5,844 research outputs found
Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model
We consider the self-assembly of fractals in one of the most well-studied
models of tile based self-assembling systems known as the Two-handed Tile
Assembly Model (2HAM). In particular, we focus our attention on a class of
fractals called discrete self-similar fractals (a class of fractals that
includes the discrete Sierpi\'nski carpet). We present a 2HAM system that
finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1.
Moreover, the 2HAM system that we give lends itself to being generalized and we
describe how this system can be modified to obtain a 2HAM system that finitely
self-assembles one of any fractal from an infinite set of fractals which we
call 4-sided fractals. The 2HAM systems we give in this paper are the first
examples of systems that finitely self-assemble discrete self-similar fractals
at scale factor 1 in a purely growth model of self-assembly. Finally, we show
that there exists a 3-sided fractal (which is not a tree fractal) that cannot
be finitely self-assembled by any 2HAM system
Self-assembly of the discrete Sierpinski carpet and related fractals
It is well known that the discrete Sierpinski triangle can be defined as the
nonzero residues modulo 2 of Pascal's triangle, and that from this definition
one can easily construct a tileset with which the discrete Sierpinski triangle
self-assembles in Winfree's tile assembly model. In this paper we introduce an
infinite class of discrete self-similar fractals that are defined by the
residues modulo a prime p of the entries in a two-dimensional matrix obtained
from a simple recursive equation. We prove that every fractal in this class
self-assembles using a uniformly constructed tileset. As a special case we show
that the discrete Sierpinski carpet self-assembles using a set of 30 tiles
Fractals, Randomization, Optimal Constructions, and Replication in Algorithmic Self-Assembly
The problem of the strict self-assembly of infinite fractals within tile self-assembly is considered. In particular, tile assembly algorithms are provided for the assembly of the discrete Sierpinski triangle and the discrete Sierpinski carpet.
The robust random number generation problem in the abstract tile assembly model is introduced. First, it is shown this is possible for a robust fair coin flip within the aTAM, and that such systems guarantee a worst case O(1) space usage. This primary construction is accompanied with variants that show trade-offs in space complexity, initial seed size, temperature, tile complexity, bias, and extensibility.
This work analyzes the number of tile types t, bins b, and stages necessary and sufficient to assemble n × n squares and scaled shapes in the staged tile assembly model.
Further, this work shows how to design a universal shape replicator in a 2-HAM self-assembly system with both attractive and repulsive forces
Self-Assembly of Infinite Structures
We review some recent results related to the self-assembly of infinite
structures in the Tile Assembly Model. These results include impossibility
results, as well as novel tile assembly systems in which shapes and patterns
that represent various notions of computation self-assemble. Several open
questions are also presented and motivated
Scaled tree fractals do not strictly self-assemble
In this paper, we show that any scaled-up version of any discrete
self-similar {\it tree} fractal does not strictly self-assemble, at any
temperature, in Winfree's abstract Tile Assembly Model.Comment: 13 pages, 3 figures, Appeared in the Proceedings of UCNC-2014, pp
27-39; Unconventional Computation and Natural Computation - 13th
International Conference, UCNC 2014, London, ON, Canada, July 14-18, 2014,
Springer Lecture Notes in Computer Science ISBN 978-3-319-08122-
- …