TY - JOUR
AU - Summers, Scott M.
AB - In this paper, we investigate the power of systems in the abstract Tile Assembly Model (aTAM) to self-assemble shapes having fractal dimensions between 1 and 2. We introduce the concept of sparsity as a tool for investigating such systems and demonstrate its utility by proving how it relates to fractal dimension. We then prove several results regarding the strict self-assembly of certain classes of fractal shapes in the aTAM including the construction of a universal tileset which, given the correct seed assembly, strictly self-assembles with nearly any desired fractal dimension. Additionally, we discuss a long standing conjecture in tile-assembly, that a class of fractals called discrete self-similar fractals cannot strictly self-assemble in the aTAM, and provide evidence that sparsity, rather than fractal dimension, is a more promising differentiating factor between shapes that can and cannot strictly self-assemble.
TI - Fractal dimension of assemblies in the abstract tile assembly model
JF - Natural Computing
DO - 10.1007/s11047-023-09942-5
DA - 2023-04-26
UR - https://www.deepdyve.com/lp/springer-journals/fractal-dimension-of-assemblies-in-the-abstract-tile-assembly-model-bnY0M6uC8V
SP - 1
EP - 16
VL - OnlineFirst
IS - 
DP - DeepDyve
ER -