Dilatation of self-assembling tilings

We introduce a new notion in self-assembly, that of transforming the dynamics of assembly. This notion allows us to have transformation of the plane computed within the assembly process. More specifically, we study a zooming transformation. First we show that the possibility of doing that transformation depends on the regularity of the assembly process, as expressed by the order condition. Then we give two ways of realizing that transformation. Our first construction works by doing a “cartesian product” of the original tileset and of a tileset which computes the transformation. Our second tileset works as an union of the original tileset and additional independent tiles for the zoom. This kind of constructions opens the possibility of designing a self-assembling system by putting together dedicated sub-tilesets.