Helicalised fractals

TL;DR

This paper introduces the helicaliser, a method to generate fractal structures by winding curves around themselves, and analyzes their geometric properties, including Hausdorff dimension, with potential applications in modeling DNA packaging.

Contribution

It formulates the helicaliser, relates it to self-similar fractals, and derives bounds on Hausdorff dimension for various helical fractals, advancing understanding of their geometric complexity.

Findings

Hausdorff dimension for straight line and circle cases derived

Upper bounds on dimension approach 2 as windings increase

Potential application in modeling chromosome topology and DNA dynamics

Abstract

We formulate the helicaliser, which replaces a given smooth curve by another curve that winds around it. In our analysis, we relate this formulation to the geometrical properties of the self-similar circular fractal (the discrete version of the curved helical fractal). Iterative applications of the helicaliser to a given curve yields a set of helicalisations, with the infinitely helicalised object being a fractal. We derive the Hausdorff dimension for the infinitely helicalised straight line and circle, showing that it takes the form of the self-similar dimension for a self-similar fractal, with lower bound of 1. Upper bounds to the Hausdorff dimension as functions of $\omega$ have been determined for the linear helical fractal, curved helical fractal and circular fractal, based on the no-self-intersection constraint. For large number of windings $\omega\rightarrow\infty$, the upper bounds all have the limit of 2. This would suggest that carrying out a topological analysis on the structure of chromosomes by modelling it as a two-dimensional surface may be beneficial towards further understanding on the dynamics of DNA packaging.