Related papers: Scaled pier 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.
In this paper, we search for {\it absolute} limitations of the Tile Assembly Model (TAM), along with techniques to work around such limitations. Specifically, we investigate the self-assembly of fractal shapes in the TAM. We prove that no…
In this paper, we present high-level overviews of tile-based self-assembling systems capable of producing complex, infinite, aperiodic structures known as discrete self-similar fractals. Fractals have a variety of interesting mathematical…
This paper gives a (polynomial time) algorithm to decide whether a given Discrete Self-Similar Fractal Shape can be assembled in the aTAM model.In the positive case, the construction relies on a Self-Assembling System in the aTAM which…
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…
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…
This paper answers a long-standing open question in tile-assembly theory, namely that it is possible to strictly assemble discrete self-similar fractals (DSSFs) in the abstract Tile-Assembly Model (aTAM). We prove this in 2 separate ways,…
A general class of finitely ramified fractals is that of P.C.F. self-similar sets. An important open problem in analysis on fractals was whether there exists a self-similar energy on every P.C.F. self-similar set. In this paper, I solve the…
We demonstrate existence of a tile assembly system that self-assembles the statistically self-similar Sierpinski Triangle in the Winfree-Rothemund Tile Assembly Model. This appears to be the first paper that considers self-assembly of a…
In this paper we consider the problem of the strict self-assembly of infinite fractals within tile self-assembly. In particular, we provide tile assembly algorithms for the assembly of the discrete Sierpinski triangle and the discrete…
We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire…
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions…
The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a…
The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the $q$-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative…
We give a direct, pointwise proof for the tube formula of Lapidus-Pearse for self-similar fractals, where we allow non-convex, non-Steiner-like generators.
We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not…
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…
The self-assembly of hard polyhedral particles confined to a flat interface is studied using Monte Carlo simulations. The particles are pinned to the interface by restricting their movement in the direction perpendicular to it while…
Much is known in the analysis of a finitely ramified self-similar fractal when the fractal has a harmonic structure: a Dirichlet form which respects the self-similarity of a fractal. What is still an open question is when such structure…
Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern…