Reflections on tiles (in self-assembly)

  • Jacob Hendricks
  • Matthew J. Patitz
  • Trent A. Rogers
Article

DOI: 10.1007/s11047-017-9617-2

Cite this article as:
Hendricks, J., Patitz, M.J. & Rogers, T.A. Nat Comput (2017). doi:10.1007/s11047-017-9617-2
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Abstract

We define the Reflexive Tile Assembly Model (RTAM), which is obtained from the abstract Tile Assembly Model (aTAM) by allowing tiles to reflect across their horizontal and/or vertical axes. We show that the class of directed temperature-1 RTAM systems is not computationally universal, which is conjectured but unproven for the aTAM, and like the aTAM, the RTAM is computationally universal at temperature 2. We then show that at temperature 1, when starting from a single tile seed, the RTAM is capable of assembling \(n \times n\) squares for n odd using only n tile types, but incapable of assembling \(n \times n\) squares for n even. Moreover, we show that n is a lower bound on the number of tile types needed to assemble \(n \times n\) squares for n odd in the temperature-1 RTAM. The conjectured lower bound for temperature-1 aTAM systems is \(2n-1\). Finally, we give preliminary results toward the classification of which finite connected shapes in \(\mathbb {Z}^2\) can be assembled (strictly or weakly) by a singly seeded (i.e. seed of size 1) RTAM system, including a complete classification of which finite connected shapes can be strictly assembled by mismatch-free singly seeded RTAM systems.

Keywords

Self-assembly Tile assembly model Molecular computing 

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Jacob Hendricks
    • 1
  • Matthew J. Patitz
    • 2
  • Trent A. Rogers
    • 2
  1. 1.Department of Computer Science and Information SystemsUniversity of Wisconsin - River FallsRiver FallsUSA
  2. 2.Department of Computer Science and Computer EngineeringUniversity of ArkansasFayettevilleUSA