In the abstract Tile Assembly Model, self-assembling systems consisting of tiles of different colors can form structures on which colored patterns are ``painted.'' We explore the complexity, in terms of the numbers of unique tile types required, of assembling various patterns. We first demonstrate how to efficiently self-assemble a set of simple patterns, then show tight bounds on the tile type complexity of self-assembling 2-colored patterns on the surfaces of square assemblies. Finally, we demonstrate an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes.

翻译：在抽象Tile组装模型中，由不同颜色Tile组成的自组装系统能够形成表面“绘制”出彩色图案的结构。我们从所需独特Tile类型数量这一复杂度角度，探讨了不同图案的组装过程。首先展示了如何高效自组装一组简单图案，随后给出了正方形组装体表面自组装双色图案的Tile类型复杂度的紧界。最后，我们证明了在限制于单平面与允许双平面的系统之间，自组装无限序列图案所需Tile类型复杂度存在指数级差距。