Patterned Self-Assembly Tile Set Synthesis

Years and Authors of Summarized Original Work

2008; Ma, Lombardi

2013; Seki

2014; Göös, Lampiäinen, Czeizler, Orponen

2014; Kari, Kopecki, Meunier, Patitz, Seki

Problem Definition

A tile type is a colored unit square each of whose four sides is provided with a glue. An assembly is a partial function from $\mathbb{Z}^{2}$ (2D-grid) to a tile type set T. A (rectangular) pattern P (of width w and height h) is a function from the rectangular domain [w] × [h] to a set of colors, where [m] = { 1, …, m} for $m \in \mathbb{N}$. If at most k colors appear on P, we say P is k-colored. Tiles being colored, an assembly of domain [w] × [h] induces a unique pattern of width w and height h.

Fig. 1

(Left) Four tile types implement together the half-adder with two inputs A, B from the west and south, the output S to the north, and the carryout C to the east (Right) Copies of the “half-adder” tile types turn the L-shape seed into the binary counter pattern