# Cyclic Oritatami Systems Cannot Fold Infinite Fractal Curves

## TL;DR

This paper proves that cyclic Oritatami Systems cannot generate certain infinite fractal curves like Koch and Minkowski, highlighting limitations in modeling infinite self-similar structures through this computational framework.

## Contribution

It establishes a negative result showing the impossibility of folding specific infinite fractals with cyclic OS and provides conditions under which cyclic OS cannot produce infinite aperiodic curves.

## Key findings

- Cyclic OS cannot generate Koch or Minkowski fractals.
- Sufficient conditions are identified where cyclic OS cannot fold infinite aperiodic curves.
- The study highlights fundamental limitations of cyclic OS in modeling infinite fractal structures.

## Abstract

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.