Abstract

This paper serves as a seamless continuation of the previous work titled “Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers”. Upon revisiting the data, a noteworthy observation emerged: the special spirals can be meticulously aligned with the prime spirals on a point-by-point basis through careful scaling and rotation for the parameter ρ remaining a constant greater than one. As ρ increases, both scaling factor and rotation angle exhibit a corresponding augmentation, albeit with an intriguing caveat—the rotation angle asymptotically approaches its limit of 90°. Notably, the galactic spiral can be perceived as a remarkably close approximation of the Archimedean spiral, with the degree of proximity increasing as ρ grows larger. Furthermore, an intriguing correspondence arises when ρ takes on a value of 1; at this value, the galactic spiral bears a striking resemblance to the Fermat spiral, the space between two consecutive spiral loops undergoes a continuous reduction as the loop’s radius expands, yet it never diminishes to zero; simultaneously, the area between two successive loops experiences an augmentation with the value of ρ. The galactic spiral will resemble the logarithmic spiral when the ρ change with time following the exponential function.

Keywords: Galactic spiral, Galactic prime spiral, Prime spiral, ROTASE model, X-matter