Figure 1.

Tiles, ISs, and structures formed by simulation. (A) C tiles. (B) Periodic IS with a minimal number of unmatched edges. (C) Simulation results. Few errors (0.2% of abutting edges are mismatched, n = 40) but many unbound edges are present. (D) Schematic shows tiles binding by a single (s) bond along a face or by a double (d) bond into a corner. Errors form only by mechanical trapping (t). Vacancies (v) also form. (E) P tiles. (F) Aperiodic IS (approximate). (G) Simulation results. A moderate number of errors (2.2%) and three separate structures are present. (H) Schematic shows tiles binding by single (s) or double (d) bonds. Matching rules allow incompletable sites (inc) into which no tile can fit without error (*), which occurs when such a site is filled. (I) XOR tiles encode the binary function XOR (X⊕Y = Z). Concave lower edges denote a pair of input bits X and Y (yellow = 1, blue = 0). Convex upper edges denote two copies of an output bit Z. The top tile encodes 1⊕1 = 0, the leftmost tile 1⊕0 = 1, the rightmost tile 0⊕1 = 1 and the bottom tile 0⊕0 = 0. (J) IS. (K) Simulation results (rotated 45° for viewing convenience). Many errors (14.7%) are present. (L) Schematic shows XOR tiles assembling on an input tile, outlined in red, denoting the string 00100 continued on both sides with zeros. Successive rows correspond to successive time steps (t = 0  … t = 4) in a simulation of the corresponding one-dimensional blocked cellular automaton (ref. 25, rule 60). Arrow indicates three tiles competing for a corner site, two of which (s) would form an error (*) if incorporated. (Inset) A larger, perfect section of the intended structure, Pascal's triangle mod 2.