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Hierarchical Growth Is Necessary and (Sometimes) Sufficient to Self-assemble Discrete Self-similar Fractals

  • Jacob Hendricks
  • Joseph Opseth
  • Matthew J. PatitzEmail author
  • Scott M. Summers
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11145)

Abstract

In this paper, we prove that in the abstract Tile Assembly Model (aTAM), an accretion-based model which only allows for a single tile to attach to a growing assembly at each step, there are no tile assembly systems capable of self-assembling the discrete self-similar fractals known as the “H” and “U” fractals. We then show that in a related model which allows for hierarchical self-assembly, the 2-Handed Assembly Model (2HAM), there does exist a tile assembly systems which self-assembles the “U” fractal and conjecture that the same holds for the “H” fractal. This is the first example of discrete self similar fractals which self-assemble in the 2HAM but not in the aTAM, providing a direct comparison of the models and greater understanding of the power of hierarchical assembly.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Jacob Hendricks
    • 1
  • Joseph Opseth
    • 2
  • Matthew J. Patitz
    • 3
    Email author
  • Scott M. Summers
    • 4
  1. 1.Department of Computer Science and Information SystemsUniversity of Wisconsin - River FallsRiver FallsUSA
  2. 2.Department of MathematicsUniversity of Wisconsin - River FallsRiver FallsUSA
  3. 3.Department of Computer Science and Computer EngineeringUniversity of ArkansasFayettevilleUSA
  4. 4.Computer Science DepartmentUniversity of Wisconsin–OshkoshOshkoshUSA

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