Strict self-assembly of discrete self-similar fractals
Contents
Strict Self-Assembly of Discrete Self-Similar Fractals
Discrete self-similar fractals are infinite aperiodic shapes that have received much interest in the domain of tile self-assembly. For instance, in [1] it was shown that the infinite Sierpinski triangle cannot strictly self-assemble in the aTAM, but a fibered approximation can be. In [2] it was shown that additional DSSFs cannot strictly self-assemble in the aTAM, but their fibered approximations can. Just a few of the additional results related to DSSFs, including more impossibility results, other approximations, and strict self-assembly in other models can be found in [3][4][5][6][7].
While in [7] it was conjectured that no DSSF can strictly self-assemble in the aTAM, that conjecture has been proven false. In fact, two different methods have been demonstrated for developing aTAM systems that strictly self-assemble DSSFs, and the following two sections give overviews of each.
DSSFs via Self-describing circuits
TODO: describe the work by Florent Becker in [8]
DSSFs via Quines
TODO: describe the work by Hader and Patitz, in progress...
A quine is a computer program that, with no input, outputs a full copy of its own source code. In the context of the aTAM, a quine was defined, with respect to an intrinsically universal tile set \(U\), as a system \(\mathcal{Q} = (Q, \sigma, \tau)\) that begins from a single seed tile, (i.e. \(|\sigma| = 1)\) and grows into a terminal assembly, \(\alpha_\mathcal{Q}\), that is correctly formatted to be the seed assembly for a system \(\mathcal{S} = (U,\alpha_\mathcal{Q}, \tau)\) that uses \(U\) and its associated representation function \(R\) such that \(R\) maps \(\alpha_\mathcal{Q}\) to \(\sigma\) (i.e. \(R(\alpha_\mathcal{Q}) = \sigma)\) and \(\mathcal{S}\) intrinsically simulates \(\mathcal{Q}\).
References
- ↑
James I. Lathrop, Jack H. Lutz, Scott M. Summers - Strict Self-Assembly of Discrete Sierpinski Triangles
- Theoretical Computer Science 410:384--405,2009
- BibtexAuthor : James I. Lathrop, Jack H. Lutz, Scott M. Summers
Title : Strict Self-Assembly of Discrete Sierpinski Triangles
In : Theoretical Computer Science -
Address :
Date : 2009
- ↑
Matthew J. Patitz, Scott M. Summers - Self-assembly of discrete self-similar fractals
- ↑
Jack H. Lutz, Brad Shutters - Approximate self-assembly of the sierpinski triangle
- ↑
Steven M. Kautz, Brad Shutters - Self-assembling rulers for approximating generalized sierpinski carpets
- Lecture Notes in Computer Science 6842:284--296,2011
- BibtexAuthor : Steven M. Kautz, Brad Shutters
Title : Self-assembling rulers for approximating generalized sierpinski carpets
In : Lecture Notes in Computer Science -
Address :
Date : 2011
- ↑
Hendricks, Jacob, Olsen, Meagan, Patitz, Matthew, Rogers, Trent, Thomas, Hadley - Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles
- ↑
Chalk, Cameron T, Fernandez, Dominic A, Huerta, Alejandro, Maldonado, Mario A, Schweller, Robert T, Sweet, Leslie - Strict self-assembly of fractals using multiple hands
- ↑ 7.0 7.1
Hader, Daniel, Patitz, Matthew J, Summers, Scott M - Fractal dimension of assemblies in the abstract tile assembly model
- ↑
Florent Becker - Strict Self-Assembly of Discrete Self-Similar Fractal Shapes