This work describes the spherical curves that are generated from two of the spin axis from the above illustration. A common example that we are very familiar with is the simple gyroscope and the property it is often used for, the tendency to resist being turned through its axis. A classical experiment was devised in order to show the trajectory of a single point (dimensionless point) on the perimeter of a spinning disk. This was achieved by placing a small light on the rim of a disk. The disk material was of a clear plastic so that observations were made possible through all possible degrees. Compressed air was the source of power chosen as this means of propulsion obviated the need to add excessive weight to the experimental apparatus and provided an easily variable source of power needed to drive each axis spin. A machine was eventually built making use of a spherical clear plastic ball (see Photos), which served the same purpose as the original disk model.

All patterns shown in the examples below appear in terms of their spin orientations corresponding to each combined spin state relating to the number of turns for each of the two variable-aligned spin-axis.

The illustrations above (approximations only) demonstrate the observed spin configurations from a 2D perspective and viewed along or parallel to the primary spin axis. These are all familiar Rose curves with the exception of the first vertical row, which shows a progression of increasing diameter circles. The three-dimensional view reveals a figure-eight pattern as variations of the Viviani-Curve. See next illustration. All such configurations expand by the action of varying the angle relationship between the primary axis and the secondary (Differential Angle). For any angle below ninety degrees, the spin two directions cancel and conversely for angles higher than ninety degrees, the spin directions reinforce.

The three dimensional geometry shown above is a few samples of the possible configurations that can be generated of a single dimensional point with this simple mechanical method. The middle figure bottom is one geometric pattern from the one-to-one ratio with the secondary axis set at ninety degrees and this creates the well-known Viviani curve. However, a whole range of figure eights can be produced by varying the secondary axis angle through 180 degrees.

These 2D star patterns demonstrate the relationship between the ratios and the plane geometry where odd numbered ratios have the distinctive Mobius character where the dimensionless point’s trajectory cycles twice before returning to its starting point plus direction but retain the geometry-to-ratio configuration. For example, a one-to-five ratio produces a five point star pattern while the even numbered ratios pass through double the number of points, so a one-to-two ratio has a geometric star pattern with four points around its perimeter.

The above configuration is a two-to-one ratio with primary axis set at ninety degrees. This geometric configuration demonstrates four rotational axes in the shape of a tetrahedron.

Further axis configurations are possible for higher ratios.

These apparent fractional ratios exist between integer ratios and may represent precession due to the friction experienced in the machines bearings etc.

One of several machines build to extract the spherical-curves geometrical configurations. This machine is constructed of clear plastic for the exclusive reason to increase the observables through all possible angles. Compressed air was the source of power as it negated the use of heavy motors etc. In addition, the compressed air could be delivered being infinitely variable.