Abstract
In this paper we consider the problem of the strict self-assembly of infinite fractals within tile self-assembly. In particular, we provide tile assembly algorithms for the assembly of a Sierpinski triangle and the discrete Sierpinski carpet within a class of models we term the h-handed assembly model (h-HAM), which generalizes the 2-HAM to allow up to h assemblies to combine in a single assembly step. Despite substantial consideration, no purely growth self-assembly model has yet been shown to strictly assemble an infinite fractal without significant modification to the fractal shape. In this paper we not only achieve this, but in the case of the Sierpinski carpet are able to achieve it within the 2-HAM, one of the most well studied tile assembly models in the literature. Our specific results are as follows: We provide a 6-HAM construction for a Sierpinski triangle that works at scale factor 1, 30 tile types, and assembles the fractal in a near perfect way in which all intermediate assemblies are finite-sized iterations of the recursive fractal. We further assemble a Sierpinski triangle within the 3-HAM at scale factor 3 and 990 tile types. For the Sierpinski carpet, we present a 2-HAM result that works at scale factor 3 and uses 1216 tile types. We further include analysis showing that the aTAM is incapable of strictly assembling the Sierpinski triangle considered in this paper, and that multiple hands are needed for the near-perfect assembly of a Sierpinski triangle and the Sierpinski carpet.
Similar content being viewed by others
References
Abel, Z., Benbernou, N., Damian, M., Demaine, E.D., Demaine, M.L., Flatland, R., Kominers, S.D., Schweller, R.T.: Shape replication through self-assembly and rnase enzymes. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1045–1064 (2010)
Aggarwal, G., Cheng, Q., Goldwasser, M.H., Kao, M., de Espanés, P.M., Schweller, R.T.: Complexities for generalized models of self-assembly. SIAM J. Comput. 34(6), 1493–1515 (2005). doi:10.1137/S0097539704445202
Barth, K., Furcy, D., Summers, S.M., Totzke, P.: Scaled tree fractals do not strictly self-assemble. In: Unconventional Computation and Natural Computation—13th International Conference, UCNC 2014, London, ON, Canada, July 14-18, 2014, Proceedings, pp. 27–39 (2014). doi:10.1007/978-3-319-08123-6_3
Cannon, S., Demaine, E.D., Demaine, M.L., Eisenstat, S., Patitz, M.J., Schweller, R.T., Summers, S.M., Winslow, A.: Two hands are better than one (up to constant factors): self-assembly in the 2ham vs. atam. In: 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27–March 2, 2013, Kiel, Germany, pp. 172–184 (2013). doi:10.4230/LIPIcs.STACS.2013.172
Carbone, A., Seeman, N.C.: A route to fractal dna-assembly. Nat. Comput. 1(4), 469–480 (2002)
Chen, H., Doty, D.: Parallelism and time in hierarchical self-assembly. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17–19, 2012, pp. 1163–1182 (2012). http://portal.acm.org/citation.cfm?id=2095208&CFID=63838676&CFTOKEN=79617016
Demaine, E., Patitz, M., Rogers, T., Schweller, R., Summers, S., Woods, D.: The two-handed tile assembly model is not intrinsically universal. In: Fomin, F., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) Automata, Languages, and Programming, Lecture Notes in Computer Science, vol. 7965, pp. 400–412. Springer, Berlin (2013). doi:10.1007/978-3-642-39206-1_34
Demaine, E.D., Demaine, M.L., Fekete, S.P., Ishaque, M., Rafalin, E., Schweller, R.T., Souvaine, D.L.: Staged self-assembly: nanomanufacture of arbitrary shapes with O (1) glues. Nat. Comput. 7(3), 347–370 (2008). doi:10.1007/s11047-008-9073-0
Demaine, E.D., Demaine, M.L., Fekete, S.P., Patitz, M.J., Schweller, R.T., Winslow, A., Woods, D.: One tile to rule them all: simulating any tile assembly system with a single universal tile. In: Automata, Languages, and Programming—41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8–11, 2014, Proceedings, Part I, pp. 368–379 (2014). doi:10.1007/978-3-662-43948-7_31
Demaine, E.D., Patitz, M.J., Schweller, R.T., Summers, S.M.: Self-assembly of arbitrary shapes using rnase enzymes: meeting the kolmogorov bound with small scale factor (extended abstract). In: 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011, March 10–12, 2011, Dortmund, Germany, pp. 201–212 (2011). doi:10.4230/LIPIcs.STACS.2011.201
Doty, D.: Producibility in hierarchical self-assembly. In: Unconventional Computation and Natural Computation—13th International Conference, UCNC 2014, London, ON, Canada, July 14–18, 2014, Proceedings, pp. 142–154 (2014). doi:10.1007/978-3-319-08123-6_12
Doty, D., Patitz, M.J., Reishus, D., Schweller, R.T., Summers, S.M.: Strong fault-tolerance for self-assembly with fuzzy temperature. In: 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23–26, 2010, Las Vegas, Nevada, USA, pp. 417–426 (2010). doi:10.1109/FOCS.2010.47
Fu, B., Patitz, M.J., Schweller, R.T., Sheline, R.: Self-assembly with geometric tiles. In: Automata, Languages, and Programming—39th International Colloquium, ICALP 2012, Warwick, UK, July 9–13, 2012, Proceedings, Part I, pp. 714–725 (2012). doi:10.1007/978-3-642-31594-7_60
Furcy, D., Summers, S.M.: Scaled pier fractals do not strictly self-assemble. Computing Research Repository abs/1406.4197 (2014). http://arxiv.org/abs/1406.4197
Kautz, S.M., Lathrop, J.I.: Self-assembly of the discrete sierpinski carpet and related fractals. In: DNA Computing and Molecular Programming, 15th International Conference, DNA 15, Fayetteville, AR, USA, June 8–11, 2009, Revised Selected Papers, pp. 78–87 (2009). doi:10.1007/978-3-642-10604-0_8
Kautz, S.M., Shutters, B.: Self-assembling rulers for approximating generalized sierpinski carpets. Algorithmica 67(2), 207–233 (2013). doi:10.1007/s00453-012-9691-x
Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete sierpinski triangles. Theor. Comput. Sci. 410(4–5), 384–405 (2009). doi:10.1016/j.tcs.2008.09.062
Lutz, J.H., Shutters, B.: Approximate self-assembly of the sierpinski triangle. Theory Comput. Syst. 51(3), 372–400 (2012). doi:10.1007/s00224-011-9345-4
Meunier, P., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5–7, 2014, pp. 752–771 (2014). doi:10.1137/1.9781611973402.56
Padilla, J.E., Patitz, M.J., Schweller, R.T., Seeman, N.C., Summers, S.M., Zhong, X.: Asynchronous signal passing for tile self-assembly: fuel efficient computation and efficient assembly of shapes. Int. J. Found. Comput. Sci. 25(4), 459–488 (2014). doi:10.1142/S0129054114400061
Patitz, M.J., Summers, S.M.: Self-assembly of discrete self-similar fractals. Nat. Comput. 9(1), 135–172 (2010). doi:10.1007/s11047-009-9147-7
Patitz, M.J., Summers, S.M.: Self-assembly of decidable sets. Nat. Comput. 10(2), 853–877 (2011). doi:10.1007/s11047-010-9218-9
Rothemund, P.W., Papadakis, N., Winfree, E.: Algorithmic self-assembly of dna sierpinski triangles. PLoS Biol. 2(12), e424 (2004)
Schweller, R.T., Sherman, M.: Fuel efficient computation in passive self-assembly. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6–8, 2013, pp. 1513–1525 (2013). doi:10.1137/1.9781611973105.109
Wang, H.: Dominoes and the aea case of the decision problem. In: Computation, Logic, Philosophy, pp. 218–245. Springer, Berlin (1990)
Winfree, E.: Algorithmic self-assembly of dna. Ph.D. thesis, California Institute of Technology (1998)
Winfree, E., Bekbolatov, R.: Proofreading tile sets: error correction for algorithmic self-assembly. In: DNA Computing, 9th International Workshop on DNA Based Computers, DNA9, Madison, WI, USA, June 1–3, 2003, revised papers, pp. 126–144 (2003). doi:10.1007/978-3-540-24628-2_13
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors’ research was supported in part by National Science Foundation Grant CCF-1117672.
Rights and permissions
About this article
Cite this article
Chalk, C.T., Fernandez, D.A., Huerta, A. et al. Strict Self-Assembly of Fractals Using Multiple Hands. Algorithmica 76, 195–224 (2016). https://doi.org/10.1007/s00453-015-0022-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-015-0022-x