The helicaliser : toroidal fractal, wiggleiser : string wiggles, and revolutioniser : curved traversable wormholes.

Abstract

This research begins with the formulation of the helicaliser, which replaces a regular curve by another regular curve that winds around it. Modifying it into the revolutioniser generates a surface of revolution. We develop the 3-d and 4-d formalisms, generalise to n-d, before applying it to three major fields. Firstly, iterative helicalisations to a curve produce a set of helicalisations, with the in finite level being a fractal. These fractals are not self-similar, but we define a parameter d, and prove it reduces to the form of the self-similar dimension for self-similar fractals. We calculate the upper bound to d, preventing self-intersections. Next, we incorporate the crucial wiggling properties of strings from string theory to the toroidal helicalisations, generating the wiggleised toroidal helicalisations. We then derive analytically and provide numerical results to show that they share similar geometrical properties with strings. Finally, as revolutionised manifolds, such objects represent traversable wormholes satisfying the Einstein field equations. We study a class of (2+1)-d wormholes obtained by the revolutioniser and show explicitly that the helical wormhole must be supported by exotic matter. Since it is non-spherically (or non-axially) symmetric, it is significant as there are regions in the helical wormhole not requiring exotic matter, permitting safe human travel.