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Helicalised fractals

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  • Saw, Vee-Liem
  • Chew, Lock Yue

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 ω 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 ω→∞, 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.

Suggested Citation

  • Saw, Vee-Liem & Chew, Lock Yue, 2015. "Helicalised fractals," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 191-203.
    • Handle: RePEc:eee:chsofr:v:75:y:2015:i:c:p:191-203
    DOI: 10.1016/j.chaos.2015.02.012
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    References listed on IDEAS

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    1. Tejera, E. & Machado, A. & Rebelo, I. & Nieto-Villar, J., 2009. "Fractal protein structure revisited: Topological, kinetic and thermodynamic relationships," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(21), pages 4600-4608.
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