I shall now recall to mind that the motion of the heavenly bodies is circular, since the motion appropriate to a sphere is rotation in a circle.

Nicolaus Copernicus

**Cyclical Axial Rotations (CARs) – Returning To The Point of Origin A Theoretical Analysis of Axial Rotational Trajectory within a Measured Unit Circle Domain**

Ever wonder how many rotations around a circle it takes for the rotation of a fixed angle to return to its point of origin? For angles that are divisors of 360 degrees it takes one rotation, for those that are not it is greater than one circle rotation.

This led to the discovery and definition of a new mathematical formula from which I developed a python script to generate the calculations and to generate the graphical representations of what such rotations would look like.

This provides results that for any given fixed angle of rotation **n,** there are **nr** number of rotations that traverse **ncr** number of circular rotations before it returns to its point of origin.

For example say we want to calculate the cyclical axial rotation for Earth which has roughly an axial rotation of 23º (varies between 22º to 24.5º). Because 23 does not divide equally into 360 and is a prime number its number of rotations nr = 360 and its number of circular rotations ncr = 23.

Lets take some other examples:

Pluto where n = 57.5º its nr = 144 and ncr = 23

Moon where n = 6.5º its nr = 720 and ncr = 13

Sun where n = 7.25º its nr = 1440 and ncr = 29

Which can be verified by the function n x nr / ncr = 360

It should be noted that the mathematical formula does not make use of Euler’s or Pythagoras theorems, in fact the math is fairly basic, however the mathematical results generated could be used to generate a vector for presenting graphical representations. It is clear from this research and analysis that certain numbers follow certain rules which provide different results based on the characteristics of the number, the incremental measure and maximum measurement of the unit circle domain. Hence, this formula relates to measured spaces.

The graphical patterns generated and shown in this video include Cluster, Spiral, Starburst and Floral Startburst patterns which are somewhat reminiscent of Fibonacci and Lucas spirals or patterns found in nature. Each black dot at the end of each line illustrates one fixed angle rotation, the red dots show the end of each complete circular rotation.

This project evolved into a digital arts project which combined mathematics, programming using Python and writing up a mathematical research paper (which has yet to be reviewed/approved mathematically).

A video showing the results from the analysis and research can be found on my YouTube channel here Cyclical Axial Rotations

**Copyright:** @2024 B. Mullally (Midge) BSc MA

**Licence:** CC-NY-ND 4.0