Coprimality, i.e., the situation where the GCD of 2 integers is 1 is one of the fundamental expressions of complexity. In that situation, two numbers can never contain the other within themselves or in multiples of them by numbers smaller than the other. In other words, their LCM is the product of the 2 numbers. There are numerous geometric expressions of this complexity inherent in coprime numbers. One way to illustrate it is by the below class of parametric curves defined by trigonometric functions:

$latex x=a_1cos(c_1t+k_1)+a_2cos(c_2t+k_2)[5pt]

y=b_1sin(c_3t+k_3)+b_2sin(c_4t+k_4)$

The human mind perceives symmetry and certain optimal complexity as the hallmarks of aesthetics. Hence, we adopt the following conditions:

1) $latex a_1, a_2, b_1, b_2$ are in the range $latex tfrac{3}{14}$..1 for purely aesthetic considerations.

2) $latex k_1, k_2, k_3, k_4$ are orthogonal rotation angles that are in the range $latex [0, 2pi]$

3) $latex c_1$, for aesthetic purposes relating to optimal…

View original post 112 more words