  RNA cotranscriptional folding is the phenomenon in which an RNA transcript
folds upon itself while being synthesized out of a gene. The oritatami system
(OS) 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. The OS is an
useful computational model for predicting and simulating an RNA folding as well
as constructing a biological structure. A fractal is an infinite pattern that
is self-similar across different scales, and is an important structure in
nature. Therefore, the fractal construction using self-assembly is one of the
most important problems. We focus on the problem of generating an infinite
fractal instead of a partial finite fractal, which is much more challenging. We
use a cyclic OS, which has an infinite periodic transcript, to generate an
infinite structure. We prove a negative result that it is impossible to make a
Koch curve or a Minkowski curve, both of which are fractals, using a cyclic OS.
We then establish sufficient conditions of infinite aperiodic curves that a
cyclic OS cannot fold.
