Abstract
We discuss theoretical aspects of the self-assembly of triangular tiles; in particular, right triangular tiles and equilateral triangular tiles. Contrary to intuition, we show that triangular tile assembly systems and square tile assembly systems are not comparable in general. More precisely, there exists a square tile assembly system S such that no triangular tile assembly system that is a division of S produces the same final supertile. There also exists a deterministic triangular tile assembly system T such that no square tile assembly system produces the same final supertiles while preserving border glues. We discuss the assembly of triangles by triangular tiles and show triangular systems with Θ(logN/loglogN) tiles that can self-assemble into a triangular supertile of size Θ(N 2). Lastly, we show that triangular tile assembly systems, either right-triangular or equilateral, are Turing universal.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adleman, L.: Toward a mathematical theory of self-assembly (1999) (manuscript), https://eprints.kfupm.edu.sa/72519/1/72519.pdf
Adleman, L., Cheng, Q., Goel, A., Huang, M.: Running time and program size for self-assembled. In: Proc. 33rd Ann. ACM Symp. Theor. of Comp (STOC 2001), pp. 740–748 (2001)
Berger, R.: The undecidability of the domino problem. Mem. Amer. Math. Soc. 66, 1–72 (1966)
Kao, M., Schweller, R.: Reducing tile complexity for self-assembly through temperature programming. In: Proc. 7th Ann. ACM-SIAM Symp. Discrete Algorithm, pp. 571–580 (2006)
Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Math. 12, 177–209 (1971)
Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares. In: Proc. 32nd Ann. ACM Symp. Theor. of Comp (STOC 2000), pp. 459–468 (2000)
Wang, H.: Proving theorems by pattern recognition II. Bell System Technical Journal 40, 1–42 (1961)
Winfree, E.: On the computational power of DNA annealing and ligation. In: DNA Based Computers: DIMACS Workshop, pp. 199–221 (1996)
Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394, 539–544 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kari, L., Seki, S., Xu, Z. (2011). Triangular Tile Self-assembly Systems. In: Sakakibara, Y., Mi, Y. (eds) DNA Computing and Molecular Programming. DNA 2010. Lecture Notes in Computer Science, vol 6518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18305-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-18305-8_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18304-1
Online ISBN: 978-3-642-18305-8
eBook Packages: Computer ScienceComputer Science (R0)