1992)

    Google Scholar 

  2. Bondarenko, B.A.: Generalized Pascal Triangles and Pyramids, Their Fractals, Graphs and Applications. The Fibonacci Association (1993)

  3. Cahen, E.: Sur la fonction ζ(s) de Riemann et sur des fonctions analogues. Annales Scientifiques de L’École Normale Supérieure 11(3), 75–164 (1894)

    MathSciNet  Google Scholar 

  4. Doty, D., Gu, X., Lutz, J.H., Mayordomo, E., Moser, P.: Zeta-dimension. In: Proc. of the Thirtieth International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3618, pp. 283–294. Springer, Berlin (2005)

    Google Scholar 

  5. Euler, L.: Variae observationes circa series infinitas. Comment. Acad. Sci. Imp. Petropolitanae 9, 160–188 (1737)

    Google Scholar 

  6. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  7. Hueter, I., Peres, Y.: Self-affine carpets on the square lattice. Comb. Probab. Comput. 6, 197–204 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kautz, S.M., Lathrop, J.I.: Self-assembly of the Sierpinski carpet and related fractals. In: Proc. of the Fifteenth International Meeting on DNA Computing and Molecular Programming. Lecture Notes in Computer Science, vol. 5877, pp. 78–87. Springer, Berlin (2009)

    Chapter  Google Scholar 

  9. Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpinski triangles. Theor. Comput. Sci. 410, 384–405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Olsen, L.: Distribution of digits in integers: fractal dimension and zeta functions. Acta Arith. 105(3), 253–277 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Patitz, M.J.: Simulation of self-assembly in the abstract tile assembly model with ISU TAS. In: Proc. of the Sixth Annual Conference on Foundations of Nanoscience: Self-Assembled Architectures and Devices (2009)

    Google Scholar 

  12. Patitz, M.J., Summers, S.M.: Self-assembly of discrete self-similar fractals. Nat. Comput. 9(1), 135–172 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Roco, M.C., Mirkin, C.A., Hersam, M.C.: Nanotechnology research directions for societal needs in 2020, NSF/WTEC report. Springer, Berlin (2011)

    Book  Google Scholar 

  14. Rothemund, P.W.: Theory and experiments in algorithmic self-assembly. Ph.D. thesis, University of Southern California, Los Angeles, California (2001)

  15. Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2(12) (2004)

  16. Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares. In: Proc. of the Thirty Second Annual ACM Symposium on Theory of Computing (2000)

    Google Scholar 

  17. Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)

    Article  Google Scholar 

  18. Seeman, N.C.: DNA in a material world. Nature 421, 427–431 (2003)

    Article  MathSciNet  Google Scholar 

  19. Sierpiński, W.: Sur une courbe dont tout point est un point de ramification. C. R. Math. 160, 302–305 (1915)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Stewart, I.: Four encounters with Sierpinski’s gasket. Math. Intell. 17(1), 52–64 (1995)

    Article  MATH  Google Scholar 

  22. Wang, H.: Proving theorems by pattern recognition, ii. Bell Syst. Tech. J. 40, 1–41 (1961)

    Google Scholar 

  23. Wang, H.: Dominoes and the AEA case of the decision problem. In: Proc. of the Symposium on Mathematical Theory of Automata (1962)

    Google Scholar 

  24. Willson, S.J.: The equality of fractional dimensions for certain cellular automata. Physica D 24, 179–189 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology, Pasadena, California (1998)

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    Jack H. Lutz & Brad Shutters

Authors
  1. Jack H. Lutz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brad Shutters
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brad Shutters.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lutz, J.H., Shutters, B. Approximate Self-Assembly of the Sierpinski Triangle. Theory Comput Syst 51, 372–400 (2012). https://doi.org/10.1007/s00224-011-9345-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-011-9345-4

Keywords

Search

Navigation