Configuration of a toroidal helix

Visualization in 3D of Dynamics of Toroidal Helical Coils (Part #3)

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As noted in concluding the earlier experiments, there appeared to be value in exploring configurations suggested by the following. However it also became clear that there was extensive literature on helical coils (and springs) but relatively limited references to their toroidal form -- especially with respect to their rendering using 3D applications.

Illustrative depictions of a circular helix
Rope quoit	Double-spun helix	Circular helix	Torus knot

Following from the earlier experiments with 5 nonagons, the approach envisaged was to have 5 helical windings around a common torus -- each winding to be understood as a nonagon. However, rather than the nonagons being of polygonal (or polyhedral) form, the concern was how to combine the 5-fold and 9-fold patterns into an aesthetically agreeable visualization, for reasons articulated in the earlier paper (Meaningful configuration engendered only by tacit aesthetic entanglement, 2016).

As noted with regard to the previous experiments, a degree of guidance was kindly provided by Sergey Bederov, Senior Developer of Cortona3D. The latter is a web plugin which enables renderings in web browsers of 3D models according to the legacy virtual reality modelling standard (VRML). The models were however developed for these experiments using the X3D Edit application (namely according to X3D, a more recent standard), and exported into VRML. Bederov provided models (presented in the previous paper) to show that the Discordian 2D representation could not be presented in 3D in a manner consistent with the Borromean condition. However he provided an alternative which evoked the possibility of a sinusoidal form of nonagon -- the focus of this document.

To that end Bederov provided a formula through which a toroidal knot could be constructed, consistent with the image on the right above. As is perhaps more evident in the image below, this seemingly involves 2 windings which are however continuous, not separate. It is therefore merely indicative of the need for a distinct formula to pursue the configuration of nonagons as envisaged here.

The (p,q)-torus knot winds q times around a circle in the interior of the torus, and p times around its axis of rotational symmetry. If p and q are not relatively prime, then we have a torus link with more than one component. where r = cos(qφ) + 2 and r=cos(qφ)+2 and 0 < φ < 2π and 0 < φ < 2 π *

Visualization of a toroidal knot (using 201 rows of xyz coordinates)
Formula		x=rCos(pφ)	y=rSin(pφ)	z= -Sin(pφ)
Spreadsheet (A1=0) (namely 12/201= 0.062832)	A2=A1+0.062832	B1=SIN(A1)*(1+COS(A1*4.5)*0.3)	C1=COS(A1)*(1+COS(A1*4.5)*0.3)	D1=SIN(A1*4.5)*0.3

Continuous toroidal knot with 9 windings (Interactive variants: X3D or VRML)

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