Self-Assembly of Squares and Scaled Shapes

Years and Authors of Summarized Original Work

• 2000; Rothemund, Winfree

• 2001; Adleman, Cheng, Goel, Huang

• 2005; Cheng, Aggarwal, Goldwasser, Kao, Schweller, Espanes

• 2007; Soloveichik, Winfree

Problem Definition

Abstract Tile Assembly Model

The abstract Tile Assembly Model (aTAM) [3] is a mathematical model of self-assembly in which system components are four-sided Wang tiles with glue types assigned to each tile edge. Any pair of glue types are assigned some nonnegative interaction strength denoting how strongly the pair of glues bind. An aTAM system is an ordered triplet $(T,\tau ,\sigma )$ consisting of a set of tiles T, a positive integer threshold parameter τ called the system’s temperature, and a special tile $\sigma \in T$ denoted as the seed tile. Assembly proceeds by attaching copies of tiles from Tto a growing seed assembly whenever the placement of a tile on the 2D grid achieves a total strength of attachment from abutting edges, determined by the sum of pairwise glue...