Vee-Liem Saw, Lock Yue Chew
We formulate the helicaliser, which replaces a given smooth curve by another curve that winds around it. Iterative applications of the helicaliser to a given curve yields a set of helicalisations, with the infinitely helicalised object being a fractal. We calculate the lengths of the curves in the set of helicalisations by approximating them as straight helices, and proceed to define the helicaliser dimension. The helicaliser dimension for an infinitely helicalised straight line or circle is shown to take the form of the self-similar dimension for a self-similar fractal, with lower bound of 1. Upper bounds to the helicaliser dimension as functions of number of windings have been determined for the helical, toroidal and circular fractals, based on the no-self-intersection constraint. The formalism of the helicaliser can be generalised in two directions: to incorporate fluctuations in the windings and compare it with the geometrical features of fundamental strings, as well as to generate manifolds of revolution that represent curved traversable wormholes as solutions to the field equations of general relativity.
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http://arxiv.org/abs/1306.4502
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