Fig. 1: Broken action–reaction symmetry and multifarious self-organization model.

a, Schematic of the realization of a sequence of programmed transitions between three different predefined target configurations. b, Specific reciprocal interactions between tiles A and B on a two-dimensional (H–V) lattice with specific binding energy U. c, Specific non-reciprocal interactions between A and B, and the corresponding interaction rates R. d, Non-reciprocity in the presence of a structure; A approaching B is different from B approaching A. e, Schematic of shape-shifting structures leveraging non-reciprocal interactions. Two desired structures are random permutations of 3 × 3 lattice configurations composed of nine distinct tiles. The non-reciprocal interactions induce transitions to the next structures. For arriving tiles, the red arrows and black lines show non-reciprocal and reciprocal interactions with the neighbouring tiles, respectively. The red boundary that emerges, grows and finally disappears represents the specific bonds that are lost due to the red arrows. This line shows how a small seed of the new structure emerges due to non-reciprocal interactions and eventually conforms to the full structure with the help of reciprocal interactions.