Impossibility of strict assembly of infinite fractals by oritatami

RNA cotranscriptional folding is the phenomenon in which an RNA transcript folds upon itself while being synthesized out of a gene. The oritatami system is a computation model of this phenomenon, which lets its sequence (transcript) of beads (abstract molecules) fold cotranscriptionally by the interactions between beads according to the binding ruleset. In such models based on self-assembly, one of the key questions is the ability to construct fractal structures. We focus on the problem of generating an infinite fractal curves using a cyclic oritatami system, which has an infinite periodic transcript. We first establish a formal definition of drawing a curve using an oritatami system, proposing conditions and restrictions with reference to prior oritatami designs for possibly infinite conformations. Under such definition, we prove that it is impossible to draw a Koch curve or a Minkowski curve infinitely. We then establish sufficient conditions of infinite aperiodic curves that a cyclic oritatami system cannot fold.