Strict Self-Assembly of Discrete Sierpinski Triangles
| dc.creator | Lathrop, James I. | |
| dc.creator | Lutz, Jack H. | |
| dc.creator | Summers, Scott M. | |
| dc.date | 2009-03-10 | |
| dc.date.accessioned | 2026-07-07T12:50:56Z | |
| dc.date.available | 2026-07-07T12:50:56Z | |
| dc.description | Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above self-assemblies tile completely filled-in, two-dimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else. We first prove that the standard discrete Sierpinski triangle cannot strictly self-assemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, non-strict self-assemblies, our strict self-assembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of Soloveichik and Winfree (2007). | |
| dc.identifier | https://arxiv.org/abs/0903.1818 | |
| dc.identifier | http://arxiv.org/abs/0903.1818 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/222837 | |
| dc.subject | Discrete Mathematics | |
| dc.title | Strict Self-Assembly of Discrete Sierpinski Triangles | |
| dc.type | text |