Movement of spheres along the pathways of a toroidal helix

Visualization in 3D of Dynamics of Toroidal Helical Coils (Part #6)

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As part of both exercises, the possibility of moving a sphere along the pathway defined by each coil was explored. The coordinates of the "spine" of the extruded pathway were used. The result is evident from the images above (and especially the animations). It is then also of interest to switch the visualization into wireframe mode. One purpose of this exercise is partially enabled by consideration of the spheres as a focus of attention, whether individually or collectively. This relates to concerns with eye-scanning of an image.

In the first approach, the speed of movement of the spheres was timed to match that of the rotational cycle of the animation (namely using the same "clock"). Clearly the exercise can be taken further by increasing the speed (namely using a second "clock"). Of particular interest is using a much greater speed, to the degree that the location of a sphere cannot then be determined at any given time. Visually the pathway of each coil is then defined primarily by the movement of the spheres rather than by the "tunnel" constituted by each coil -- especially in wireframe mode. The coils could in fact be rendered transparent, if only for a portion of the cycle (obviating the need to use wireframe mode).

One purpose of this approach follows from the possibility that a "Borromean condition" of a kind would be defined by the movement of the spheres rather than by that of the coiled pathways themselves. In other words, rather than expecting a condition in which the coils would not touch during their cycle, the focus would then be on whether the spheres collided during the course of their movement along the coiled pathways.

Using the spheres to define the cycle is consistent with the sinusoidal pathway of the coils, effectively then even more suggestive of a sine wave with its implications.