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One Tile to Rule Them All: Simulating Any Tile Assembly System with a Single Universal Tile

  • Erik D. Demaine
  • Martin L. Demaine
  • Sándor P. Fekete
  • Matthew J. Patitz
  • Robert T. Schweller
  • Andrew Winslow
  • Damien Woods
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

In the classical model of tile self-assembly, unit square tiles translate in the plane and attach edgewise to form large crystalline structures. This model of self-assembly has been shown to be capable of asymptotically optimal assembly of arbitrary shapes and, via information-theoretic arguments, increasingly complex shapes necessarily require increasing numbers of distinct types of tiles.

We explore the possibility of complex and efficient assembly using systems consisting of a single tile. Our main result shows that any system of square tiles can be simulated using a system with a single tile that is permitted to flip and rotate. We also show that systems of single tiles restricted to translation only can simulate cellular automata for a limited number of steps given an appropriate seed assembly, and that any longer-running simulation must induce infinite assembly.

Keywords

DNA computing algorithmic self-assembly hexagonal tiles 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Martin L. Demaine
    • 1
  • Sándor P. Fekete
    • 2
  • Matthew J. Patitz
    • 3
  • Robert T. Schweller
    • 4
  • Andrew Winslow
    • 5
  • Damien Woods
    • 6
  1. 1.Massachussetts Institute of TechnologyCambridgeUSA
  2. 2.TU BraunschweigGermany
  3. 3.University of ArkansasFayettevilleUSA
  4. 4.University of Texas–Pan AmericanEdinburgUSA
  5. 5.Tufts UniversityMedfordUSA
  6. 6.California Institute of TechnologyPasadenaUSA

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