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Self-Assembly of Patterns in the abstract Tile Assembly Model

Phillip Drake, Matthew J. Patitz, Scott M. Summers, Tyler Tracy

TL;DR

This paper demonstrates how to efficiently self-assemble a set of simple patterns, then shows tight bounds on the tile type complexity of self-assembling multi-colored patterns on the surfaces of square assemblies, and demonstrates an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes.

Abstract

In the abstract Tile Assembly Model, self-assembling systems consisting of tiles of different colors can form structures on which colored patterns are ``painted.'' We explore the complexity, in terms of the numbers of unique tile types required, of assembling various patterns. We first demonstrate how to efficiently self-assemble a set of simple patterns, then show tight bounds on the tile type complexity of self-assembling 2-colored patterns on the surfaces of square assemblies. Finally, we demonstrate an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes.

Self-Assembly of Patterns in the abstract Tile Assembly Model

TL;DR

This paper demonstrates how to efficiently self-assemble a set of simple patterns, then shows tight bounds on the tile type complexity of self-assembling multi-colored patterns on the surfaces of square assemblies, and demonstrates an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes.

Abstract

In the abstract Tile Assembly Model, self-assembling systems consisting of tiles of different colors can form structures on which colored patterns are ``painted.'' We explore the complexity, in terms of the numbers of unique tile types required, of assembling various patterns. We first demonstrate how to efficiently self-assemble a set of simple patterns, then show tight bounds on the tile type complexity of self-assembling 2-colored patterns on the surfaces of square assemblies. Finally, we demonstrate an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes.
Paper Structure (22 sections, 9 theorems, 3 equations, 17 figures, 12 algorithms)

This paper contains 22 sections, 9 theorems, 3 equations, 17 figures, 12 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary Definitions and Models
  3. The abstract Tile-Assembly Model
  4. Patterns
  5. Simple Patterns
  6. Tight Bounds for Patterns on $n \times n$ Squares
  7. Repeated Patterns
  8. Barely-3DaTAM patterns
  9. Proof sketch
  10. Extending a pattern $p_n$ to infinitely cover $\mathbb{Z}^2$
  11. Technical Details of the Proofs of Section \ref{['sec:simple']}
  12. Proof of Theorem \ref{['thm:multi-pixel']}
  13. Proof of Theorem \ref{['thm:stripes']}
  14. Technical Details of the Proofs of Lemma \ref{['lem:random-lower']}
  15. Technical Details of the Construction for Theorem \ref{['thm:repeated-pattern']}
  16. ...and 7 more sections

Key Result

theorem thmcountertheorem

For all $n, i, j \in \mathbb{N}$ such that $n \ge i,j$, there exists an aTAM system $\mathcal{T} = (T,\sigma,2)$ such that $|\sigma|=1$, $|T| = O(\log(n))$, and $\mathcal{T}$ weakly self-assembles $\texttt{SinglePixel}(n, i, j)$.

Figures (17)

  • Figure 1: (a) An example of a single-pixel pattern. The black pixel is located at (10, 2). (b) The same single-pixel pattern but with the counter box and counter tiles colored for demonstration. The counter box is colored red. The counters are colored blue. The white locations are filled by generic filler tiles.
  • Figure 2: (a) An example of a multi-pixel pattern with three black pixels. (b) A tree of counters is constructed to grow to each pixel and the edges of the square. The counter boxes are colored red. The counters are colored blue. The white locations are filled by generic filler tiles.
  • Figure 3: (a) An example of a stripes pattern. (b) The blue tiles count to the next stripe, while the red tiles count the number of stripes. Green tiles represent the starting rows for the counters (with the seed tile being at the corner where the green row and column intersect). Dark grey tiles represent counter tiles that are colored black
  • Figure 4: A schematic example of the construction of the proof of Lemma \ref{['lem:random-upper']}. Instead of showing the black and white colors corresponding to the pattern, we color the tiles to show the pieces of the construction to which they belong.
  • Figure 5: An example of an assembly that repeats a pattern $m = 5$ times horizontally and vertically. Each spine is colored solely for clarity of presentation, and in the actual construction, the colors of the tiles on the spines would match the pixels of the pattern. Red spines represent a 1, and blue spines represent a 0. The spines count upwards until the counter is finished
  • ...and 12 more figures

Theorems & Definitions (19)

  • definition thmcounterdefinition: Single-Pixel Pattern Class
  • theorem thmcountertheorem
  • proof
  • definition thmcounterdefinition: Multi-Pixel Pattern Class
  • theorem thmcountertheorem
  • definition thmcounterdefinition: Stripes Pattern Class
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • ...and 9 more